Fortin, Sebastian - Holik, Federico - Lombardi, Olimpia - López, Cristian - Quantum Worlds - Perspectives On The Ontology of Quantum Mechanics-Cambridge University Press (2019)
Fortin, Sebastian - Holik, Federico - Lombardi, Olimpia - López, Cristian - Quantum Worlds - Perspectives On The Ontology of Quantum Mechanics-Cambridge University Press (2019)
Fortin, Sebastian - Holik, Federico - Lombardi, Olimpia - López, Cristian - Quantum Worlds - Perspectives On The Ontology of Quantum Mechanics-Cambridge University Press (2019)
Quantum theory underpins much of modern physics, and its implications draw the
attention of industry, academia, and public funding agencies. However there are
many unsettled conceptual and philosophical problems in the interpretation of
quantum mechanics, which are a matter of extensive debate. These hotly debated
topics include the meaning of the wave function, the nature of the quantum objects,
the role of the observer, the nonlocality of the quantum world, and the emergence
of classicality from the quantum domain. Containing chapters written by eminent
researchers from the fields of physics and philosophy, this book provides interdis-
ciplinary, comprehensive, and up-to-date perspectives of the problems related to
the interpretation of quantum theory. It is ideal for academic researchers in physics
and philosophy who are working on the ontology of quantum mechanics.
Edited by
OLIMPIA LOMBARDI
University of Buenos Aires
SEBASTIAN FORTIN
University of Buenos Aires
CRISTIAN LÓPEZ
University of Buenos Aires
FEDERICO HOLIK
National University of La Plata
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Information on this title: www.cambridge.org/9781108473477
DOI: 10.1017/9781108562218
© Cambridge University Press 2019
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First published 2019
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A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Lombardi, Olimpia, editor. | Fortin, Sebastian, 1979– editor. | López, Cristian, editor. |
Holik, Federico, editor.
Title: Quantum worlds : perspectives on the ontology of quantum mechanics / edited by
Olimpia Lombardi (Universidad de Buenos Aires, Argentina), Sebastian Fortin
(Universidad de Buenos Aires, Argentina),
Cristian López (Universidad de Buenos Aires, Argentina), Federico Holik
(Universidad Nacional de La Plata, Argentina).
Description: Cambridge ; New York, NY : Cambridge University Press, 2019. |
Includes bibliographical references and index.
Identifiers: LCCN 2018045102 | ISBN 9781108473477 (hardback)
Subjects: LCSH: Quantum theory.
Classification: LCC QC174.12 .Q385 2019 | DDC 530.12–dc23
LC record available at https://lccn.loc.gov/2018045102
ISBN 978-1-108-47347-7 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents
v
vi Contents
Index 393
Contributors
Jonas R. B. Arenhart
Federal University of Santa Catarina
Guido Bellomo
University of Buenos Aires – CONICET
Gustavo M. Bosyk
National University of La Plata – CONICET
Otávio Bueno
University of Miami
Dennis Dieks
Utrecht University
Michael Esfeld
University of Lausanne
Sebastian Fortin
University of Buenos Aires – CONICET
Manuel Gadella
University of Valladolid
Hans Halvorson
Princeton University
vii
viii List of Contributors
Nathan Harshman
American University
Alejandro Hnilo
CITEDEF – CONICET
Federico Holik
National University of La Plata – CONICET
Ruth Kastner
University of Maryland
Décio Krause
Federal University of Santa Catarina
James Ladyman
University of Bristol
Roberto Laura
National University of Rosario
Olimpia Lombardi
University of Buenos Aires – CONICET
Cristian López
University of Buenos Aires – University of Lausanne – CONICET
Marcelo Losada
National University of Rosario – CONICET
Wayne C. Myrvold
University of Western Ontario
List of Contributors ix
Alyssa Ney
University of California, Davis
Lev Vaidman
Tel Aviv University
Leonardo Vanni
University of Buenos Aires
Nino Zanghì
University of Genova
Preface
xi
xii Preface
the Faculty of Exact and Natural Sciences and in the Faculty of Philosophy and
Letters of the University of Buenos Aires: Hernán Accorinti, Guido Bellomo,
Martín Bosyk, Mariana Córdoba, María José Ferreira Ruiz, Manuel Herrera, Jesús
Jaimes Arriaga, Martín Labarca, Marcelo Losada, Juan Camilo Martínez González,
Erick Rubio, Leonardo Vanni, and Alfio Zambon. Their strong commitment and
unlimited enthusiasm made the organization of the meeting an enjoyable task.
Although the presentations from the workshop were the basis for this book, we
are also grateful to Jonas Arenhart, Otávio Bueno, and Manuel Gadella, who
graciously accepted the invitation of the editors to contribute to the present volume
in different chapters. Last, but not least, we want to express our gratitude to
Cambridge University Press, in the person of Simon Capelin, Editorial Director
(physical sciences) and Sarah Lambert, Editorial Assistant, for their support and
assistance during all the stages of this project.
Introduction
In its original meaning, ‘ontology’ is the study of what there is – not only of what
entities exist but also of the very structure of reality. For the most part of the history
of philosophy, ontology was the core of metaphysics, perhaps the major branch of
philosophy. Nowadays, however, the word has different meanings and nuances. In
the analytic tradition, for instance, ontology is the study not only of what there is,
but also of the most general features of and the relations among what there is. This
study commonly starts out from our intuitions about reality or from an a priori
reasoning. Yet, another, increasingly growing sense of ontology has to do with
reality itself in relation to our best scientific theories: When one asks for “the
ontology of” a certain scientific theory, the question is about what reality would be
like if the theory were true. Although this second meaning does not exactly match
the etymology of the word (from Ancient Greek: on, what is; logos, discourse,
account), the meaning drift is completely natural in the light of the fact that, at least
after the Renaissance, scientific knowledge was crucial with respect to how the
structure of reality and the nature of its entities were conceived.
Quantum mechanics is probably the most successful and the least understood
physical theory that we have ever had. Even though this claim has become almost a
cliché, its frequent repetition does not make it less true. Indeed, after almost a
century of its first formulations, quantum mechanics is still posing unsolved
puzzles with respect to our understanding of the microscopic world. Of course,
numerous important results have been obtained during years of research, and many
of them are relevant to the foundations and the interpretation of the theory.
However, it is not completely clear yet how the ontology of the theory is, in
particular, how reality would be if quantum mechanics were true. Not only does
the question remain in force just as in the first decades of the twentieth century, but
a century of philosophical and scientific discussions has brought to light that
quantum reality is far more complex than originally supposed. The numerous
and varied perspectives developed up to the present time just manifest this
1
2 Quantum Worlds
contexts for quantum histories, in which the contexts of properties at each time
must satisfy compatibility conditions given by commutation relations in the Hei-
senberg representation; any family of histories satisfying these conditions is
organized in a distributive lattice with well-defined probabilities obtained by a
natural generalization of the Born rule.
Part II, “Realism, Wave function, and Primitive Ontology,” is devoted to the
question of realism in the quantum domain in general and, in particular, of realism
regarding the wave function. In the first chapter, “What Is the Quantum Face of
Realism?,” James Ladyman explores the interaction between different forms of
realism and different forms of quantum physics, showing the tension between usual
arguments for scientific realism in the philosophy of science literature and the
invocation of realism in certain interpretations of quantum mechanics. In the second
chapter of this part, “To Be a Realist about Quantum Theory,” Hans Halvorson takes
a closer look at the distinction between realist and antirealist views of the quantum
state, and argues that this binary classification should be reconceived as a continuum
of different views about which properties of the quantum state are representationally
significant. The final chapter of the second part, “Locality and Wave Function
Realism” by Alyssa Ney, advocates for wave function realism, according to which
the fundamental quantum entity is the wave function, understood as a scalar field on a
high-dimensional space with the structure of a configuration space; according to her,
this kind of realism is an attempt to explain nonlocal influences, instead of taking
them as brute facts of the world.
In Part III, “Individuality, Distinguishability, and Locality,” the ontological
problems related to the identity and nature of quantum particles are addressed.
This part begins with the chapter by Jonas Arenhart, Otávio Bueno, and Décio
Krause, “Making Sense of Nonindividuals in Quantum Mechanics,” which focuses
on a very specific question: Assuming that quantum theories deal with “particles”
of some kind, what kind of entities can such particles be? The authors respond that
quantum entities are nonindividuals and that a metaphysics of nonindividuals
requires a system of logic where the basic items have no identity. In the second
chapter of this part, entitled “From Quantum to Classical Physics: The Role of
Distinguishability,” Ruth Kastner reviews the derivations of the classical and the
quantum statistics in order to argue that a form of separability is a key feature of
the quantum-to-classical transition; on this basis, she considers the question of
what allows separability to serve as a form of distinguishability in the classical
limit. The third chapter, “Individuality and the Account of Nonlocality: The Case
for the Particle Ontology in Quantum Physics” by Michael Esfeld, examines
different solutions to the measurement problem to conclude that the particle
ontology of Bohmian mechanics provides the least deviation from the ontology
of classical mechanics that is necessary so as to accommodate quantum physics,
4 Quantum Worlds
both in the case of quantum mechanics and in the case of quantum field theory. This
third part closes with the chapter by Alejandro Hnilo, “Beyond Loophole-Free
Experiments: A Search for Nonergodicity,” where he analyzes the experiments
designed to measure violations of Bell’s inequalities and argues that, besides locality
and realism, the measurement of the inequalities implicitly assumes the ergodic
hypothesis; therefore, in order to save the validity of local realism in nature, it is
necessary to search for evidence of nonergodic dynamics in Bell’s experiments.
The chapters composing Part IV, “Symmetries and Structure in Quantum
Mechanics,” deal with structural features of the quantum theory. The first chapter
of this part, “Spacetime Symmetries in Quantum Mechanics” by Cristian López
and Olimpia Lombardi, stresses the relevance of symmetries to interpretation; on
this basis, the authors consider the behavior of nonrelativistic quantum mechanics
under the Galilean group and critically analyze the widely-accepted view about the
invariance of the Schrödinger equation under time reversal. In the second chapter,
“Symmetry, Structure, and Emergent Subsystems,” Nathan Harshman focuses on
the particular structures called irreducible representations of symmetry groups, in
order to explore the connections between the mathematical units of symmetry
embodied by those irreducible representations and the conceptual units of reality
that form the basis for the interpretation of quantum theories. Finally, the third
chapter of this fourth part, “Majorization, across the (Quantum) Universe” by
Guido Bellomo and Gustavo Bosyk, reviews the wide applicability of majorization
in the quantum realm and stresses that such applicability emerges as a consequence
of deep connections among majorization, partially ordered probability vectors,
unitary matrices, and the probabilistic structure of quantum mechanics.
The chapters in the fifth and final part of this volume, “The Relationship
between the Quantum Ontology and the Classical World,” address the classical
limit of quantum mechanics from different perspectives. In the first chapter,
“A Closed-System Approach to Decoherence,” Sebastian Fortin and Olimpia
Lombardi argue that the conceptual difficulties of the orthodox approach to
decoherence are the result of its open-system perspective; so, they propose a
closed-system approach that not only solves or dissolves the problems of the
orthodox approach, but is also compatible with a top-down view of quantum
mechanics. In the second chapter of this part, “A Logical Approach to the
Quantum-to-Classical Transition,” Sebastian Fortin, Manuel Gadella, Federico
Holik, and Marcelo Losada present a logical approach to the emergence of
classicality based on the Heisenberg picture, which describes how the logical
structure of the elementary properties of a quantum system becomes classical
when the classical limit is reached. In the chapter, “Quantum Mechanics and
Molecular Structure: The Case of Optical Isomers,” which closes the last part
and the volume, Juan Camilo Martínez González, Jesús Jaimes Arriaga, and
Introduction 5
1.1 Introduction
Dynamical collapse theories, such as the Ghirardi–Rimini–Weber (GRW) theory
(Ghirardi, Rimini, and Weber 1986), the Continuous Spontaneous Localization
theory, or CSL (Pearle 1989, Ghirardi, Pearle, and Rimini 1990), Quantum
Mechanics with Universal Position Localization, or QMUPL (Diósi 1989), and
their respective relativistic extensions (Dove 1996, Dove and Squires 1996,
Tumulka 2006, Bedingham 2011a, b, Pearle 2015), modify the usual deterministic,
unitary quantum dynamics such as to produce something like the textbook collapse
process. See Bassi and Ghirardi (2003), Bassi et al. (2013), and Ghirardi (2016) for
overviews.
If some sort of dynamical collapse theory is correct, what might the world be
like? Can a theory of that sort be a quantum state monist theory, or must such
theories supplement the quantum state ontology with additional beables? In a
previous work (Myrvold 2018), I defended quantum state monism. The view
defended involves a natural extension of the usual eigenstate-eigenvalue link,
which provides a sufficient condition for a quantum state to be one in which a
system has a definite value of some dynamical variable, namely, that the quantum
state be an eigenstate of that variable. The usual eigenstate-eigenvalue link leaves
open the question of what to say about states that are not eigenstates. A state that is
not an eigenstate of some dynamical variable, but is very close to an eigenstate,
exhibits behaviour that closely approximates that of the eigenstate. In accordance
with a proposal of Ghirardi, Grassi, and Pearle (1990), in such a case the quantity
may be treated as if it were definite. However specification of the quantities that are
definite or near-definite does not exhaustively specify the condition of the physical
world, as there are matters of fact about such things as the spread of values of a
dynamical variable in a given state. The natural ontology for a collapse theory is a
distributional ontology along the lines advocated by Philip Pearle (2009). On such
9
10 Wayne C. Myrvold
be changed by goings-on outside the region unless those goings-on have an effect
on local matters within the region. By contrast, if the contents of your office are
approximately 150 million kilometers from the nearest star, this fact is clearly a
fact about relations between the things in your office and the world outside of it.
A symptom of this fact is that it can be changed by making changes outside your
office that do not affect anything within it.
By a local beable, I will mean something that is, in this sense, local to a
bounded spacetime region. The ontology of a physical theory might contain both
local and nonlocal beables. If it is the case that, for an arbitrarily fine covering of
spacetime with open sets, the full ontology of the theory supervenes on beables
that are local to elements of that covering, we will say that the ontology satisfies
the condition of separability (see Myrvold 2011 for further discussion).
Quantum state realism entails rejection of separability. It does not follow
that there are no local beables. For one thing, there could be local beables
postulated in addition to the quantum state. Additionally, some aspects of the
quantum state – in particular, the reduced state that is the restriction of the state to
observables pertaining to a bounded spacetime region – might be counted as local
beables.
Need there be any local beables at all? If we are willing to countenance a
rejection of separability, might we not go all the way and accept a radically holistic
view in which there are no beables intrinsic to any region short of the whole of
spacetime?
The difficulty with this is that, if the theory is meant to be one that is in principle
comprehensive, it must have room for such things as experimental apparatus that is
subject to local manipulations and whose experimental readouts are, presumably,
matters of fact local to the laboratory. In the absence of things like these, the theory
runs the risk of undermining its own evidential base (see Maudlin 2007 for a lucid
discussion of this point).
A brief comment, before we continue. What it means to say that a structure
found within a physical theory plays the role of spacetime for that theory is that it
has the appropriate connections with dynamics. In speaking of spacetime, I will
always mean that structure that plays the role in the theory of affording spatio-
temporal relations, such as distances, temporal intervals, causal connectability and
the like, distances and temporal intervals and causal relations that are relevant to
the dynamics. It is necessary to say this because it has been claimed that quantum
theory motivates the introduction of a so-called fundamental space, or fundamental
arena, a high-dimensional space that would be such that quantum states involve
nothing more than assignments of local beables to points in that fundamental space
(see Albert 1996 and the various contributions to Ney and Albert 2013). In a
quantum theory, even if such a space can be found, that space is not the structure
Ontology for Relativistic Collapse Theories 13
on which the distances, temporal intervals, and causal relations relevant to the
dynamics are defined. For that reason, such a space, even if it were to exist, is not
spacetime in the sense of the word being used in this chapter. Hence, even if a
fundamental space of the sort sought by Albert and others did exist, a quantum
state realist ontology violates separability as we are using the term.
This modification has been dubbed, by Clifton and Monton (1999), the fuzzy
link.
To say that we can ascribe a property to a system when the quantum state is such
that its variance is negligibly small requires that there be a matter of fact about
what the variance is. Considerations of this sort suggest a revision of the way we
think about dynamical quantities, along the lines advanced by Pearle (2009). On
this view, dynamical variables typically do not take on sharp values as they would
classically. What they have, instead, is a distribution associated with them. These
distributions, though having the formal characteristics of probability distributions,
are to be thought of not as a probability distribution over precise but unknown
possessed values, but as reflecting a physical, ontological lack of determinacy
about what the value is. A limiting case would be the classical case, in which the
distribution is a delta function.
Ontology for Relativistic Collapse Theories 15
X
n X
m
xi ψ i ¼ yj φj :
i¼1 j¼1
Then
X
n X
m
xi T ðψ i Þ ¼ yj T φj :
i¼1 j¼1
probability distribution over the possible values of γ. Suppose that, with respect
to some background measure μ, this probability distribution is represented by a
density function pðγÞ.
With this apparatus in place, we can define a mixed state ρ ðσ0 ; σÞ as the
weighted average over the possibilities for the state on σ0 , given jψ ðσ Þi.
ð
ρ ðσ ; σÞ ¼ pðγÞ ψγ ðσ0 Þ ψγ ðσ0 Þ dμ:
0
(1.3)
γ
This would be the state used by someone who knows the state on σ and
the possible state transitions from jψ ðσ Þi to jψ ðσ 0 Þi and their respective
probabilities, but does not know the outcome of the process that occurs between
σ and σ0 .
Gisin’s proof, mentioned earlier, generalizes to stochastic theories. If we take
T to be the mapping that takes a pure state on σ to a mixed state ρ ðσ0 ; σÞ, no-
signalling entails that this map must satisfy the linearity condition (Simon,
Bužek, and Gisin 2001, Bassi and Hejazi 2015), and from this, together with
the condition that, applied to subsystems in entangled states, the mapping
extends to a positive map on the state space of the wider system, it entails that
the map from the state on σ to the mixed state ρ ðσ0 ; σÞ be a completely
positive map.
We will therefore take the mapping from a pure state on σ to the mixed state
ρ ðσ0 ; σÞ to be a nonselective completely positive map, which is a mixture of
0
selective completely positive maps that takes us from jψ ðσ Þi to ψ γ ðσ Þ . This
entails that there is a set of operators K γ ; γ 2 Γ , which we will call evolution
operators, such that, for some γ,
jψ ðσ 0 Þi ¼ ψ γ ðσ 0 Þ ¼ K γ jψ ðσ Þi= K γ jψ ðσ Þi , (1.4)
Ontology for Relativistic Collapse Theories 19
Any probabilities that depart from these would lead to signalling. The condition
that p always be normalized is the condition that
ð
K †γ K γ dμ ¼ 1: (1.7)
Γ
The evolutions should also satisfy the semi-group property, which requires that, for
Cauchy surfaces σ σ0 σ00 , the possible evolutions from σ to σ00 be the compos-
itions of evolutions from σ to σ0 with evolutions from σ0 to σ00 .
The theory of quantum dynamical semi-groups is well studied (see Bassi and
Ghirardi 2003 or Alicki and Lendi 2007 for an introduction). Provided that the
evolution satisfies an appropriate continuity condition, the mixed-state density oper-
ators on Cauchy surfaces σ to the future of some surface σ0 will satisfy a Lindblad
equation. We consider the change in ρ ðσ; σ0 Þ as we pass from one surface σ to
another σ0 differing by a small deformation about a point x on σ with spacetime
volume δσ. Let H ðxÞ be the Hamiltonian density, that is, the component of the
energy-momentum density along the normal to σ at x. For a Lindblad-type evolution,
there is also a countable set fLα ðxÞg of operators, such that the change δρ satisfies
!
δρ 1 X 1 X † X
† †
¼ ½H ðxÞ; ρ þ Lα ðxÞρ Lα ðxÞ L ðxÞLα ðxÞρ þ ρ Lα ðxÞLα ðxÞ :
δσ iℏc α
2 α α α
(1.8)
s¢
d d¢
s
0 0
Figure 1.1 Cauchy surfaces σ, σ , with σ σ , that coincide everywhere except on
the boundaries of two bounded regions δ and δ0 .
20 Wayne C. Myrvold
s²
s¢
s
a
s²
is contained as a common part of all σn and for all n, σnþ1 σn , and the sequence
converges on the past light cone of α (that is, the set of points that are to the past of
all σn is precisely the causal past of α). Define the past light-cone state of α as the
limit, if it exists, of ρα ðσn Þ, as n increases indefinitely. Though a state derived from
a Cauchy surface with events to its past that are spacelike separated from α cannot
be regarded as the intrinsic state of α, its past light-cone state can.
states. We must take care to formulate the condition in a manner that is independ-
ent of assumptions such as these.
We assume a von Neumann algebra RðαÞ, whose self-adjoint elements represent
the bounded observables pertaining to the forward domain of dependence of α. Let
ρ be a normal state of RðαÞ (that is, a completely additive state). We define the
support projection for ρ as the orthogonal complement of the union of all projec-
tions in RðαÞ to which ρ assigns expectation value zero.
With these definitions in hand, it can be shown that, given a set fσ; σ0 ; σ00 ; . . .g
of Cauchy surfaces containing α, then on the assumption that there is a Cauchy
surface containing α that is nowhere to the past of any of them (which, in
particular, will always be the case for any finite set of Cauchy surfaces), the
corresponding set of states fρα ðσÞ; ρα ðσ0 Þ; ρα ðσ00 Þ; . . .g have nonzero overlapping
support. See Appendix for details.
The 00-component of this is the relativistic analogue of the mass density that has
been proposed as an appropriate ontology for nonrelativistic collapse theories.
For a bounded region α contained in distinct Cauchy surfaces σ and σ0 , the
reduced states ρα ðσÞ and ρα ðσ0 Þ may yield differing values for T μν ðxÞ, with x
within α. But, obviously, these do not yield rival accounts of local beables within
α, as they are defined via the extrinsic states ρα ðσÞ and ρα ðσ0 Þ, which are not
themselves local beables.
Where then, may we find local beables for a relativistic collapse theory? There
are, in the literature, two proposals for extending the fuzzy link to a relativistic
context. One is what might be called the agreement criterion, formulated by
Ghirardi, Grassi, and Pearle (1991):
We think that the appropriate attitude is the following: when considering a local observable
A with its associated support we say that an individual system has the objective property a
(a being an eigenvalue of A), only when the mean value of Pa is extremely close to one,
when evaluated on all spacelike hypersurfaces containing the support of A.
(Ghirardi et al. 1991: 1310).
Ontology for Relativistic Collapse Theories 23
Figure 1.3 Two spin-½ particles, located in world-tubes A and B. The initial
conditions are taken on a Cauchy surface σ0 , and the particle in A is undisturbed
in the interval between σ0 and some later Cauchy surface σ1 . α is a spacelike slice
of A between σ0 and σ1 . β0 , β1 are the intersections of B with σ0 and σ1 ,
respectively, such that they are to the past and future, respectively, of α.
1.5 Conclusion
There is a sensible ontology for collapse theories in a relativistic context. More-
over, considerations of what it takes for a theory to represent a world that contains,
among other things, objects like our experimental apparatus, to be thought of a
local beables, determine the form that this ontology takes. It is one on which all
dynamical quantities are distributional in character. In spite of this distributional
character, dynamical quantities may have effectively precise values (in the sense
that they behave, to a high degree of approximation, as if they have precise values);
it is the goal of a collapse theory to ensure that the properties of macroscopic
objects almost always have this character. Beables local to a bounded spacetime
region are to be evaluated via the past light-cone state of that region.
Appendix
Proof. Any positive operator E has a square root E 1=2 . Suppose that
hψ ðσ ÞjEjψ ðσ Þi ¼ 0: (A1.2)
Therefore, since
2
hψ ðσ ÞjEjψ ðσ Þi ¼ E1=2 jψ ðσ Þi , (A1.3)
26
Ontology for Relativistic Collapse Theories 27
then
E1=2 jψ ðσ Þi ¼ 0: (A1.4)
For some K γ that commutes with all self-adjoint E 2 RðαÞ,
jψ ðσ 0 Þi ¼ K γ jψ ðσ Þi= K γ jψ ðσ Þi : (A1.5)
Therefore,
E1=2 jψ ðσ 0 Þi ¼ E1=2 K γ jψ ðσ Þi= K γ jψ ðσ Þi ¼ K γ E1=2 jψ ðσ Þi= K γ jψ ðσ Þi ¼ 0 (A1.6)
and so
hψ ðσ 0 ÞjE jψ ðσ 0 Þi ¼ 0 (A1.7)
□.
Lemma 1 gives us a relation between the supports of ρα ðσÞ and ρα ðσ0 Þ when
σ σ0 .
Proposition 1. Let σ and σ0 be Cauchy surfaces containing a common
open subset α. If σ σ0 , then Null½ρα ðσÞ Null½ρα ðσ0 Þ; equivalently,
Supp½ρα ðσ0 Þ Supp½ρα ðσÞ.
Proof. We can construct a Cauchy surface σþ that contains α and is such that
σi σþ for each i, by taking the least upper bound of the set fσ1 ; σ2 ; . . . ; σn g under
the ordering . Consider, now, the state ρα ðσþ Þ. By Proposition 1,
Supp½ρα ðσþ Þ Supp½ρα ðσi Þ for all i, and hence the support of σþ lies within the
intersection of the supports of ρα ðσ1 Þ, ρα ðσ2 Þ, . . . , ρα ðσn Þ.
Remark I. The restriction to a finite set is unnecessary; the result holds for an infinite
set of Cauchy surfaces provided that there is a Cauchy surface that is the upper
bound of all of them. Furthermore, if there is a future light-cone state that is the limit
of an increasing (in the ordering ) set of Cauchy surfaces that converges on the
future light cone of α, then the support of the future light-cone states is the intersec-
tion of the supports of all ρα ðσÞ for all σ containing α.
Remark II. A quantum field is an assignment of a field operator φ ^ ðxÞ to each point of
spacetime. In standard quantum field theories on Minkowksi spacetime, it is assumed
that there is a unitary representation of the group of spacetime translations, with
infinitesimal generators Pμ that satisfy the spectrum condition:
For any future-directed timelike vector a, the spectrum of Pa is in Rþ
This ensures positivity of the energy, with respect to any reference frame.
28 Wayne C. Myrvold
We assume a unique vacuum state that is invariant under all spacetime symmet-
ries. Define the standard Hilbert space of the theory as the closure in norm of the
set of all vectors that can be obtained by operating on the vacuum state with
operators constructed from standard fields. It follows from the Reeh–Schlieder
theorem that, for any state ρ that is analytic in energy, for any α that is such that the
set of points spacelike separated from α contains an open set, the null space of ρ is
empty. If ρα ðσ1 Þ, ρα ðσ2 Þ, . . . , ρα ðσn Þ are all states in the standard Hilbert space of
the theory that are analytic in the energy, each of their null spaces consists solely of
the zero vector, and hence Proposition 2 holds trivially.
The proposition is less trivial for theories that introduce nonstandard fields and
whose states go beyond the standard Hilbert space, as do the relativistic versions of
CSL due to Bedingham (2011a, b) and Pearle (2015). It can be shown, for any
indeterministic theory formulated within the framework sketched in Section 1.3
that is set in Minkowski spacetime, it is necessary to go beyond the standard
Hilbert space (see Myrvold 2017).
Acknowledgments
Many thanks are due to the organizers of the workshop Identity, indistinguish-
ability and non-locality in quantum physics (Buenos Aires, June 2017) and to the
participants in that workshop, for their helpful comments. I would also like to
thank Philip Pearle for comments and advice. I am grateful to Graham and Gale
Wright, who generously sponsor the Graham and Gale Wright Distinguished
Scholar Award at the University of Western Ontario, for financial support of
this work.
References
Albert, D. Z. (1996). “Elementary quantum metaphysics,” pp. 277–284 in J. T. Cushing,
A. Fine, and S. Goldstein (eds.), Bohmian Mechanics and Quantum Mechanics: An
Appraisal. Dordrecht: Kluwer Academic Publishers.
Albert, D. Z. (2015). After Physics. Cambridge, MA: Harvard University Press.
Albert, D. Z. and Loewer, B. (1991). “Wanted dead or alive: Two attempts to solve
Schrödinger’s paradox,” pp. 278–285 in A. Fine, M. Forbes, and L. Wessels (eds.),
PSA 1990: Proceedings of the 1990 Biennial Meeting of the Philosophy of Science
Association, Volume One: Contributed Papers. East Lansing, MI: Philosophy of
Science Association.
Alicki, R. and Lendi, K. (2007). Quantum Dynamical Semigroups and Applications, 2nd
ed. Berlin: Springer.
Allori, V., Goldstein, S., Tumulka, R., and Zanghì, N. (2008). “On the common structure
of Bohmian mechanics and the Ghirardi-Rimini-Weber theory,” The British Journal
for the Philosophy of Science, 59: 353–389.
Ontology for Relativistic Collapse Theories 29
Ghirardi, G. C., Grassi, R., Butterfield, J., and Fleming, G. N. (1993). “Parameter depend-
ence and outcome dependence in dynamical models for state vector reduction,”
Foundations of Physics, 23: 341–364.
Ghirardi, G. C., Grassi, R., and Pearle, P. (1990). “Relativistic dynamical reduction
models: General framework and examples,” Foundations of Physics, 20: 1271–1316.
Ghirardi, G. C., Grassi, R., and Pearle, P. (1991). “Relativistic dynamical reduction models
and nonlocality,” pp. 109–123 in P. Lahti and P. Mittelstaedt (eds.), Symposium on
the Foundations of Modern Physics 1990. Singapore: World Scientific.
Ghirardi, G. C., Pearle, P., and Rimini, A. (1990). “Markov processes in Hilbert space and
continuous spontaneous localization of systems of identical particles,” Physical
Review A, 42: 78–89.
Ghirardi, G., Rimini, A., and Weber, T. (1986). “Unified dynamics for microscopic and
macroscopic systems,” Physical Review D, 34: 470–491.
Gisin, N. (1989). “Stochastic quantum dynamics and relativity,” Helvetica Physica Acta,
62: 363–371.
Goldstein, S. (1998). “Quantum theory without observers – Part two,” Physics Today, 51:
38–42.
Kent, A. (2005). “Nonlinearity without superluminality,” Physical Review A, 72:
012108.
Maudlin, T. (1996). “Space-time in the quantum world,” pp. 285–307 in J. T. Cushing,
A. Fine, and S. Goldstein (eds.), Bohmian Mechanics and Quantum Theory: An
Appraisal. Dordrecht: Kluwer Academic Publishers.
Maudlin, T. (2007). “Completeness, supervenience and ontology,” Journal of Physics A:
Mathematical and Theoretical, 40: 3151–3171.
Myrvold, W. C. (2003). “Relativistic quantum becoming,” The British Journal for the
Philosophy of Science, 53: 475–500.
Myrvold, W. C. (2011). “Nonseparability, classical and quantum,” The British Journal for
the Philosophy of Science, 62: 417–432.
Myrvold, W. C. (2016). “Lessons of Bell’s theorem: Nonlocality, yes; action at a distance,
not necessarily,” pp. 238–260 in S. Gao and M. Bell (eds.), Quantum Nonlocality and
Reality: 50 Years of Bell’s Theorem. Cambridge: Cambridge University Press.
Myrvold, W. C. (2017). “Relativistic Markovian dynamical collapse theories must employ
nonstandard degrees of freedom,” Physical Review A, 96: 062116.
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Collapse of the Wave Function: Models, Ontology, Origin, and Implications. Cam-
bridge: Cambridge University Press.
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Quantum Mechanics. Oxford: Oxford University Press.
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Ontology for Relativistic Collapse Theories 31
2.1 Introduction
In the seventies, Bas van Fraassen (1972, 1974) proposed an approach to quantum
mechanics different than those of the best known interpretations. According to
him, although the quantum state always evolves unitarily (with no collapse), it is a
modal element of the theory: It describes not what is the case but what may be the
case. This idea led several authors since the eighties to propose the so-called modal
interpretations (Kochen 1985, Dieks 1988, 1989, Vermaas and Dieks 1995, Dieks
and Vermaas 1998, Bacciagaluppi and Dickson 1999, Bene and Dieks 2002), that
is, realist, noncollapse interpretations of the standard formalism of quantum mech-
anics, according to which the quantum state assigns probabilities to the possible
values of all the properties of the system. But since the contextuality of quantum
mechanics (Kochen and Specker 1967) implies that it is not possible to consist-
ently assign definite values to all the properties of a quantum system at a single
time, it is necessary to pick out, from the set of all observables of a quantum system
the subset of definite-valued properties. The different modal interpretations differ
from each other mainly with respect to their rule of definite-value ascription (see
Lombardi and Dieks 2017 and references therein).
Like most interpretations of quantum mechanics, the traditional modal
interpretations were specifically designed to solve the measurement problem. In
fact, they successfully reached this goal in the case of ideal measurements.
However, a series of articles of the nineties (Albert and Loewer 1990, 1991,
1993, Elby 1993, Ruetsche 1995) showed that those traditional approaches based
on the modal views did not pick out the right properties for the apparatus in
nonideal measurements, that is, in measurements that do not introduce a perfect
correlation between the possible states of the measured system and the possible
states of the measuring apparatus. As ideal measurements can never be achieved in
practice, this shortcoming was considered a “silver bullet” for killing modal
32
The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 33
Galilei group. Finally, in Section 2.5, the ontological picture suggested by the MHI
will be described, stressing how that picture supplies a conceptually clear solution
to some traditional interpretive problems of quantum mechanics.
With respect to the definite-valued observables, the basic idea of the MHI is that
the Hamiltonian of the system, with its own symmetries, defines the subset of the
observables that acquire definite actual values. The group of the transformations
that leave the Hamiltonian invariant is usually called Schrödinger group (Tinkham
1964). In turn, each symmetry of the Hamiltonian leads to an energy degeneracy.
The degeneracies with origin in symmetries are called “normal” or “systematic,”
and those that have no obvious origin in symmetries are called “accidental”
(Cohen-Tannoudji, Diu, and Lalöe 1977). However, a deeper study usually shows
either that the accidental degeneracy is not exact or else that a hidden symmetry in
the Hamiltonian can be found that explains the degeneracy. For example, the
degeneracy in the hydrogen atom of states of different angular momentum l but
the same principal quantum number n arises from a four-dimensional rotational
symmetry of the Hamiltonian in momentum space (Fock 1935). For this reason it is
assumed that once all the symmetries of the Hamiltonian have been considered, a
basis for the Hilbert space of the system is obtained and the “good quantum
numbers” are well defined.
Once these symmetry considerations are taken into account, the basic idea
of the MHI can be expressed by the classical Latin maxim Ubi lex non
distinguit, nec nos distinguere debemus (where the law does not distinguish,
neither ought we to distinguish). The Hamiltonian of the system, with its
symmetries, is what determines which observables acquire definite values.
This means that any observable whose eigenvalues would distinguish among
eigenvectors corresponding to a single degenerate eigenvalue of the Hamilto-
nian does not acquire definite value, because such an acquisition would
introduce in the system an asymmetry not contained in the Hamiltonian. Once
this idea is understood, the rule of definite-value ascription can be formulated
in a very simple way:
Actualization rule: Given an elemental quantum system S : ðO; H Þ, the actual
definite-valued observables of S are H, and all the observables commuting with H
and having, at least, the same symmetries as H.
While definite records of the apparatus’ pointer are obtained even in nonideal
situations, one can legitimately ask whether all nonideal measurements are equally
unsatisfactory. The MHI supplies a clear criterion to distinguish between reliable
and nonreliable frequency measurements (for a detailed explanation and applica-
tions, see Lombardi and Castagnino 2008: section 6). In the former case, the
coefficients of the measured system’s state can be computed on the basis of the
frequencies of the pointer’s readings is spite of the imperfect correlation; in the
latter case, the same computation would give inaccurate results (for a presentation
of the reliability criterion in informational terms, see Lombardi, Fortin, and López
2015). Albert and Loewer (1990, 1993) were right in claiming that the ideal
measurement is a situation that can never be achieved in practice: The interaction
between measured system and measurement apparatus never introduces a com-
pletely perfect correlation; in spite of this, physicists usually perform successful
measurements. The MHI account of the quantum measurement shows that perfect
correlation is not a necessary condition for “good” measurements: The coefficients
of the system’s state at the beginning of the process can be approximately obtained
even when the correlation is not perfect, if the reliability condition is satisfied.
Nevertheless, both in the reliable and in the nonreliable case, a definite reading of
the apparatus’ pointer is obtained in each single measurement.
definite value to Lz ; the definite value of Lz would break the symmetry of the
Hamiltonian of the free hydrogen atom in a completely arbitrary way.
If we want to have empirical access to Lz , we need to apply a magnetic field B
along the z-axis, which breaks the isotropy of space and, as a consequence, the
space-rotation symmetry of the atom’s Hamiltonian. In this case, the symmetry
breaking removes the energy degeneracy in ml : Now Lz is not arbitrarily chosen
but selected by the direction of the magnetic field. However this, in turn, implies
that the atom is no longer free: The Hamiltonian of the new system includes the
magnetic interaction. As a consequence, the original degeneracy of the ð2l þ 1Þ-
fold multiplet of fixed n and l is now removed, and the energy levels turn out to be
displaced by an amount Δωnlml , which is also function of ml : This is the manifest-
ation of the so-called Zeeman effect. This means that the Hamiltonian, with
eigenvalues ωnlml , is now nondegenerate: It constitutes by itself the CSCO fH g
that defines the preferred basis fjn; l; ml ig. According to the MHI’s rule of definite-
value ascription, in this case H and all the observables commuting with H are
definite-valued: Since this is the case for L2 and Lz , in the physical conditions
leading to the Zeeman effect, both observables acquire definite values.
Besides the free hydrogen atom and the Zeeman effect, the MHI was applied to
many other physical situations, leading to the results expected from a physical
viewpoint; e.g., the free-particle with spin, the harmonic oscillator, the fine struc-
ture of atoms, the Born-Oppenheimer approximation (see Lombardi and Castag-
nino 2008: section 5). Recently, the interpretation was applied to solve the problem
of optical isomerism (Fortin, Lombardi, and Martínez González 2018), which is
considered one of the deepest problems for the foundations of molecular
chemistry.
All those physical situations show that we have no empirical access to the
observables that are generators of the symmetries of the system’s Hamiltonian;
and, in the context of measurement, the observable A of the measured system S
may be one of those observables. This is also the case in the Stern–Gerlach
experiment, where Sz is a generator of the space-rotation symmetry of
H spin ¼ k S2 ; it is the interaction with the magnetic field B ¼ Bz that breaks the
isotropy of space by privileging the z-direction and, as a consequence, breaks the
space-rotation symmetry of H spin . Therefore, when the observable A to be meas-
ured is a generator of a symmetry of the Hamiltonian of S, the interaction with the
apparatus M must not only establish a correlation between A and the pointer P, but
also must break that symmetry. Therefore, from a physical viewpoint, measure-
ment can be conceived as a process that breaks the symmetries of the Hamiltonian
of the system to be measured and, in this way, turns an otherwise nondefinite-
valued observable into a definite-valued and empirically accessible observable.
This means that the formal von Neumann model of quantum measurement must be
The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 39
quantum system are (i) the observables C i represented by the Casimir operators of
the Galilei group in the corresponding irreducible representation, and (ii) all the
observables commuting with the Ci and having, at least, the same symmetries as
the Ci (Lombardi, Castagnino, and Ardenghi 2010). Therefore, the interpretation
should be more precisely referred to by the name “modal-Casimir interpretation,”
although in the case of nonrelativistic quantum mechanics the original name is also
adequate.
guaranteed. This fact suggests the possibility of generalizing the idea of relying on
symmetry groups in two senses.
It cannot be expected that relativistic quantum mechanics be invariant under
the Galilei group, given the fact that it includes the action of electric and
magnetic fields described by a theory that is not Galilei invariant, but Poincaré
invariant. In turn, in quantum field theory, fields are quantum items, not
external fields acting on a quantum system; as a consequence, the generators
of the Poincaré group do not need to be reinterpreted in the presence of external
factors. These facts lead to generalize the group-based interpretive ideas: The
realist interpretation, expressed in terms of the Casimir operators of the Galilean
group in nonrelativistic quantum mechanics, can be transferred to the relativistic
domain by changing the symmetry group accordingly – the definite-valued
observables of a system in relativistic quantum mechanics and in quantum field
theory would be those represented by the Casimir operators of the Poincaré
group. Since the mass operator M and the squared-spin operator S2 are the only
Casimir operators of the Poincaré group, they would always be definite-valued
observables. This conclusion agrees with a usual physical assumption: Elemen-
tal particles always have definite values of mass and spin, and those values are
precisely what define their different kinds. Moreover, the classical limit of
relativistic theories manifests the limit of the corresponding Casimir operators
(see Ardenghi, Castagnino, and Lombardi 2011): There is a meaningful limiting
relation between the observables that acquire definite values according to
relativistic theories and those that acquire definite values according to nonrela-
tivistic quantum mechanics.
These group-based interpretive ideas can be further generalized in a second
sense. If invariance is a mark of objectivity, there is no reason to focus only on
spacetime global symmetries. Internal or gauge symmetries should also be con-
sidered as relevant in the definition of objectivity and, as a consequence, in the
identification of the definite-valued observables of the system. For instance, in
relativistic quantum mechanics a gauge symmetry is what identifies the charge as
an objective quantity. Therefore, a realist interpretation can be extended to the
gauge symmetries of the theory: The observables represented by operators invari-
ant under those symmetries are also definite-valued observables according to the
theory.
In summary, besides its wide applicability in the nonrelativistic quantum
domain, the MHI opens the way for a general interpretive strategy, valid for any
realistic view of quantum theories – the definite-valued observables of a system,
whose behavior is governed by a certain theory, are the observables invariant under
all the transformations corresponding to the symmetries of the theory, both exter-
nal and internal.
The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 43
depend on the order in which the original identical bundles are considered; the
combination of identical bundles must be commutative. This commutativity is
manifested by the fact that the observables that constitute the resulting bundle-
system are represented by operators symmetric with respect to the permutation of
the indices coming from the original identical bundles. Here symmetry is not an ad
hoc assumption but a consequence of an ontological feature. When the expectation
values of these symmetric observables are computed, only the symmetric part of
the state has an effect. The nonsymmetric part is superfluous, because it plays no
role in the physically measurable magnitudes (see details in da Costa et al. 2013).
Therefore, symmetrization is not the result of an ad hoc strategy, but is due to
ontological reasons: The symmetry properties of states are a consequence of the
symmetry of the observables of the whole composite system, which is, in turn, a
consequence of the ontological picture supplied by the interpretation. In other
words, from the perspective given by the modal-Hamiltonian interpretation,
indistinguishability is not a relation between particles manifested in statistics, but
rather an internal symmetry of a single bundle of properties.
In summary, according to the MHI, the talk of individual entities and their
interactions can be retained only in a metaphorical sense. In fact, even the number
of particles is represented by an observable, and superpositions of different particle
numbers are theoretically possible. This fact, puzzling from an ontology populated
by individuals, involves no mystery in an ontology of properties: If quantum
systems are bundles of possible properties, the particle picture, with a definite
number of particles, is only a contextual picture valid exclusively when the number
of particles satisfies the constraints of the rule of definite-value ascription. In other
cases, wave packets may remain narrow and more or less localized during a
relatively long time. In this way, particle-like behavior can temporarily emerge –
wave packets can represent approximately definite positions and can follow an
approximately definite trajectory (see Lombardi and Dieks 2016). Moreover, the
MHI has proved to be compatible with the theory of decoherence (Lombardi 2010,
Lombardi, Ardenghi, Fortin, and Castagnino 2011, Lombardi, Ardenghi, Fortin,
and Narvaja 2011). Nevertheless, those particular situations do not undermine the
fact that quantum systems are nonindividual bundles of properties.
There are several issues that can still be faced from this interpretive frame-
work. A very interesting question is that related to the interpretation of external
fields in a theory that, as quantum mechanics, does not treat fields as quantized
entities. In particular, the Aharonov-Bohm effect is worthy of being analyzed
from an ontology-of-properties view. Another topic to be examined is how the
MHI is in resonance with a closed-system view of decoherence (Castagnino and
Lombardi 2004, 2005a, b, Castagnino, Laura, and Lombardi 2007, Castagnino,
Fortin, and Lombardi 2010, 2014), according to which decoherence is a process
relative to the selected partition of a closed system and how this leads to a top-
down view of quantum mechanics based on an algebraic view that turns
entanglement and discord also into relative phenomena (for initial ideas, see
Lombardi, Fortin, and Castagnino 2012, Fortin and Lombardi 2014, Lombardi
and Fortin 2016). Finally, the natural subsequent interpretive step consists in
extending the MHI to quantum field theory, not only regarding the definite-
valued observables, but also with respect to the ontology referred to by the
theory. In particular, an ontology-of-properties view seems to favor a field view
in the debate on fields vs. particles, but without representing an obstacle to
explaining the emergence of the nonrelativistic quantum ontology. These differ-
ent problems will guide the future research on the further development and
extrapolation of the MHI.
Acknowledgments
I am grateful to the participants of the workshop Identity, indistinguishability and
non-locality in quantum physics (Buenos Aires, June 2017) for their useful com-
ments. This work was made possible through the support of Grant 57919 from the
John Templeton Foundation and Grant PICT-2014–2812 from the National
Agency of Scientific and Technological Promotion of Argentina.
References
Albert, D. and Loewer, B. (1990). “Wanted dead or alive: Two attempts to solve Schrö-
dinger’s paradox,” pp. 277–285 in A. Fine, M. Forbes, and L. Wessels (eds.),
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Association.
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Synthese, 88: 87–98.
Albert, D. and Loewer, B. (1993). “Non-ideal measurements,” Foundations of Physics
Letters, 6: 297–305.
Ardenghi, J. S., Castagnino, M., and Lombardi, O. (2009). “Quantum mechanics: Modal
interpretation and Galilean transformations,” Foundations of Physics, 39: 1023–1045.
48 Olimpia Lombardi
Lombardi, O., Fortin, S., and Castagnino, M. (2012). “The problem of identifying the
system and the environment in the phenomenon of decoherence,” pp. 161–174 in
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Weyl, H. (1952). Symmetry. Princeton, NJ: Princeton University Press.
3
Quantum Mechanics and Perspectivalism
dennis dieks
51
52 Dennis Dieks
The properties of a quantum system according to Bohr only become well defined in
the context of the system’s coupling to a measuring device which points in the
direction of a relational nature of physical properties.
A more formal analysis of quantum measurements, close to von Neumann’s
account, was given by London and Bauer in their 1939 booklet on the Theory of
Observation in Quantum Mechanics. London and Bauer consider three interacting
systems: x, the object system, y, a measuring device, and z, the observer. As a
result of the unitary evolution of the combined object-device system, an entangled
P
state will result: k ck jxik jyik . When the observer reads off the result of the
measurement, a similar unitary evolution of the x,y,z system takes place, so that
P
the final state becomes: jΨi ¼ k ck jxik jyik jzik . London and Bauer (1939: 41–42)
comment:
“Objectively” that is, for us who consider as “object” the combined system x,y,z the
situation seems little changed compared to what we just met when we were only
considering apparatus and object. . . . The observer has a completely different viewpoint:
for him it is only the object x and the apparatus y which belong to the external world, to
that what he calls “objective.” By contrast, he has with himself relations of a very special
character: he has at his disposal a characteristic and quite familiar faculty which we can call
the “faculty of introspection.” For he can immediately give an account of his own state. By
virtue of this “immanent knowledge” he attributes to himself the right to create his own
objectivity,
P namely, to cut the chain of statistical correlations expressed by
c
k k jx i k jy i k zik by stating “I am in the state jzik ,” or more simply “I see yk ” or even
j
directly “X ¼ yk .” [Here X stands for the observable whose value is measured by the
apparatus.]
It is clear from this quote and the further context that London and Bauer
believed that there is a role for human consciousness in bringing about a definite
measurement outcome even though they also assumed, like von Neumann, that
“from the outside” the observer, including his consciousness, can be described in a
physicalist way, by unitary quantum evolution (see Jammer 1974). The appeal to
consciousness can hardly be considered satisfactory, though: It appears to invoke a
deus ex machina, devised for the express purpose of reconciling unitary evolution
with definite measurement results. More generally, the hypothesis that the definite-
ness of the physical world only arises as the result of the intervention of (human?)
consciousness does not sit well with the method of physics.
Although certain elements of London and Bauer’s solution are therefore hard to
accept, the suggestion that it should somehow be possible to reconcile universal
unitary evolution and the resulting omnipresence of entangled states, with the
occurrence of definite values of physical quantities, appears plausible. Indeed,
the theoretical framework of quantum mechanics itself (as opposed to modifica-
tions of the quantum formalism, as in the Ghirardi–Rimini–Weber [GRW] theory)
54 Dennis Dieks
does not in a natural way leave room for another dynamical process beside unitary
evolution; e.g., there is no time scale or scale of complexity at which this alterna-
tive evolution could set in. As already mentioned, empirical results support this
verdict. Accordingly, in the next sections we shall investigate whether the early
intuitions about the universality of unitary evolution, excluding collapse as a
dynamical process, can be salvaged in a purely physicalist way. We shall argue
that “perspectival” noncollapse interpretations capture the intuitions behind the
London and Bauer and von Neumann analyses, without an appeal to consciousness
or human observers.
superposed state of the room and its contents), the formalism tells us that we shall
find the result “1” with certainty; this is different from what a mixed state would
predict. Experiment confirms predictions of this kind.
But we also possess robust experience about what happens when we watch an
experiment while finding ourselves inside a closed laboratory room: There will be
a definite outcome. It therefore seems inevitable to accept that during the experi-
ment our friend becomes aware of exactly one spin value. As stated by London and
Bauer, our friend will be justified in saying either “the spin is up” or “the spin is
down” after the experiment. The dilemma is that we, on the outside, can only
derive an “improper mixture” as a state for the particle spin and that well-known
arguments forbid us to think that this mixture represents our ignorance about the
actually realized spin-eigenstate (indeed, if the spin state actually was one of the up
or down eigenstates, it would follow that the total system of room and its contents
had to be an ignorance mixture as well, which conflicts with the premise
supported both theoretically and empirically that the total state is a
superposition).
Our proposed perspectival way out of this dilemma is to ascribe more than one state
to the same physical system. In the case under discussion, with respect to us,
representing the outside point of view, the contents of the laboratory room are
correctly described by an entangled pure state so that we should ascribe improper
mixtures (obtained by “partial tracing”) to the inside observer, the measuring device
and the spin particle. But with respect to the inside observer (or with respect to the
measuring device in the room) the particle spin is definite-valued. So the inside
observer assigns a state to his environment that appropriately reflects this definiteness.
This line of thought leads to the idea of assigning relational or perspectival
states, i.e., states of a physical system A from the perspective of a physical system
B. This step creates room for the possibility that the state and physical properties of
a system A are different in relation to different “reference systems” B. As suggested
by the examples, this move may make it possible to reconcile the unitary evolution
during a quantum measurement with the occurrence of definite outcomes. The
properties associated with the superposition and the definite outcomes, respect-
ively, would relate to two different perspectives the idea already suggested by
von Neumann and by London and Bauer. Of course, we should avoid the earlier
problems associated with consciousness. The different perspectives, and different
relational states, should therefore be defined in purely physical terms.
The idea as just formulated was tentative: We spoke in a loose way of “states,”
thinking of wave functions (or vectors in Hilbert space) without specifying what
the attribution of quantum states to physical systems means on the level of physical
quantities, i.e., in terms of physical properties of the systems concerned. In fact,
this physical meaning is interpretation-dependent.
Quantum Mechanics and Perspectivalism 57
intended to possess an ontological status: It is not the case that for an outside
observer the internal spin values are definite but unknown. The proposal is that the
spin really is indeterminate with respect to the world outside the laboratory room.
This perspectivalism with respect to properties does not seem an inevitable
feature of all noncollapse interpretations, however. In particular, those interpret-
ations of quantum mechanics in which it is assumed that there exists an a priori
given set of preferred observables that is always definite in all physical systems,
at all times, and in all circumstances are by construction at odds with the
introduction of a definiteness that is merely relative. The Bohm interpretation is
a case in point. According to this interpretation all physical systems are composed
of particles that always possess a definite position, as a monadic attribute inde-
pendent of any perspective. So in our sealed-room experiment the instantaneous
situation inside is characterized by the positions of all particles in the room, and
this description is also valid with respect to the outside world even though an
outside observer will usually lack information about the exact values of the
positions. Thus for an external observer there exists one definite outcome of the
experiment inside, corresponding to one definite particle configuration. The out-
come of any measurement on the room as a whole that the outside observer might
perform again corresponds to a definite configuration of particles with well-defined
positions. The fact that this value is not what we would classically expect (for
example, when we measure jΨihΨj) is explained by the Bohm theory via the
nonclassical measurement interaction between the external observer’s measuring
device and the room. The quantum states that in perspectival schemes encode
information about which physical properties are definite, in the Bohm types of
interpretations only play a role in the dynamics of a fixed set of quantities, so that
the possibility of relational properties or perspectivalism does not suggest itself.
However, it has recently been argued that all interpretations of this unitary kind,
characterized by definite and unique (i.e., one-world) outcomes at the end of each
successful experiment even though the total quantum state always evolves unitar-
ily, cannot be consistent (Araújo 2016, Frauchiger and Renner 2016). This argu-
ment is relevant for our theme, and we shall discuss it in some detail.
they do not fully take into account that in unitary interpretations only the total
(noncollapsed) state can be used for predicting the probabilities of results obtained
by the Assistant and Wigner (The bone of contention is statement 4 in Araújo’s
(2016) reconstruction of the F-R inconsistency, in which Friend 1 argues that her
coin measurement result is only compatible with one single later result obtained by
Wigner in the final measurement. However, in unitary interpretations previous
measurement results do not always play a role in the computation of probabilities
for future events. Indeed, a calculation on the basis of the total uncollapsed
quantum state, as given by Araújo (2016: 4), indicates that Wigner may find either
one of two possible outcomes, with equal probabilities, even given the previous
result of Friend 1 this contradicts the assumption made by Frauchiger and
Renner).
The situation becomes more transparent when we make use of an elegant
version of the F-R thought experiment recently proposed by Bub (2017). Bub
replaces Friend 1 by Alice and Friend 2 by Bob; Alice and Bob find themselves at a
great distance from each other. Alice has a quantum coin which she subjects to a
measurement of the observable pffiffiffiA with eigenstates
pffiffiffiffiffiffiffiffi jhiA , jt iA ; the coin has been
prepared in the initial state 1= 3 jhiA þ 2=3jt iA . Alice
pthen ffiffiffi prepares a qubit in
the state j0iB if her outcome is h and in the state 1= 2 j0iB þ j1iB if her
outcome is t. She subsequently sends this qubit to Bob this is the only
“interaction” between Alice and Bob. After Bob has received the qubit, he subjects
it to a measurement of the observable B with eigenstates j0iB , j1iB .
In accordance with the philosophy of noncollapse interpretations, we assume
that Alice and Bob obtain definite outcomes for their measurements, but that
the total system of Alice, Bob, their devices and environments, and the coin
and the qubit, can nevertheless be described by the uncollapsed quantum state,
namely:
1
jΨi ¼ pffiffiffi jhiA j0iB þ jt iA j0iB þ jt iA j1iB (3.1)
3
For ease of notation, the quantum states of Alice and Bob themselves, plus the
measuring devices used by them, and even the states of the environments that have
become correlated to them, have here all been included in the states
jhiA , jt iA , j0iB , j1iB (so that these states no longer simply refer to the coin and the
qubit, respectively, but to extremely complicated many-particles systems!).
Now we consider two external observers, also at a great distance from each other,
who take over the roles of Wigner and his Assistant, and are going to perform
measurements on Alice and Bob (and their entire experimental setups), respectively.
The external observer pffiffiwho
ffi focuses on Alice measures an
pffiffiobservable
ffi X with
eigen-
states jfailiA ¼ 1= 2 jhiA þ jtiA and jokiA ¼ 1= 2 jhiA jt iA , and the
Quantum Mechanics and Perspectivalism 61
1 1
jΨi ¼ pffiffiffiffiffi jokiA jokiB pffiffiffiffiffi jokiA jfailiB þ
12 12
rffiffiffi (3.2)
1 3
þ pffiffiffiffiffi jfailiA jokiB þ jfailiA jfailiB
12 4
rffiffiffi
2 1
jΨi ¼ jfailiA j0iB þ pffiffiffi jt iA j1iB (3.3)
3 3
rffiffiffi
1 2
jΨi ¼ pffiffiffi jhiA j0iB þ jt i jfailiB (3.4)
3 3 A
From Eq. (3.2), we see that the outcome fok; okg in a joint measurement of X and
Y has a nonzero probability: This outcome will be realized in roughly 1/12 th of all
cases if the experiments are repeated many times. From Eq. (3.3) we calculate that
the pair fok; 0g has zero probability as a measurement outcome, so fok; 1g is the
only possible pair of values for the observables X, B in the cases in which X has the
value ok. However, from Eq. (3.4) we conclude that the pair ft; okg has zero
probability, so h is the only possible value for the observable A if Y has the value
ok and A and Y are measured together. So this would apparently lead to the pair of
values fh; 1g as the only possibility for the observables A and B, if X and Y are
jointly measured with the result fok; okg. But this pair of values has zero prob-
ability in the state jΨi so it is not a possible pair of measurement outcomes for
Alice and Bob in that state. So although the outcome fok; okg for X and Y is
certainly possible, the (seemingly) necessarily associated outcome fh; 1g for A and
B is not this seems an inconsistency.
In this inconsistency argument there is a silent use of nonperspectivalism
conditions. For example, if Bob’s measurement outcome is 1 from the perspective
of the X measurement, it is assumed that this outcome also has to be 1 as judged
from the perspective of the Y observer. However this assumption does not sit well
with what the quantum formulas show us: The relative state of Bob with respect to
the Y outcome “ok” is not j1iB , but jokiB (see Eq. (3.5)).
To see what is wrong with the inconsistency argument from a perspectival point
of view that closely follows the quantum formalism, it is helpful to note that the
states in Eqs. (3.1), (3.2), (3.3), and (3.4) are all states of Alice and Bob, including
their devices and environments, but without the external observers. In a consistent
noncollapse interpretation we must also include the external observer states in the
62 Dennis Dieks
3.4 Relativity
The diagnosis of the previous section is confirmed when we directly study the
consequences of relativity for interpretations of quantum mechanics. In particular,
when we attempt to combine special relativity with unitary interpretational
schemes, new hints of perspectivalism emerge. As mentioned, the Bohm interpret-
ation has difficulties in accommodating Lorentz invariance. Bohmians have there-
fore generally accepted the existence of a preferred inertial frame in which the
equations assume their standard form a frame resembling the ether frame of
prerelativistic electrodynamics. Accepting such a privileged frame in the context of
what we know about special relativity and Minkowski spacetime is, of course, not
Quantum Mechanics and Perspectivalism 65
their particles, in a certain direction; thereafter, at a second instant, each of the three
outside observers performs a measurement on his or her room. This measurement
is of a “whole-room” observable, like in the Frauchiger-Renner thought experi-
ment discussed in the previous section. As a result of the internal measurements by
the three friends the whole system, consisting of the rooms and their contents, has
ended up in an entangled GHZ-state. Leegwater is able to show that this entails that
the assumption that the standard rules of quantum mechanics apply to each of three
differently chosen sets of simultaneity hyperplanes, gives rise to a GHZ-
contradiction: The different possible measurement outcomes (all +1 or 1) cannot
be consistently chosen such that each measurement has the same outcome irre-
spective of the simultaneity hyperplane on which it is considered to be situated
(and so that all hyperplanes mesh). As in the original GHZ-argument (Greenberger
et al. 1990), the contradiction is algebraic and does not involve the violation of
probabilistic (Bell) inequalities.
One way of responding to these results is the introduction of a preferred inertial
system (a privileged perspective!), corresponding to a state of absolute rest,
perhaps defined with respect to an ether. This response is certainly against the
spirit of special relativity, in particular because the macroscopic predictions of
quantum mechanics are such that they make the preferred frame undetectable.
Although this violation of relativistic invariance does not constitute an inconsist-
ency, it certainly is attractive to investigate whether there exist other routes to
escape the no-go theorems. Now, as we have seen, a crucial assumption in these
theorems is that properties of systems are monadic, independent of the presence of
other systems and independent of the hyperplane on which they are considered.
This suggests that a transition to relational or perspectival properties offers an
alternative way out.
In fact, that unitary evolution in Minkowski spacetime leads naturally to a
hyperplane-dependent account of quantum states if one describes measurements
by effective collapses has been noted in the literature before (see e.g. Dieks 1985,
Fleming 1996, Myrvold 2002b). The new light that we propose to cast on these and
similar results comes from not thinking in terms of collapses, and of a dependence
on hyperplanes or foliations of Minkowski spacetime as such, but instead of
interpreting them as consequences of the perspectival character of physical prop-
erties: that the properties of a system are defined with respect to other systems.
What we take the considerations in the previous and present sections to suggest is
that it makes a difference whether we view the physical properties of a system from
one or another system – or from one or another temporal stage in the evolution of a
system. In the case of the (more-or-less) localized systems that figure in the
relativistic no-go theorems that we briefly discussed, this automatically leads to
property ascriptions that are different on the various hyperplanes that are
Quantum Mechanics and Perspectivalism 67
is natural to make the description of the measurement and its result perspective
dependent: For the two friend-branches inside the room there is a definite outcome,
but this is not so for the external observer. So perspectivalism as a consequence of
holding fast to unitarity and Lorentz invariance seems more basic than the further
choice of interpreting measurements in terms of many worlds; even the many-
worlds interpretation must be committed to perspectivalism. But perspectivalism
on its own is already sufficient to evade the anti-single-world arguments of Section
3.3, so for this purpose we do not need the further assumption of many worlds.
Finally, the introduction of perspectivalism opens the door to several new
questions. In everyday circumstances we do not notice consequences of
perspectivalism, so we need an account of how perspectival effects are washed
out in the classical limit. It is to be expected that decoherence plays an important
role here, as alluded to in the Introduction however, this has to be further worked
out (compare Bene and Dieks 2002). Further, there is the question of how the
different perspectives hang together; for example, in Section 3.3 it was shown that
perspectives of distant observers cannot be simply combined in the case of
entanglement, which may be seen as a nonlocal aspect of perspectivalism. By
contrast, it has been suggested in the literature that perspectivalism makes it
possible to give a purely local description of events in situations of the Einstein-
Podolsky-Rosen type, and several tentative proposals have been made in order to
substantiate this (Rovelli 1996, Bene and Dieks 2002, Smerlak and Rovelli 2007,
Dieks 2009, Laudisa and Rovelli 2013). These and other questions constitute
largely uncharted territory that needs further exploration.
References
Araújo, M. (2016). “If your interpretation of quantum mechanics has a single world but no
collapse, you have a problem,” http://mateusaraujo.info/2016/06/20/if-your-interpret
ation-of-quantum-mechanics-has-a-single-world-but-no-collapse-you-have-a-problem/
Bacciagaluppi, G. (2002). “Remarks on space-time and locality in Everett’s interpretation,”
pp. 105–122 in T. Placek and J. Butterfield (eds.), Non-Locality and Modality.
Dordrecht: Springer.
Bene, G. and Dieks, D. (2002). “A perspectival version of the modal interpretation of
quantum mechanics and the origin of macroscopic behavior,” Foundations of Physics,
32: 645–671.
Berndl, K., Dür, D., Goldstein, S., and Zanghì, N. (1996). “Nonlocality, Lorentz invari-
ance, and Bohmian quantum theory,” Physical Review A, 53: 2062–2073.
Bohm, D. (1952). “A suggested interpretation of the quantum theory in terms of ‘hidden’
variables, I and II,” Physical Review, 85: 166–193.
Bohr, N. (1928). “The quantum postulate and the recent development of atomic theory,”
Nature, 121: 580–590.
Bohr, N. (1935). “Can quantum-mechanical description of physical reality be considered
complete?,” Physical Review, 48: 696–702.
Quantum Mechanics and Perspectivalism 69
71
72 Nino Zanghì
that traditional scientific realism was childish and nonscientific, and he proposed
what it is still called the Copenhagen interpretation of quantum mechanics. On the
basis of this doctrine, the physical laws no longer have to do with the question of
how the world is made, but with our ability to know it, which is intrinsically
limited: The quantum mechanics of Copenhagen refuses in principle to provide a
consistent history of what happens to microscopic objects. From the point of view
of Copenhagen, reality is divided into two worlds, the microscopic and the
macroscopic, the classical and the quantum, the world regulated by classical
logic and the one regulated by quantum logic. Although it is not clear where the
boundary between these two worlds lies and how this duality can be compatible
with the fact that apples and chairs consist of electrons and other particles, the
Copenhagen doctrine has become orthodoxy. That is to say, it has become not only
the majority viewpoint among physicists, but also the dogma.
In more recent years, a version of quantum mechanics based on information
theory has grown in popularity. It is a dress that seems new, packed on the wave of
the theoretical and experimental development of quantum information and quan-
tum computation, but in reality it is a used dress, which was already tailored in
Copenhagen. Already in 1952, Schrödinger warned against the idea of reducing
quantum mechanics to a simple representation of our knowledge (Schrödinger
1995).
In spite of the pragmatic tribute reserved to the dogma, the peculiar role of the
observer in the formulation of the theory has always puzzled many physicists, as
can be seen for example, from the following considerations by Richard Feynman:
This is all very confusing, especially when we consider that even though we may
consistently consider ourselves to be the outside observer when we look at the rest of
the world, the rest of the world is at the same time observing us, and that often we agree on
what we see in each other. Does this then mean that my observations become real only
when I observe an observer observing something as it happens? This is a horrible
viewpoint. Do you seriously entertain the idea that without the observer there is no
reality? Which observer? Any observer? Is a fly an observer? Is a star an observer? Was
there no reality in the universe before 10⁹ B.C. when life began? Or are you the observer?
Then there is no reality to the world after you are dead? I know a number of otherwise
respectable physicists who have bought life insurance.
(Feynman, Morinigo, and Wagner 2003: 14)
Feynman is putting his finger on the most commonly cited conceptual difficulties
that plague quantum mechanics – the measurement problem, or what amounts to
more or less the same thing, the paradox of Schrödinger’s cat. The problem can be
rephrased as follows: Suppose that the wave function of any individual system
provides a complete description of that system. When we analyze the process of
measurement in quantum mechanical terms, we find that the after-measurement
Quantum Physics Grounded on Bohmian Mechanics 73
wave function for system and apparatus arising from the Schrödinger equation
typically involves a superposition over terms corresponding to what we would like
to regard as the various possible results of the measurement, e.g., different pointer
orientations. It is difficult to discern in this description of the after-measurement
situation the actual result of the measurement, e.g., some specific pointer orienta-
tion. In brief, quantum mechanics does not account for the obvious fact that
measurements do have results.
Bohr considered that philosophy was very important to understand quantum
mechanics and introduced the notion of complementarity, a many-purpose notion
good for solving the wave-particle duality, the measurement problem, and indeed,
all interpretative problems of quantum mechanics. This attitude sustained the idea
that with the problem of measurement we are facing a purely philosophical
problem. This idea was then nurtured and nourished by a sort of naive realism
about the operators – the idea that in quantum mechanics the observables and the
properties acquire a radically new, highly nonclassical meaning, reflected in the
noncommutative structure of the algebra of quantum observables.
4.2 Noncommutativity
The Hilbert space of quantum states is a vector space with a scalar product rule,
and it would be surprising if the operators on this space did not play an important
role in the formulation of quantum theory. And indeed, it is obviously so: The
temporal evolution of the states is given by a unitary operator that is generated by a
self-adjoint operator, the Hamiltonian. Not only are the temporal translations
governed by a self-adjoint operator, but so also are all the other symmetries of
the system. For example, the momentum operator is the generator of spatial
translations and the angular momentum operators govern the change of states as
a consequence of a rotation of the physical space. In quantum field theory, the
operators of creation and annihilation, operators that transform the state of a system
with a certain number of particles into another with a different number of particles,
play an extremely important role, as basic bricks of the Hamiltonian. In brief, linear
operators play an important role in quantum mechanics. And the main algebraic
feature of the operators is not to commute. So far, everything is clear and nothing is
mysterious.
The mystery arises when it is argued that the association of quantum observ-
ables with self-adjoint operators is to be considered a direct generalization of the
notion of classical observables and that quantum theory should not be conceptually
more problematic than classical physics once this fact is appreciated.
The classical observables – for a particle system, their positions, their momenta
and the functions of these variables, that is, functions on phase space – form a
74 Nino Zanghì
4.3 Contextuality
A milestone in the foundations of quantum mechanics is Bell’s nonlocality analy-
sis (Bell 1964). It has a by-product that is interesting in itself: The incompatibility
of Bell’s inequality with the predictions of quantum mechanics is a demonstration
Quantum Physics Grounded on Bohmian Mechanics 75
are not properties at all, do not exist, and their inadequacy to carry out the role of
properties is in the strongest sense possible.
In short, contextuality means nothing more than the fact that the result of an
experiment depends on the experiment itself, and this applies equally to both
classical physics and quantum physics: For any experiment, be it classical or
quantum, it would be a mistake to assume that any device involved in the experi-
ment plays only a passive role, unless the experiment is not the genuine measure-
ment of a property of the system, in which case the result is determined by the
initial condition of the system only. In classical physics, it is traditionally assumed
that it is in principle possible to measure any property without sensitively disturb-
ing the measured object, but this is false in quantum mechanics – and should be
questioned in classical physics, too.
So, the orthodox vision ends up providing a response to the problem of measure-
ment that many orthodoxy enthusiasts still struggle to accept: the wave function
does not provide a complete representation of the state of affairs of the world; in
addition, you need to specify the values of classic variables – which for conveni-
ence will be named here “Z-variables.”
According to Bohr, the Z-variables are precisely those that establish quantum
mechanics and make the quantum formalism coherent and applicable to the
study of the phenomena we observe in the laboratory or in nature. In other
words, according to the orthodox interpretation, the complete description of a
state of affairs of the world is given by the pair ðΨ; Z Þ, where Ψ is the wave
function and the Z are in some sense macroscopic classical variables. So,
although according to the orthodox view Ψ does not represent anything real –
thus the famous motto “there is no quantum world, there is only an abstract
quantum description” – the role of the wave function is to govern the statistics
of the Z-variables that indeed do represent what is to be considered real, or at
least “concrete.”
Quantum Physics Grounded on Bohmian Mechanics 77
As pointed out by John Bell (1990) in the original formulation of the Copen-
hagen interpretation, the sense in which the complete description of a state of
affairs of the world is given by the pair ðΨ; Z Þ is not clearly specified. In addition,
the dynamics of the pair is not specified in a clear and unambiguous way:
Sometimes the dynamics of the wave function is given by the Schrödinger
equation and sometimes the dynamics of the macroscopic variables is that fixed
by the laws of classical mechanics and classical electromagnetism. However, when
the classical variables interact with the quantum variables, the dynamical laws
change: The wave function no longer evolves according to the Schrödinger
equation, instead it evolves according to the collapse rule and the Z-variables
undergo random leaps that are statistically governed by the wave function.
The difficulty raised by Feynman in the passage quoted in Section 4.1 might be
so rephrased: Where is the borderline between what is classical and what is
quantum? When can we treat an object as classical and when must we treat it as
quantum? In other words, the distinction between microscopic and macroscopic, as
well as that between the classical world and the quantum world, lacks a precise
definition and introduces a fundamental ambiguity that cannot have any place in
any theory that claims physical accuracy.
Indeed, if the completion is achieved in what is really the most obvious way – by
simply including the positions of the particles of a quantum system as part of the
state description of that system, allowing these positions to evolve in the most
natural way – one arrives at the theory developed by David Bohm (1952). This
theory is nowadays known as Bohmian mechanics, de Broglie-Bohm’s theory, the
wave-pilot theory, or the causal interpretation of quantum mechanics.
In the theory proposed by Bohm, a particle system is described in part by its
wave function Ψ, which evolves, as usual, according to the Schrödinger equation.
However, the wave function only provides a partial description of the system. This
78 Nino Zanghì
with U the corresponding classical potential energy; rk is the gradient with respect
to the coordinates of the k-th particle; for a complex-valued wave function Ψ, Ψ∗
denotes its complex conjugate; if Ψ has spinorial values, the products in the
numerator and in the denominator of the guiding equation should be understood
as scalar products in the spinor space; if external magnetic fields are present, the
gradient should be understood as a covariant derivative, involving the potential
vector; “Im” means taking the imaginary part of a complex number.
To gain familiarity with the guiding equation, it is useful to consider the
extremely simple case of a single free particle of mass m guided by a monochro-
matic wave with wave vector k (and thus with wavelength λ ¼ 2π=k), ideally
80 Nino Zanghì
approximated by the plane wave Ψðq; tÞ / eiðk:qωtÞ , ω ¼ ℏk2 =ð2mÞ. The l.h.s. of
the guiding equation is the velocity v of the particle. A simple calculation shows
that the right-hand side r.h.s. of the guiding equation is ðℏ=mÞk. Thus the guiding
equation of Bohmian mechanics turns out to be precisely the relation p ¼ ℏk
which de Broglie proposed in late 1923 and which quickly led Schrödinger, during
the end of 1925 and the beginning of 1926, to the discovery of his wave equation.
Now consider the guiding equation in the case of two particles. The wave
function Ψ ¼ Ψðq1 ; q2 ; tÞ generates the possible speeds of the two particles, i.e.,
ℏ Ψ∗ ðq1 ; q2 ; tÞr1 Ψðq1 ; q2 ; tÞ
v1 ¼ Im (4.4)
m1 Ψ∗ ðq1 ; q2 ; tÞΨðq1 ; q2 ; t Þ
for particle 1 and
ℏ Ψ∗ ðq1 ; q2 ; tÞr2 Ψðq1 ; q2 ; tÞ
v2 ¼ Im (4.5)
m2 Ψ∗ ðq1 ; q2 ; tÞΨðq1 ; q2 ; t Þ
for particle 2. These formulas show that the velocity of a particle, at a certain
instant of time, depends in general on where, at the same time, the other particle
is. An exception is the case in which the wave function factorizes, i.e., is of the
form Ψ ¼ Ψðq1 ; t ÞΨðq2 ; t Þ. In this case, the velocity of particle 1 does not
depend on the position q2 of particle 2, and vice versa. However, for a general
wave function, i.e., for an entangled quantum state, there is a corresponding
entanglement of the velocities which persists, without attenuating in any way,
even when the distance between the two particles is very large. This important
property of the guiding law shows that the velocity of a particle can depend on
the position of the other even when the module of the wave function is very
small (which, in fact, is what happens when the distance between the two
particles is very large).
Thus, in Bohmian mechanics, the law that governs the motion of particles in
physical space makes manifest the most dramatic effect of quantum mechanics,
quantum non locality – the fact that physical events can mutually influence each
other more quickly than the speed of light, even at arbitrarily large distances,
without this mutual influence being mediated by physical fields (such as, for
example, the electromagnetic or gravitational field) or by particles (or energy or
signals) that can somehow travel from one event to another. Far from being a
defect, it is a remarkable merit of the theory, since, as Bell has shown,
nonlocality is a basic property of nature. One could say that Bohmian mechanics
does nothing but to inherit and to make explicit the nonlocal character implicit in
the notion, common to all the formulations and interpretations of quantum
mechanics, of a wave function on the configuration space of a system with many
particles.
Quantum Physics Grounded on Bohmian Mechanics 81
d 2 Qk
mk ¼ rk U rk W (4.6)
dt 2
where W is a certain function of the positions of all the particles, which is
determined uniquely by the square of the wave function in a rather complicated
way that, for the purposes of the present discussion, it is not necessary to make
explicit; we just mention an important property of this function: If the wave
function is multiplied by a constant, the value of W does not change. Bohm called
the function W quantum potential (energy).
Eq. (4.4) has a Newtonian form: Its left-hand side (l.h.s.) is the second derivative
with respect to the time of the position of the k-th particle, i.e., its acceleration,
multiplied by the mass of the particle. Therefore, it has the structure of the first
member of the classical Newton equation (mass acceleration). Now consider the
r.h.s. of the equation: U is the potential energy of N interacting particles that
appears in the Schrödinger equation, a function that has exactly the same form as in
the corresponding classical situation. Thus the first term on the right is actually the
force, derived from the potential energy U, which in the corresponding classical
situation would act on the k-th particle. If we interpret the second term as a force
82 Nino Zanghì
derived from the quantum potential W, the r.h.s. of Eq. (4.6) can be interpreted as a
force – a force of classical origin added to a force of quantum origin.
However, this formulation of Bohmian mechanics in classical clothes has a cost.
The most obvious cost is an increase in complexity: The quantum potential is
neither simple nor natural. Moreover, it does not seem very satisfactory to think
that the quantum revolution is reduced to the understanding that, after all, nature is
classical, with an additional force term, all in all fairly ad hoc, the term that
corresponds to the quantum potential.
Furthermore, Bohmian mechanics is not simply a reformulation of quantum
mechanics in classical terms with an additional force term. In Bohmian mechan-
ics, the velocities are not independent of the positions, as in the classical case, but
are constrained by the guiding equation. Despite the apparent form of the second
order of Newton’s quantum equation, its solutions are not characterized by
positions and velocities at some initial time, but only by the initial positions,
because the initial speeds are determined by the guiding law at the initial time. In
other words, having derived the members of the guiding equation with respect to
time, and thus obtained the quantum Newton equation, does not change the
nature of the theory; at most it makes it more complicated and less transparent.
It is as if in classical mechanics we derived both members of the Newton
equation and decreed that the equation of the third order thus obtained should
be considered the classical law of motion. In this way we would not obtain a
genuinely different theory, that is, a third-order theory, with positions, velocities,
and accelerations as initial independent conditions, because the initial acceler-
ations would in any case be bound by the Newton equation to be functions of
initial positions and velocities.
Because the dynamics of Bohmian mechanics is completely defined by the
Schrödinger equation and the guiding equation, there is no need for further axioms
involving a quantum potential. Therefore the quantum potential, together with the
quantum Newton equation in which it appears, should not be considered funda-
mental. The correct way to look at Bohmian mechanics is as a first-order theory, in
which the fundamental quantities are the positions of the particles, whose dynam-
ics is specified by the guiding equation. Second-order concepts, such as acceler-
ation and force, work and energy, play no fundamental role in the theory. The
artificiality suggested by the quantum potential is the price you pay when dressing
a highly nonclassical theory with classical clothes. This does not mean that these
second-order concepts cannot play any role in Bohmian mechanics. These are
emerging notions, which are fundamental for the theory to which Bohmian
mechanics converges in the classical limit – that is, Newtonian mechanics: When
the quantum force is negligible, one has in effect classical behavior (Allori, Dürr,
Goldstein, and Zanghì 2002).
Quantum Physics Grounded on Bohmian Mechanics 83
4.8 The Classical Variables of Bohr out of the Classical Variables of Bohm
The statistical character of quantum theory was first fully acknowledged in
1926 by Max Born, shortly after Schrödinger discovered his famous equation.
Born interpreted Schrödinger’s function in a statistical sense and postulated that
the configuration Q of a quantum system is random, with probability distribution
given by the density jΨðqÞj2 . Under the influence of the growing consensus in
favor of the Copenhagen interpretation, the density jΨðqÞj2 began to be considered
the probability of finding the configuration Q, whatever it was measured, rather
than the probability that the configuration was really Q, a notion that was believed
to be meaningless. In accordance with these quantum probabilities, measurements
performed on a quantum system with a defined wave function typically provide
random results.
The density jΨðqÞj2 takes on a particular importance in Bohmian mechanics. As
an elementary consequence of the Schrödinger equation and the guiding equation,
it is equivariant, in the sense that these equations are compatible with respect to the
distribution jΨðqÞj2 . More precisely, this means that if the configuration Q of a
system is random, with distribution jΨðqÞj2 at some time, then this will also be true
for any other time. This distribution is therefore called the quantum equilibrium
distribution.
A Bohmian universe, although deterministic, evolves in such a way that an
appearance of randomness emerges, precisely as described by quantum formalism.
In order to understand how this comes about, one should realize that, in a world
governed by Bohmian mechanics, the measuring apparatuses are also made of
particles. In a Bohmian universe, the measurement apparatus, tables, chairs, cats,
and other objects of our daily experience are simply agglomerations of particles,
described by the positions they occupy in physical space, and whose evolution is
governed by Bohmian mechanics.
The following theorem is crucial:
The “largest system required for the analysis of the experiment” means the
composite system that includes the system on which the experiment is performed,
as well as the measuring apparatus and all the other instruments used in the
execution of the experiment, together with all the other systems that have signifi-
cant interaction with them during the experiment.
84 Nino Zanghì
The condition of reproducibility means that the quantum experiment should give
the same result if immediately repeated (after having brought the apparatus back to
its READY state). It should be noted that if it is not assumed that the experiment is
reproducible, the statistics of the experiment is given by a more general operator
structure than that of the self-adjoint operators, namely that of generalized quantum
observables that has been introduced to describe quantum measurements in par-
ticular situations encountered in quantum optics, and more generally, in the theory
of open quantum systems (these are the positive-operator-valued measures
[POVM], also used in quantum information). Thus, Bohmian mechanics provides
an immediate understanding of this generalization of the notion of quantum
observable and a clarification of the type of idealization involved in the notion of
“operator as observable.”
The Stern-Gerlach experiment is particularly illuminating to clarify the content
of Theorem 2. In the Stern-Gerach experiment, as a consequence of the interaction
with the magnetic field, the parts of the wave function that are in the different
autospaces of the relevant spin operator (for example, the component along z)
become spatially separated and the particle (the silver atom), which moves
according to the guiding equation, ends up being in the support of only one of
these parts. The end result (“up” or “down”) is therefore a function of the final
position of the particle, which is revealed on the screen. Of this position, what we
can predict is only that it is random and distributed according to jΨðqÞj2 at the final
time. By calibrating the results of the experiment with numerical values, for
example +1 for a top detection and 1 for a bottom detection, it is not difficult
to show (by solving the Pauli-Schrödinger equation and the guiding equation for
this situation) that the probability distribution of these values is expressed in terms
of the usual quantum spin operators – the Pauli matrices.
It is important to observe that, since the results of a Stern-Gerlach experiment
depend not only on the position and initial wave function of the particle, but also
on the choice of the different possible magnetic fields that could be used to
measure the same spin operator, this experiment is not a genuine measurement in
the literal sense; that is, it does not reveal a preexisting value associated with the
spin operator. Indeed, there is nothing mysterious or nonclassical about the non-
existence of such values associated with the operators. Bell said that (for Bohmian
mechanics) spin is not real. Perhaps he should have said, “even spin is not real,”
not simply because among all the quantum observables, the spin is considered to
be the paradigmatic quantum observable, but also because spin is treated in
orthodox quantum mechanics in a fairly similar way to position, as a “degree of
Quantum Physics Grounded on Bohmian Mechanics 87
If at certain time we have that X 2 M dead (as it results from appropriate condi-
tions of the external environment Y, for example, from the temperature value of the
cat measured with a thermometer), the relevant wave function for the subsequent
temporal evolution of the configuration of the cat is ψ dead ðxÞ. In other words, the
guiding law provides the same temporal evolution for the cat configuration
whether we use the whole wave function ψ, or just its ψ dead ðxÞ. Using only the
latter, that is, applying the rule of collapse, is therefore, in Bohmian mechanics,
only a practical matter that does not change anything of what actually happens in
the world – the actual history of the cat and, more generally, of the physical
systems that, like cats, populate the world.
The following question arises: Can a dead cat become alive in the future? What
is it that forbids the X configuration, originally in M dead , to enter at a certain
moment of the future in the region M alive ? Strictly speaking, nothing. In Bohmian
mechanics, such an event is possible, although the probability of its occurrence is
frighteningly small, ridiculously small. However, the mere possibility that such an
event is realized does not involve any interpretative problem for the theory. On the
contrary, it only emphasizes that the explanatory structure of Bohmian mechanics
is quite analogous to that of classical statistical mechanics.
In fact, an event of this type, extraordinary and highly improbable, is also
possible in the dynamical scheme provided by classical mechanics, as it is possible
that all the air in this room spontaneously leaves the window and I die from
suffocation, and if it were possible to imagine a cat such that the particles that
compose it were governed by the laws of classical mechanics, this hypothetical
dead cat could return alive. Events of this type are possible, but highly improbable
because they would entail a decrease in the entropy of the universe. The explan-
ation of the impossibility for the cat to return alive is, in Bohmian mechanics,
exactly of the same type: It is guaranteed by simple entropic reasons, that is, by the
typical behavior of the physical systems according to the second law of
thermodynamics.
The difference with the classical case concerns only how the macroscopic
thermodynamic laws are stabilized by the underlying microscopic dynamics. In
the quantum case, quantum entanglement plays a very important role in the process
of stabilizing macroscopic properties of physical systems and makes (if ever
needed) it extremely unlikely that dead cats can resurrect. In fact, even if the
regions M alive and M dead were not macroscopically disjoint, but only sufficiently
disjoint, the interaction of the system with the external environment and the
consequent formation of entangled states in superpositions containing many terms,
would produce a partition of the configuration space of the larger system (which
includes the starting system and its external environment) in distinct regions, each
in a two-way correspondence with a term of the superposition of the wave function
90 Nino Zanghì
of the largest system. These regions would be macroscopically disjoint, and the
greater the number of external environment systems that come into play, the
greater the macroscopic separation that is achieved, and thus the greater the degree
of irreversibility of the process.
4.12 Relativity
Many of the objections to Bohmian mechanics – if not the overwhelming major-
ity – are very weak and often arise from a gross misunderstanding of the theory.
Some of them arise from the lack of understanding that Bohmian mechanics should
be considered, mathematically and conceptually, as a theory profoundly different
from Newtonian mechanics.
The most serious objection raised against Bohmian mechanics is that this theory
does not account for those phenomena, such as the creation and destruction of
particles, which are characteristic of the quantum theory of fields. Actually, this is
not, in itself, an objection to the Bohmian mechanics, but simply the observation
that quantum field theory, for what concerns what happens in the physical world,
explains much more than the nonrelativistic theory, be it in the orthodox or the
Bohmian form. However, this objection has the merit of highlighting how import-
ant and necessary finding an adequate Bohmian version of quantum field theory is.
We have already mentioned proposals in this direction, involving a stochastic
dynamics according to which particles can be created and destroyed (Dürr, Gold-
stein, Tumulka, and Zanghì 2004).
An objection in some way connected to the previous one is that Bohmian
mechanics cannot be made invariant under Lorentz transformations, with what,
presumably, it is meant that it is not possible to find any Bohmian theory – a theory
that could be considered as a natural extension of Bohmian mechanics – that is
invariant under Lorentz transformations. If it were correct, this objection should be
seriously considered. However, it is not supported by any argument that makes
plausible the alleged impossibility of finding an extension of the Bohmian mech-
anics invariant under Lorentz transformations. The reason for this widespread
belief is the manifest nonlocality of Bohmian mechanics. But, as Bell has shown,
nonlocality is a fact established by the experiments.
Moreover, as regards the equally widespread belief that conventional quantum
theories would have no difficulty in incorporating Einstein’s relativity, whereas
Bohmian mechanics would have it, there is, even in this case, much less than meets
the eye. One should always bear in mind that the empirical content of conventional
quantum mechanics is based on (1) the unitary evolution of the state vector (or the
equivalent unitary evolution in the Heisenberg representation) and (2) the collapse
or reduction of the wave function (or any other equivalent artifact that allows the
Quantum Physics Grounded on Bohmian Mechanics 91
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5
Ontology of the Wave Function and the
Many-Worlds Interpretation
lev vaidman
5.1 Introduction
Quantum theory is about a century old, but as the existence of this volume shows, we
are far from consensus about its interpretation. Science does not develop in a straight
line. For a decade, quantum theory had no real basis, only phenomenological
equations found by Bohr, who made many of us believe until today that quantum
mechanics cannot be understood. The relativistic generalizations of the Schrödinger
equation, however, provide a complete, elegant physical theory that is fully consist-
ent with experimental data with precision of up to 10 significant digits. The situation
today is much better than at the time of Lord Kelvin’s speech in 1900, in which he
argued that physics is almost finished except for solving “two clouds,” which were
later to become the theory of relativity and quantum theory. This error, and the fact
that quantum equations describe several outcomes for quantum measurements
although we always see just one, are probably the main reasons why contemporary
physicists are reluctant to state that physics is close to being finished.
To deal with this second problem, we either have to add something to the wave
equation, but no proposal attractive enough to reach consensus has been found, or
to admit that what we see is only a tiny part of what is and that there are multiple
parallel worlds similar to ours. I find that this last option is the only reasonable one,
and I hope it will reach consensus in a foreseeable future.
93
94 Lev Vaidman
nature at a given instant, as well as the momentary positions of all things in the universe,
would be able to comprehend in one single formula the motions of the largest bodies as
well as the lightest atoms in the world, provided that its intellect were sufficiently powerful
to subject all data to analysis; to it nothing would be uncertain, the future as well as the past
would be present to its eyes. The perfection that the human mind has been able to give to
astronomy affords but a feeble outline of such an intelligence.
(Laplace 1820/1951)
I assume that the clouds Lord Kelvin talked about do not exist. Newton’s laws and
Maxwell’s equations are somewhat different, such that they provide a consistent
theory for microscopic particles moving on continuous trajectories, which, using
the methods of statistical mechanics, explains well all experimental data. It is also
an assumption about different experimental data, because we know that actually
observed data cannot be explained by a classical model.
In the model we consider, the three-dimensional space is given. There are
particles moving in well-defined trajectories and fields spread out in space. Par-
ticles create fields that propagate in space and change the motion of other particles
present in locations with nonvanishing fields. The laws of creation and propagation
of fields explain the existence of stable rigid objects and everything else (including
ourselves) that we experience with our senses. The behavior of objects is deter-
ministic; free will is an illusion.
Like in actual physics, our model can be presented in a different way. There is a
point in the configuration space of all particles and the configuration of fields
fulfilling some global equation of extremal action. The global laws provide the
same solution for trajectories. Both explanations are acceptable, but I feel that it is
the first presentation, with fields locally acting on particles moving on trajectories
in three-dimensional space, that is a more convincing explanation of the world.
The existence of a global mathematical representation is important, but it hides the
local causal story, which is what is considered as an explanation of motion of
micro systems as well as our behavior.
It seems to me that in a counterfactual universe with successful classical physics
as described earlier, there will be no philosophical controversy in how to describe
reality. Particle trajectories governed by local forces through fields in three dimen-
sions would be a clear consensus.
quantum mechanics. It is true that the community of physicists working in the field
of foundations of quantum mechanics has not reached any consensus over its
interpretation. Many of them (like me) are certain that their favorite view is a
satisfactory (or even an excellent) solution, but each separate group is a small
minority, so the message of the community as a whole is that currently there is no
good solution. There are many, also outside the foundations community, who feel
that we need the correct interpretation, but that it is different from all current
proposals. However, the majority of physicists really think that the problem does
not exist and that textbook quantum mechanics is satisfactory. It tells us that every
time we perform a quantum measurement there is a collapse of the quantum wave
function and that the collapsed wave function well describes all that we see around
us. Von Neumann proved that the tough question of when exactly collapse occurs
needs not to be answered, because wherever we put the cut between classical and
quantum, we will observe no contradiction with our experience. A vague statement
according to which all “macroscopic” objects are “well” localized provides a
satisfactory criterion.
Von Neumann collapse is ad hoc without any concrete mechanism. In physical
collapse theories, such as those of Pearle (1976), Ghirardi, Rimini, and Weber
(1986), Diosi (1987), and Penrose (1996), the collapsed wave function is not
completely identical to that of von Neumann, but it is very close; so proponents
of physical collapse theories also consider the collapsed wave function as a
satisfactory description of what we see. Apparently it is the promotion of the wave
function ontology in configuration space by Albert (2013) that led to strong
criticism. Maudlin (2013) understandably complained: How can a mathematical
object in high dimensional space represent our experience in three dimensions?
The key to answering this question is the understanding that our experience
supervenes on macroscopic objects. We do not directly experience the electron
wave function in the atoms of our body. Parts of our body and our neurons are
macroscopic bodies. We needed configuration space because of entanglement.
Whereas for describing classical particles we had a choice between one point in
3N dimensional space or N points in three-dimensional space, in the quantum case
with entanglement the second option does not exist, entanglement requires multi-
dimensional space. Complete descriptions of all particles separately do not provide
a complete description of entangled particles. But entanglement of quantum
systems does not exist for macroscopic systems. In von Neumann’s approach, it
is absent by definition, and in physical collapse theories, the mechanism removes
entanglement of macroscopic objects very quickly. Without entanglement, we can
describe all objects in three dimensions. To summarize, the analysis of the process
of obtaining experience from our senses when the universe is described by a
collapsed wave function of a textbook or by the collapsed wave function of a
96 Lev Vaidman
Bohmian particles describe a world that looks like the one we observe and so
also does the collapsed wave function. Because the wave functions of macroscopic
objects are well localized, the picture drawn by the expectation values of the
position vectors of all particles of these objects also provides a familiar picture.
The particles are very close to each other, so we do not observe the points. We
observe the smoothed picture of everyday objects: tables, cats, people, etc. – the
collapsed wave function provides a familiar picture of the world. The theory has a
tacit assumption, a postulate: Our everyday experience supervenes on the collapsed
wave function. Since the picture drawn by the particles and the picture we draw
based on our experience are so similar, we usually forget that we make an
assumption connecting the formalism with experience. The theory is supposed to
describe our experience. In this theory there is only one picture that looks like the
world we see, we assume that there is only one world, so naturally we connect
them. But the postulate is needed, because there can be other options: We can also
imagine the presence of Bohmian particles in a theory that makes collapses, and
attaches experience to Bohmian particles. It is a different rule, although it seems to
provide essentially the same experience.
The wave function of a world jψ i i must be of a particular type. Probably the most
informative definition is that this is the type of wave function that might appear in
the textbook as “collapsed wave function of the universe.” Without relying on a
textbook definition, this type of wave function can be defined by the property that
all macroscopic objects must be well localized. Both definitions are vague. We
usually require rigor and precision in our theories. However, it must be so when we
consider exact sciences. The many-worlds interpretation (MWI) has two clearly
separated parts: (i) a precise physical theory of evolution of the wave function of
Ontology of the Wave Function and the Many-Worlds Interpretation 99
the universe jΨiUNIVERSE and (ii) the connection of this universal wave function to
our experience(s) (see Vaidman 2002). In classical theory, in Bohmian theory, and
in quantum mechanics with collapse, the separation between the two parts of the
theory was not emphasized because the second part was very simple: The connec-
tion is natural and obvious in contrast to the MWI, where the second part is
significant. But I believe that the connection implicitly taken in single-world
theories can be directly adopted to the MWI, and vagueness of the splitting of
the worlds is much less problematic than the vagueness of the answer to the
question: When does collapse occur? Our experiences cannot be described in terms
of exact science. So, it is understandable and acceptable that the concepts
belonging to the second part of the theory are not rigorously defined. The wave
function of a world has the following form
! ! ! ! ! !
jψ i iWORLD ¼ ψ 1CM r 1CM φ1rel r1i r1j ψ 2CM r 2CM φ2rel r2i r 2j . . .
!CM M ! !
ψMCM r M φrel rMi rMj ΦREST (5.2)
!
It is a product state of wave functions ψ kCM r kCM of centers of mass of macro-
scopic
! objects, times entangled states of relative coordinates of these objects
!
φkrel rki rkj , times the wave function ΦREST of the remaining particles that are
not part of macroscopic objects. The terms “macroscopic” and “well localized”
might be chosen as more or less “fine grained,” so the decomposition Eq. (5.1) is
only approximately defined.
It seems to me that I can stop here. Textbook quantum theory is a well-
established and well-tested theory that well explains everything we see around
us. However, it includes the unphysical process of collapse, which makes it very
difficult to believe that it is true. I remove the collapse and use the same postulate
of connection to our experience. Now, the theory is a good physical theory
(deterministic, no action at a distance, and no ad hoc rules). The postulate of the
correspondence of the experience with the wave function of the type of the
universal wave function of a collapsed world makes the experience, by fiat,
identical to that of an experience of an observer living in a universe with collapsing
wave functions. However, I know that this picture is (still) not in a consensus.
I need to persuade the community that it is consistent.
that, for micro systems, there is no scattering between wave packets of the same
electron. Also note that we have no experimental evidence for the existence of
collapse. Testing MWI versus collapse theories will require a quantum experiment
up to a stage that the collapse proponents accept that collapse must have happened
and then undoing the experiment (Deutsch 1986, Vaidman 1998). If we get the
original state every time we perform this procedure, it is a proof that collapse did
not take place. Until today there is no sign of a collapse, but we are very, very far
from a decisive experiment. If collapse proponents will claim that collapse happens
only after we write the result of a quantum experiment in a notebook, it can never
be tested. Of course, it is the MWI that cannot be proved. If collapse exists at an
early stage, it will be observed (although it will not be easy to persuade opponents
that the explanation of the signal is not a failure of experimentalists to prevent
decoherence).
Slightly less naïve criticism is the problem of preferred basis. Mathematically,
one can decompose the wave function of the universe into a superposition of
orthogonal components, not just as in Eq. (5.1), but in many other ways that will
not provide a familiar world’s picture in every branch. So, critics might say that the
proposal is circular: I define by fiat what I want to explain. First, a simple definition
that is confirmed by observation sounds to me like a legitimate strategy. But there
is also a more specific answer. The basis of the decomposition is indeed preferred.
Until now I have not mentioned time evolution. Everything was considered at a
particular moment. But we cannot experience anything at zero time. We need an
order of 0.1 seconds to identify our experience. Thus, the world needs some finite
time to be defined. The world has to be stable, at least on the scale of seconds.
Locality of interactions in nature ensures that only the decomposition of wave
functions corresponding to well-localized macroscopic objects can be stable.
A quantum state describing the superposition of a macroscopic object in separate
locations with a particular phase evolves almost immediately into a mixture that
has a large component with a different phase. This obvious fact is analyzed in
numerous papers using the buzzword ‘decoherence.’
happened. In the MWI, such meaning does not exist. The event happened in this
world, but there is no alternative, it could not be otherwise. I want to say that the
traditional concepts of probability are not applicable in the framework of the MWI
when we consider the outcomes of quantum experiments to be performed. Every
time we perform a quantum experiment and it seems to us that a single result is
obtained, all possible outcomes are obtained, each in a different world. There is no
meaning to the question: In which world will I end up? In some sense I will be in
all. In no sense will I be in a particular one.
Still, we have to explain our experience of apparent random behavior and the
frequency pattern of the results of quantum experiments. My claim remains that
there is no difference between my experience if I live in one of the MWI worlds
and my experience if I live in the only world of the universe with collapses on
every measurement. How do we reconcile the difference between the existence of a
probability concept and identity of experience? To avoid that difficulty, there are
proposals to introduce uncertainty in the MWI and provide the meaning for the
probability that I will end up in a world with a particular outcome (Saunders and
Wallace 2008). In my view, adding uncertainty to the theory spoils it. The wave
function of the universe is supposed to be the whole physical ontology, and it does
not have any pointer moving from one world to another.
But do we really have a problem here? The fact that there is no meaning for the
probability of the result of a future measurement does not contradict the claim of
identical experiences. The standard approach to probability is to consider events
that will happen, but testing probability claims relies on records of experiments in
the past – frequencies of the outcomes of repeated identical measurements per-
formed in the past. So even in the framework of collapse theory, probability
assignments are confirmed or refuted by our experiences in the past. Thus, the
difficulty of the MWI to introduce the concept of probability for future outcomes is
not relevant.
Assignment of probability for future experiment relies on an additional assump-
tion, even in the framework of a collapse theory. We assume that nature will not
change its laws. Another way to make predictions about future measurement is to
assume that, after performing the measurement, the frequency of the measurement
results fits our probability assignment. This approach may also be applied in the
framework of the MWI. We expect that within the world, the frequencies of the
results of past measurements will correspond to the probability assignment.
The justification of this principle for our betting example is as follows. Every
descendant will have a genuine probability concept and would like to have a bet
according to a particular probability assignment. Since the descendants will get the
reward of the bet, the experimentalist, naturally caring for his or her descendants,
Ontology of the Wave Function and the Many-Worlds Interpretation 103
has a rational reason to put the bet for the results of the experiment. Tappenden
(2011) suggested (and I think it is a reasonable approach) that there is no need to
perform the complicated procedure with the sleeping pill. It is enough that one can
imagine performing such a procedure to justify the betting assignment of the
experimentalist who believes in the MWI. So, identical experiences lead to identi-
cal behavior, although the argumentation is different, which is not surprising in
view of different world views.
Putting aside the correctness of the proofs of the Born Rule, I argue that none of
them can be considered as unconditioned on any assumption. For the Born Rule in
the framework of collapse theories, we need an additional postulate about collapse.
It is not part of the standard formalism. A priori it need not follow any laws of
standard physical theory. We need to postulate some assumption – in my proof it is
the impossibility of superluminal signaling. In the framework of the MWI we also
need an assumption. The physics part, the evolution of the wave function of the
universe, has to be supplemented by some law connecting it to our experience.
Here, the assumption may be considered natural and minimal: Everything, includ-
ing our experiences, supervenes on the wave function of the universe. Then,
physical laws governing the evolution of the wave function are relevant to our
experience, too; so we might claim that no additional assumption was made. Still,
in all cases there is a tacit assumption that probability (or the illusion of probabil-
ity) depends on the wave function.
5.11 Conclusions
Today’s physics is quantum theory and it enjoys unprecedented success
explaining all observed phenomena. There are questions that do not have good
answers (yet). Quantum gravity, dark matter, dark energy. . . it might happen
that resolving these questions will require new revolutionary ideas. However,
quantum mechanics apparently will remain the theory explaining electromag-
netic interactions – the interactions that are responsible for almost everything
we see in everyday life. Unphysical features of collapse are the main reasons for
doubts that this is the final theory of nature. But actually, there is no evidence
for collapse. Apparently, it is just the difficult philosophical consequences of no
collapse that prevents consensus about quantum theory without collapse and
about the existence of multiple worlds. It took time before people were ready to
accept that the Earth is not the center of the universe. We also need time to
accept that we are not unique and that there are many similar copies of us. We
need time to establish the connection between the well-established mathemat-
ical part of the theory and our experience. It is an unusual situation, which we
did not encounter in old physical theories. Philosophers should play an import-
ant role in this project because it requires a dramatic change in our world view.
Observing the rapidly increasing number of publications related to the MWI in
the philosophical literature makes me optimistic. I am not sure that the large
effort to find a derivation of the Born Rule is justified (I doubt that the MWI has
significant advantage here), but this activity leads to accepting the legitimacy of
the MWI by physicists, and I believe that its advantage as a physical theory will
bring it to consensus.
Ontology of the Wave Function and the Many-Worlds Interpretation 105
Acknowledgments
I benefited greatly from numerous discussions with Kelvin McQueen, David
Albert, and the participants of the workshop Identity, indistinguishability and
non-locality in quantum physics (Buenos Aires, June 2017). This work has been
supported in part by the Israel Science Foundation Grant No. 1311/14.
References
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106 Lev Vaidman
6.1 Introduction
In the standard approach to quantum mechanics, there is no way to compute the
probability for expressions involving properties at different times. These probabil-
ities can be useful to relate a property of a microscopic system, before the
measurement process, to the value of the pointer variable of the macroscopic
apparatus after the measurement. Moreover, in the double-slit experiment, it is
important to have a suitable language to describe through which slit the particle
detected on a photographic plate has passed.
The theory of consistent histories has been introduced by Griffiths (1984),
Omnès (1988), and Gell-Mann and Hartle (1990), defining the notion of history
as a sequence of properties at different times. The probability for a history was
defined in this theory by an expression motivated by the path integral formalism,
but with no direct relation to the usual Born Rule. For a valid description of a
quantum system, the histories with well-defined probabilities should belong to a
family satisfying a state-dependent consistency condition.
In this theory, measurement is considered as a quantum interaction between the
measured microscopic system and the measuring macroscopic apparatus, and there
is no collapse postulate. Thus, the theory of consistent histories appeared to some
as a strong candidate for the realization of a quantum theory in which the act of
measurement would not have the distinguished role assigned in ordinary quantum
mechanics.
However, an important problem of this theory is that it does not provide us with
a single family of consistent histories, and the choice of different families may give
different descriptions for the time evolution of the same physical system. More-
over, different families of consistent histories can provide the prediction or the
retrodiction of contrary properties. If this is the case, it seems that the future or the
107
108 Marcelo Losada, Leonardo Vanni, and Roberto Laura
past of the quantum system could depend on the choice of the universe of discourse
(see Dowker and Kent 1996, Laloë 2001, Okon and Sudarsky 2014).
In this chapter we present a summary of our formalism of generalized contexts
for quantum histories (Laura and Vanni 2009, Losada, Vanni, and Laura 2013), in
which the ordinary contexts of properties for each different time should satisfy
compatibility conditions given by commutation relations in the Heisenberg
representation. A family of histories satisfying these compatibility conditions is
organized in a distributive lattice having well-defined probabilities obtained by a
natural generalization of the Born Rule of ordinary quantum mechanics.
In Section 6.2, our formalism is introduced for the case of histories in classical
mechanics. In Section 6.3, the formalism of generalized contexts for quantum
histories is presented in Section 6.4, it is applied to quantum measurements and to
the description of the double-slit experiment, with and without measurement
apparatuses. The conclusions are presented in Section 6.5.
in a single, more refined one. Let us consider two sets of atomic properties, pj
represented by the sets C j , (j 2 σ), and pμ , represented by the sets C μ and (μ 2 σ 0 ),
generating two different classical contexts. A more refined partition of the phase
space is obtained with the sets C j \ Cμ , representing a new set of atomic properties.
This new set generates a classical context containing the two previous ones. As an
important consequence, there is no restriction for the properties that can be
included in a classical context. This is not the case for quantum properties, as we
will discuss in Section 6.3.
This last equation can be used to obtain the probability Pr t ðpÞ of property p at time
t given in Eq. (6.1) in terms of the probability density at a reference time t0 . By
considering Eq. (6.2) with t 0 equal to an arbitrary fixed time t 0 in Eq. (6.1), we
obtain
ð ð
Pr t ðpÞ ¼ ρt ðxÞdx ¼ ρt0 ðxÞdx: (6.3)
Cp Cp, 0 St0 t Cp
110 Marcelo Losada, Leonardo Vanni, and Roberto Laura
We notice that the probability for the property p at time t can be expressed in two
different forms: either with a time-dependent probability density ρt ðxÞ together
with a time-independent set C p representing the property p, or with a time-
independent density ρt0 ðxÞ together with a time-dependent set Cp, 0 St0 t C p repre-
senting the property p. By anticipating what is found in quantum mechanics, we
will use the terms Schrödinger and Heisenberg representations to name the first
and the second forms of expressing the same probability for property p at time t.
where C 1 and C2 (C 1, 0 and C 2, 0 ) are the subsets of the phase space corresponding
to the properties p1 and p2 in the Schrödinger (Heisenberg) representation.
The previous equations strongly suggest the following definitions for the prob-
abilities corresponding to the conjunction and the disjunction of the properties p1 at
time t 1 and p2 at time t 2 :
ð
Pr½ðp1 , t 1 Þ∧ðp2 , t 2 Þ ρt0 ðxÞdx, (6.5)
C 1, 0 \ C2, 0
ð
Pr½ðp1 , t 1 Þ∨ðp2 , t 2 Þ ρt0 ðxÞdx: (6.6)
C 1, 0 [ C 2, 0
The formalism presented for the case of two times can be easily generalized to
include histories for sequences of n times.
Generalized Contexts for Quantum Histories 113
and satisfy
the compatibility
conditions when translated to the common time t 1 ,
i.e., U 1 Πaj U; Πqi ¼ 0. Therefore, the generalized context formalism allows
computing the conditional probability
Pr ðqi ; t 1 Þ∧ aj ; t 2 hψ j U 1 Πaj UΠqi j ψ 1 i
Pr qi ; t 1 jaj ; t 2 ¼ ¼ 1 ¼ δij : (6.12)
Pr aj ; t 2 hψ 1 j U 1 Πaj U j ψ 1 i
For the composite system prepared in the state j ψ 1 i ¼j φ1 i j a0 i, this result can be
interpreted by saying that if the apparatus’ pointer variable has the value aj after the
measurement, the system S had the property Q ¼ qj before the measurement. More
details of the application of this formalism to the logic of quantum measurements
can be found in our previous works (Vanni and Laura 2013, Losada, Vanni, and
Laura 2016).
to describe this process is the Hilbert space of the particle (H ¼ H particle ). The
projectors representing the particle located in each slit at time t 1 are
ð ð
Πt 1
u
d rjrihrj,
3
Πt1
d
d3 rjr ihrj , (6.13)
Vu Vd
where V u (V d ) is the volume of the upper (lower) slit, and jr i is a generalized
eigenvector of the position operator of the particle with generalized eigenvalue r .
For the later time t 2 , the projectors corresponding to the particle in small regions of
the vertical zone to the right of the double slit are
ð
Πt 2
n
d 3 rjrihrj, (6.14)
Vn
where V n is the volume of the small region of the vertical zone labelled by the
index n. We proved that the properties represented by the projectors Eq. (6.13) and
Eq. (6.14), translated to the common time t 1 , are represented by noncommuting
projectors, i.e.,
h i h i
Πut1 ; U 1 Πnt2 U 6¼ 0, Πdt1 ; U 1 Πnt2 U 6¼ 0, (6.15)
where U ¼ U ðt 2 ; t 1 Þ ¼ eiH 0 ðt2 t1 Þ=ℏ is the unitary evolution generated by the free-
particle Hamiltonian H 0 ¼ p2 =2m. Therefore, our formalism shows the well-
known fact that it is not possible to provide a description of the quantum process
suitable to specify through which slit the particle passed before reaching a region
of the vertical zone.
We also considered a modified double-slit experiment with an ideal measure-
ment apparatus A located in the slits zone, interacting with the particle during the
short time interval ½t 1 ; t 1 þ Δ1 , and with its pointer variable indicating au (ad ) if
the particle is detected passing through the upper (lower) slit. A second ideal
measurement apparatus B is located in the vertical zone to the right of the double
slit, interacting with the particle in the short time interval ½t 2 ; t 2 þ Δ2 , and with a
pointer variable indicating bn if the particle is detected in the small zone labelled by
the index n of the vertical zone. The Hilbert space for the description of this
process is the tensor product of the Hilbert space of the particle and the two Hilbert
spaces of the detectors, i.e., H ¼ H particle ⊗H A ⊗H B . The unitary time evolution is
assumed to be dominated by the interaction between the particle and apparatus A in
the short time interval ½t 1 ; t 1 þ Δ1 , by the free evolution in the time interval
½t1 þ Δ1 ; t 2 , and by the interaction of the particle and apparatus B in the time
interval ½t 2 ; t 2 þ Δ2 .
The possible pointer indications of the apparatus A at time t 1 þ Δ1 are repre-
sented by the projectors
116 Marcelo Losada, Leonardo Vanni, and Roberto Laura
We also proved that the properties corresponding to the projectors Eq. (6.16) and
Eq. (6.17), translated to a common time, are represented by commuting projectors.
Therefore, within the generalized-context formalism there is a generalized context
for the composite system’s history that includes the fact that the particle is
measured to pass through a definite slit at a certain time and the fact that the
particle is measured in a definite region of the vertical plane at a later time. The
corresponding conditional probabilities give the expected noninterference pattern.
6.5 Conclusions
We have presented our formalism of generalized contexts for quantum histories. It
was successfully applied to describe the logic of quantum measurements (Vanni
and Laura 2013, Losada et al. 2016) and the results of the double-slit experiment
with and without measurement apparatuses (Losada et al. 2013). It was also
suitable to give a discussion of the decay process (Losada and Laura 2013) and
to provide a deduction of the complementarity principle for the case of the Mach-
Zehnder interferometer (Vanni and Laura 2010).
The compatibility conditions of our formalism impose stronger conditions on
the allowed families of histories than the conditions imposed by the theory of
consistent histories (Losada and Laura 2014a). Therefore, the number of universes
of discourse (families of histories) allowed by our formalism is reduced. Two
important consequences of our formalism are that different families of histories
would not give retrodictions or predictions of contrary properties (Losada and
Laura 2014b), and that any allowed family of histories is organized in a distributive
and orthocomplemented lattice.
However, our formalism is not in position to provide a full interpretation of
quantum mechanics. There are no different allowed families of histories predicting
or retrodicting contrary properties, but there is still the freedom of choice of
different generalized contexts. The formalism in itself gives no indication about
which family should be privileged in a description of the time evolution of a
system. It seems that stronger conditions should be added to our formalism in order
to satisfy a realist perspective. In order to endow this formalism with realist
interpretive content, it is necessary to associate it with a specific interpretation that
Generalized Contexts for Quantum Histories 117
Acknowledgments
This work was made possible through the support of Grant 57919 from the John
Templeton Foundation and Grant PICT-2014–2812 from the National Agency of
Scientific and Technological Promotion of Argentina.
References
Castagnino, M., Id Betan, R., Laura, R., Liotta, R. (2002). “Quantum decay processes and
Gamov states,” Journal of Physics A, 35: 6055–6074.
Dowker, F. and Kent, A. (1996). “On the consistent histories approach to quantum
mechanics,” Journal of Statistical Physics, 82: 1575–1646.
Gell-Mann, M. and Hartle, J. B. (1990). “Quantum mechanics in the light of quantum
cosmology,” pp. 425–458 in W. Zurek (ed.), Complexity, Entropy and the Physics of
Information. Reading: Addison-Wesley.
Griffiths, R. (1984). “Consistent histories and the interpretation of quantum mechanics,”
Journal of Statistical Physics, 36: 219–272.
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paradoxes, and theorems,” American Journal of Physics, 69: 655–701.
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continuous spectrum,” Physical Review E, 57: 3948–3961.
Laura, R. and Vanni, L. (2009). “Time translation of quantum properties,” Foundations of
Physics, 39: 160–173.
Losada, M. and Laura, R. (2013). “The formalism of generalized contexts and decay
processes,” International Journal of Theoretical Physics, 52: 1289–1299.
Losada, M. and Laura, L. (2014a). “Generalized contexts and consistent histories in
quantum mechanics,” Annals of Physics, 344: 263–274.
Losada, M. and Laura, R. (2014b). “Quantum histories without contrary inferences,”
Annals of Physics, 351: 418–425.
Losada, M. and Lombardi, O. (2018). “Histories in quantum mechanics: distinguishing
between formalism and interpretation,” European Journal for Philosophy of Science,
8: 367–394.
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in classical and quantum mechanics,” Physical Review A, 87: 052128.
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contexts formalism for quantum histories,” International Journal of Theoretical
Physics, 55: 817–824.
Okon, E. and Sudarsky, D. (2014). “On the consistency of the consistent histories approach
to quantum mechanics,” Foundations of Physics, 44: 19–33.
Omnès, R. (1988). “Logical reformulation of quantum mechanics. I. Foundations,” Jour-
nal of Statistical Physics, 53: 893–932.
Vanni, L. and Laura, R. (2010). “Deducción del principio de complementariedad en la
teoría cuántica,” Epistemología e Historia de la Ciencia. Selección de trabajos de las
XX Jornadas, 16: 647–656.
Vanni, L. and Laura, R. (2013). “The logic of quantum measurements,” International
Journal of Theoretical Physics, 52: 2386–2394.
Part II
Realism, Wave Function, and Primitive Ontology
7
What Is the Quantum Face of Realism?
james ladyman
121
122 James Ladyman
documents, Bohr often made realist claims to go alongside the much quoted anti-
realist ones. For example, in his Nobel Prize lecture Bohr expressed his belief in
atoms and our knowledge of their microscopic constituents. Similarly, presumably
most, if not all, the physicists who immediately applied quantum mechanics to
problems in chemistry and solid-state physics, believed in atoms and electrons. The
fact that many physicists learned to invoke Copenhagen as a way of avoiding
worrying about philosophy does not imply their adherence to any detailed positive
interpretation, and certainly not to anything but a very localized antirealism.
A much discussed survey of physicists, asking them to which interpretation of
quantum mechanics they subscribe, has results as follows for the most discussed
interpretations: 42% Copenhagen, 18% Everett, 0% de Broglie-Bohm, and 9%
Objective Collapse (see Schlosshauer, Kofler, and Zeilinger 2013). Sean Carroll
(2013) says that the lack of agreement makes this the most embarrassing graph in
physics. However, such lack of agreement is not unusual in the history of science.
(It would be interesting to know what they would have said if asked first whether
they accepted metaphysical and scientific realism as defined below.) The physicists
who expressed their support for the Copenhagen interpretation in the poll surely
would not deny that there is a supermassive black hole at the center of the Milky
Way. In this way, quantum orthodoxy sits alongside scientific realism, and it is
plausible that many of those who opted for Copenhagen did so because for many
practicing physicists the measurement problem is irrelevant to their working lives,
and their adherence to Copenhagen orthodoxy goes as far as instrumentalism about
wave functions and collapse when pressed (the idea of wave function collapse as a
real process has nothing to do with Bohr, but is largely due to von Neumann; see
Howard 1985). It is because of the confusion and conflation surrounding the idea
of the Copenhagen orthodoxy that pragmatic attitudes to the interpretation of
quantum mechanics, by which is meant solving the measurement problem, are
taken to be tantamount to idealism or other forms of antirealism about the world in
general and even physics in particular (there is more discussion below of the
different ideas associated with the Copenhagen interpretation).
To bring more precision to the discussion consider the following:
vague without an account of reference and truth, and much ink has been spilled
about these topics without much agreement, and different forms of scientific
realism have proliferated. However, even those who defend the strongest forms
of scientific realism are not required to think that all theoretical terms successfully
refer. For example, a scientific realist may think superstrings have not yet been
shown to exist. Structural realism arose from these discussions and various other
variants including, most relevantly for the present discussion, entity realism.
Nancy Cartwright and Ian Hacking, among others, argued for this view, and it is
the fairly mainstream position in the scientific realism debate that seems to clearly
take the side of the scientific realist where it counts most for scientific practice.
Entity realism. Unobservable entities that are interacted with in the laboratory
should be taken as real, even if the theories that describe them are not taken to
be true.
It should be noted that entity realists are skeptical about claims to completeness
and fundamentality (which Section 7.3 argues are particularly dubious in the case
of quantum mechanics). Entity realism is clearly compatible with an instrumental-
ist view of the wave function and the problem of collapse. The measurement
problem might be supposed to force the adherent of scientific realism to commit
to more, but it is argued that it does not in Section 7.3.
(2) No way was found in practice to distinguish in advance among physical
situations to which quantum physics assigns only probabilities, but which have
different observable outcomes. Paradigmatically, radioactive decay is still con-
sidered random for all intents and purposes in atomic and nuclear physics. John
Stuart Mill (1843) defined determinism as the claim that for any situation there is a
description of it, such that for any other situation satisfying the same description,
the same future will unfold. This may sound overly epistemic to current ears but it
is the way the notion of determinism is applied to the world in scientific practice.
Classical phenomena pass Mill’s test insofar as there is a level of precision of
initial conditions that makes it possible to specify in advance, for example, whether
a fair coin will land heads or tails. Effective classical deterministic behavior of
macroscopic bodies emerges somehow, even though quantum systems fail Mill’s
test as far as anybody has been able to determine. This is a remarkable fact when
one considers the exponential advances and growth in measurement technology
since the discovery of radioactivity more than a century ago. When physicists first
debated the uncertainty relation, they were interested in whether it was a constraint
on practice, and Einstein argued it was not (Bohr 1949). The hidden variables
theorists lost the scientific battle long ago in this sense. Despite the unsoundness of
the no-go theorems for any form of hidden variable, the idea of Bohr – that
124 James Ladyman
quantum mechanics was complete for all practical purposes – has only been
vindicated by subsequent developments.
De Broglie, and later Bohm, showed that the idea that the world is fundamen-
tally random is not logically required by NRMPQM. It is possible to attribute
underlying deterministic trajectories to quantum particles moving in accordance
with the evolution of the wave function. However, since such hidden variables
must be nonlocal in the strong sense, namely that a preferred frame is required in
which it is unambiguously true that the adjustment of a measurement setting
changes the trajectories of particles at spacelike separation, they require revision-
ary physics for no immediate practical use just because they are hidden. These are
unlikely to have been received well by those who had learned to accept relativity,
even if their possibility had not been deliberately or mistakenly ignored by most
physicists. Cushing (1998) speculates that Bell’s theorem could have become
known in the Twenties, and sees this in terms of people accepting that quantum
mechanics is nonlocal. This is highly tendentious, as discussed later, but even
granting it, he must admit that the hidden variables would not have been used in
practice for the problems people went on to solve in all the areas of physics to
which quantum mechanics was applied. He assumes that Dirac’s operator formal-
ism would still have been available to apply quantum mechanics to problems.
Scientific realist literature abounds with the call to take the practice of science
seriously and not to only consider theoretical possibilities. Any form of scientific
realism that does not take quantum randomness at face value is at least out of step
with the practice of science.
Cushing discusses the Forman thesis linking the repudiation of the law of
causality by German quantum theorists with the neoromantic disdain for science
in Weimar culture (see Forman 1971, 1984; for a critique, see Kragh 2002: chapter
10). However, note that Peirce, in the nineteenth century, rejected determinism and
accepted brute randomness in nature within his philosophy, which was based on
his appreciation and study of science (see Ladyman and Ross 2013). Similarly,
both Cushing and Forman offer no explanation of why British physicists such as
Dirac and Mott readily embraced the new quantum mechanics. More importantly,
there is an equivocation on causality in this context, for while Kant and others took
it to be the same thing as determinism in the sense of there being necessary and
sufficient antecedent events for everything that happens, in the twentieth century,
probabilistic causality became the norm as a result of the extension of causal
modeling to the behavioral and social sciences.
(3) Quantum mechanics was successfully applied in chemistry and solid state
physics in the first months and years of its existence. It was also more or less
immediately extended to relativistic physics and to field theory. All of its
What Is the Quantum Face of Realism? 125
seemingly impossible implications that have been tested have been confirmed. It is
applied throughout the rest of physics, and quantum statistical mechanics and
semiclassical domains have been discovered. However, it has not been successfully
applied to gravity itself (though quantum field theory [QFT] has been applied to the
effects of gravity in the form of Hawking radiation) and so is not a complete theory
of reality. In philosophical discussions of physics, people often consider models in
which there is only a spacetime and some fields or particles of some particle kind.
This generates the idea of the ontology of the theory, as if it purported to be an
account of the whole of reality rather than just an aspect of it. For example,
diffraction and Stern-Gerlach experiments on beams of particles or individual ones,
and EPR-type experiments on entangled systems, are a tiny proportion of the
applications of quantum physics to reality. Any viable form of quantum realism
must not conflate NRMPQM with quantum physics, and it must be apt for phenom-
ena other than familiar, simple experiments that are not representative of quantum
physics, though they are legitimately used to understand the conceptual and math-
ematical foundations of the theory and how it represents physical systems.
constructive empiricism (van Fraassen 1980), which is the most discussed alterna-
tive to realism. It is ironic then that the positive case for Bohmian mechanics
involves the combination of an insistence on a realist interpretation of quantum
mechanics based on the theoretical empirical equivalence of the formalism, with
the claim that the reasons why Bohmian mechanics was not accepted by the
scientific community are extrascientific. Hence, the aforementioned discussion of
the Forman thesis by Cushing, and the emphasis on how discussion of the
measurement problem, and the work of de Broglie, Bohm, and Everett was
supressed. (Bohm’s ideas were not well known until after the work of Bell, and
the current relative prominence of many-worlds interpretations is due to the revival
of Everett’s ideas by others. Saunders et al. 2010 includes discussion of all this and
a very comprehensive relevant citation.)
Realist responses to the underdetermination problem usually aim at rationally
reconstructing theory choice, for example, in the case of wave versus particle
theories of light, special relativity versus Lorentzian contraction, and so on. The
theoretical virtues of novel predictive success, non-ad hocness, explanatory power,
simplicity, coherence with background metaphysics, and so on, are discussed to
explain why mere empirical equivalence is not sufficient to make the community’s
choice arbitrary and why the chosen theory scientifically preferable. In this context,
historical theories of confirmation and the idea of progressive versus degenerating
research programs are much discussed following the work of Popper, Lakatos, and
Musgrave. Among the theoretical vices are the negation of each of the virtues just
described, such as ad hocness and ontological profligacy. Another theoretical vice
that is often discussed is parasitism, which is when a theory requires another theory
for its formulation. An extreme example is the theory that says “the world is as if
QED”; it generates no predictions at all without the input of QED. The wave and
particle theories of light were not like this. Ray optics fits well with the mechanics
of particles, and diffraction fits well with waves.
The main virtue of Bohm theory for many of its current defenders is that it offers
the best prospects for realism about the quantum world. This is because Bohm
theory posits definite values for all physical quantities at all times, eliminating the
apparent indeterminacy in the values of physical quantities prior to measurement.
The Bohmian mechanics usually discussed is such that all particles have well-
defined trajectories at all times and the evolution of all quantum systems is entirely
deterministic and causal. Thus particles can be individuated by their spatiotem-
poral properties (although of course these will not be accessible to experiment). It
is also virtuous in cohering with background ideas of causality and determinism,
and of the nature of particles. However, it is important to note that, as Harvey
Brown and others (1996) argue, the “particles” of Bohm theory are not those of
classical thought. The dynamics of the theory is such that the properties normally
What Is the Quantum Face of Realism? 127
associated with particles like mass, charge, and so on, are in fact, all inherent in the
quantum wave function and not in the particles. It seems that the particles only have
position. Apart from any worries we might have about the intelligibility of this
notion of particle, it seems that they have none of the features of classical particles
other than point position; hence, there would seem to be little referential continuity
for ‘particle’ available to the realist. Furthermore, if the trajectories are enough to
individuate particles in Bohm theory, what makes the difference between an “empty”
trajectory and an “occupied” one? Since none of the physical properties ascribed to
the particle inheres in points of the trajectory, giving content to the claim that there is
actually a “particle” there would seem to require some notion of the raw stuff of the
particle; in other words, haeccities seem to be needed to make the ontology of Bohm
theory intelligible after all. This is without even beginning the discussion of the
nature of the wave function in Bohm theory (Brown et al. 1996 also argue that the
action–reaction principle is violated by Bohm theory).
Bohm theory is often criticized for its alleged ad hocness and lack of simplicity
relative to the standard formalism. These complaints aside, more problematic is that it
is nonlocal in the strong sense that, if Bohm theory is correct, then physicists will
have to rewrite their textbooks so that a Newtonian or Galilean spacetime with a
global time coordinate underlies the Lorentz invariant phenomena of electromagnet-
ism. When such theories are proposed as alternatives to relativity, they are sometimes
linked to cosmic conspiracies because they propose far-fetched hidden machinations.
Similarly, instantaneous action at a distance as a physical process has been widely
regarded as unscientific since the beginnings of field theory in the mid-nineteenth
century, but Bohm theory makes it a pervasive feature of the world, and it requires
that the change of setting on the apparatus on one side has a physical effect on the
trajectory of the particle on the other side. Unsurprisingly, Bohmians always argue
that quantum mechanics requires action at a distance anyway, but this is not correct.
Bell’s theorem does not show that there is an objective causal asymmetry between the
two wings of the Aspect experiment, only that the attribution of possessed values, or
counterfactual definiteness, or some other condition, is incompatible with locality.
There is no proof of Bell’s theorem without what is often termed a “realist”
assumption of some kind, but note that none of these realist assumptions are required
by either metaphysical or scientific realism.
The so-called pessimistic meta-induction argument against scientific realism
would be reinforced if Bohm theory is adopted by most scientists for two reasons:
(a) It would support the arguments of those who argue that we cannot learn
metaphysical lessons from science, because the orthodoxy among physicists
would have been quite wrong about the world since at least 1935, insofar as it
has been widely held and taught that quantum phenomena are genuinely
128 James Ladyman
indeterministic – that electrons and other “particles” are really neither waves
nor particles, that matter has new properties like spin, and so on.
(b) The acceptance of relativity theory as a theory of the nature of space and time,
such that there is no absolute simultaneity or privileged coordinate system, and
such that the Lorentz invariance of Maxwell’s equations has been taken to be
not merely empirical but also reflective of fundamental symmetries in the
nature of reality, would have to be regarded as quite mistaken. The more cases
there are of the ontological interpretation of scientific theories being later
abandoned, the more compelling the meta-induction becomes.
According to van Fraassen, the big problem with scientific realism is that it adds
metaphysics to scientific theories for no empirical gain, and this seems to be true in
the case of Bohm theory. In this context we might ask: What has the interpretation
of quantum mechanics ever done for us? To which the answer is a lot, most
obviously, it has more or less directly given us Bell’s theorem (Bohm), quantum
computation and information processing (Deutsch), and weak values (Aharonov).
As with the theoretical virtues, there is no support for any one interpretation
from consideration of fecundity for physics, except perhaps for the Copenhagen
interpretation, insofar as the orthodoxy, such as it was and is, has undoubtedly
accompanied the most extraordinary scientific success. Quantum physics has been
outstandingly virtuous on all criteria other than that of cohering with background
metaphysics. For all the theory’s indeterminism, when it is taken to the relativistic
domain, it is as predictively accurate in practice as any determinist could reason-
ably demand. It does not tell us what the fundamental nature of being is, in terms of
something we can think of as objectively and separately located, with intrinsic
properties in space and time. However, it is not a complete theory of the world, as
discussed in the next section.
(3) There many cases in physics and science generally in which “more is
different.”
For all these reasons it seems that the measurement problem does not compel a
choice between Everett, Bohm, and dynamical collapse. Maudlin (2018) quotes
Lakatos, saying that Bohr and his associates brought about the “defeat of reason
within modern physics.” However, all the revisionist interpretations of NRMPQM
are ultimately parasitic, in practice as well as in theory, on the great empirical
success of standard quantum physics, which they have not yet matched, even in
principle and with the benefit of hindsight. They may say there is no collapse but
they all have some notion of “effective collapse.” In the two-slit experiment, no
matter what the mechanism for detecting which slit a particle went through,
determining the position always has the same effect on the determinacy of the
momentum in accordance with the uncertainty relation.
References
Bohr, N. (1949). “Discussions with Einstein on epistemological problems in atomic
physics,” pp. 200–241 in P. A. Schilpp, Albert Einstein: Philosopher-Scientist.
Evanston: The Library of Living Philosophers.
Brown, H., Elby, A., and Weingard, R. (1996). “Cause and effect in the pilot-wave
interpetation of quantum mechanics,” pp. 309–319 in J. T. Cushing, A. Fine, and
S. Goldstein (eds.), Bohmian Mechanics and Quantum Theory: An Appraisal.
Dordrecht: Kluwer Academic Publishers.
Carroll, S. (2013). “The most embarrassing graph in modern physics,” www.prepost
erousuniverse.com/blog/2013/01/17/the-most-embarrassing-graph-in-modern-physics/
Cushing, J. T. (1998). Philosophical Concepts in Physics: The Historical Relation between
Philosophy and Scientific Theories. New York: Cambridge University Press.
Daumer, M., Dürr, D., Goldstein, S., Maudlin, T., Tumulka, R., and Zanghì, N. (2006).
“The message of the quantum?,” pp. 129–132 in A. Bassi, D. Dürr, T. Weber, and
N. Zanghì (eds.), Quantum Mechanics: Are There Quantum Jumps? and On the
Present Status of Quantum Mechanics, AIP Conference Proceedings. College Park,
Maryland: American Institute of Physics.
Folse, H. J. (1985). The Philosophy of Niels Bohr: The Framework of Complementarity.
Amsterdam-Oxford: North-Holland.
Forman, P. (1971). “Weimar culture, causality, and quantum theory: Adaptation by
German physicists and mathematicians to a hostile environment,” Historical Studies
in the Physical Sciences, 3: 1–115.
Forman, P. (1984). “Kausalität, Anschaullichkeit, and Individualität, or how cultural
values prescribed the character and lessons ascribed to quantum mechanics,”
pp. 333–347 in N. Stehr and V. Meja (eds.), Society and Knowledge. New Bruns-
wick-London: Transaction Books.
Howard, D. (1985). “Einstein on locality and separability,” Studies in History and Phil-
osophy of Science, 16: 171–201.
Kragh, H. (2002). Quantum Generations: A History of Physics in the Twentieth Century.
Princeton: Princeton University Press.
Ladyman, J. and Ross, D. (2007). Every Thing Must Go. Metaphysics Naturalized. Oxford-
New York: Oxford University Press.
Ladyman, J. and Ross, D. (2013). “The world in the data,” pp. 108–150 in D. Ross,
J. Ladyman, and H. Kincaid (eds.), Scientific Metaphysics. Oxford: Oxford University
Press.
Maudlin, T. (2018). “The defeat of reason,” Boston Review, http://bostonreview.net/sci
ence-nature-philosophy-religion/tim-maudlin-defeat-reason
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Works of John Stuart Mill. Toronto: University of Toronto Press, London: Routledge
and Kegan Paul.
132 James Ladyman
Psillos, S. (1999). Scientific Realism: How Science Tracks Truth. London: Routledge.
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Quantum Theory, and Reality. Oxford: Oxford University Press.
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van Fraassen, B. (1980). Scientific Image. Oxford: Oxford University Press.
8
To Be a Realist about Quantum Theory
hans halvorson
8.1 Introduction
There is a story that some philosophers have been going around telling. It goes
something like this:
The pioneers of quantum mechanics – Bohr, Heisenberg, Dirac, et al. – simply abandoned hope
of providing a realist theory of the microworld. Instead, these physicists settled for a
calculational recipe, or statistical algorithm, for predicting the results of measurements. In
short, Bohr et al. held an antirealist or operationalist or instrumentalist view of quantum theory.
This kind of story can be very appealing. It is the age-old “good guys versus bad guys”
or “us versus them” motif. And those “ist” words make it easy to distinguish the good
guys from the bad, sort of like the white and black hats of the classic westerns.
The story is brought into clearer focus by talking about the quantum wave
function. What divides the realists from the antirealists, it is said, is their respective
attitudes toward the wave function: Antirealists treat it as “just a bookkeeping
device,” whereas realists believe it has “ontological status.” Witness the faux-
historical account of Roger Penrose:
It was part of the Copenhagen interpretation of quantum mechanics to take this latter
viewpoint, and according to various other schools of thought also, ψ is to be regarded as a
calculational convenience with no ontological status other than to be part of the state of
133
134 Hans Halvorson
mind of the experimenter or theoretician, so that the actual results of observation can be
probabilistically assessed.
(Penrose 2016: 198)
I suppose that Penrose can be forgiven for oversimplifying matters, as well as for
propagating the myth of the “Copenhagen interpretation” (see Howard 2004).
After all, there can be great value in simple fictional tales if they get readers
interested in the issues.
I also imagine that Sean Carroll is aiming to generate some heat – rather more
heat than light – when he poses the following dilemma about the wave function:
The simplest possibility is that the quantum wave function isn’t a bookkeeping device at
all . . .; the wave function simply represents reality directly.
(Carroll 2017: 167)
This seemingly simple dilemma – ontological status: yes or no, – is a fine device
for popular science writing, which should not demand too much from the reader.
But is it really the right place to locate a pivot point? Is the question, “Ought I to
commit ontologically to the wave function?” the right one to be asking?
Popular science writers are not the only ones to have located a fulcrum at this point.
In fact, some philosophers say that if you are a scientific realist, then you are logically
compelled to accept the Everett interpretation. I am thinking of this kind of argument:
If you’re a realist about quantum theory, then you must grant ontological status to the
quantum state. If you grant ontological status to the quantum state, and if quantum
mechanics is true, then unitary dynamics is universal. Under these conditions, realism
and unitary dynamics, you have two options: either you accept the completeness of
quantum theory, or you don’t. And if you accept the completeness of quantum theory,
then the Everett interpretation is true.
the only other serious alternative (to realists) is quantum state realism, the view that the
quantum state is physically real, changing in time according to the unitary equations and,
somehow, also in accordance with the measurement postulates.
(Saunders 2010: 4)
To Be a Realist about Quantum Theory 135
In short, if you are a good realist, then you will say that the quantum state is
physically real, and from there it is a short step to the Everett interpretation.
There is something strange about this sort of argument. The notion of “realism”
is doing so much of the work – and yet, nobody has told us what it means. How
could the “if realism, then Everett” argument be valid when “realism” has not been
defined clearly? And how could the argument be convincing when realism has not
been motivated, except through its undeniable emotional appeal?
In this chapter, I will take a closer look at the distinction between realist and
antirealist views of the quantum state. I will argue that this binary classification
should be reconceived as a continuum of different views about which properties of
the quantum state are representationally significant. What is more, the extreme
cases – all or none – are simply absurd and should be rejected by all parties. In
other words, no sane person should advocate extreme realism or antirealism about
the quantum state. And if we focus on the reasonable views, it is no longer clear
who counts as a realist and who counts as an antirealist. Among those taking a
more reasonable intermediate view, we find figures such as Bohr and Carnap – in
stark opposition to the stories we have been told.
8.2 Extremists
Suppose that you were asked to list historical figures on two sheets of paper: On
the first sheet, you are supposed to list realists (about the quantum state), and on
the second sheet you are supposed to list antirealists (about the quantum
state). Suppose that you are asked to sort through all of the big names of quantum
theory – Bohr, Heisenberg, Dirac, Bohm, Everett, etc.
I imagine that this task would be difficult, and the outcome might be controver-
sial. For almost none of these people ever explicitly said, “I am a realist” or “I am
an antirealist” or “the wave function has ontological status” or anything like that.
You would have to do quite a bit of interpretative work before you could justify
assigning a person to one of the lists. You would have to assess that person’s
attitude toward the quantum state by studying their behavior and utterances with
respect to it. For example, if person X makes free use of the collapse postulate, with
no proposed physical mechanism, then you might surmise that X is either a mind-
body dualist, or an operationalist about the quantum state, or both. In other words,
an operationalist stance might serve as the best explanation for X’s utterances and
behavior.
The task of sorting people into realist and antirealist would be simpler
for contemporary figures, who seem happy to embrace one of these two labels. For
example, Sean Carroll and Lev Vaidman will tell you, with great passion, that the
wave function is just as real as – in fact, more real than! – a rock, or a tree, or your
spouse. In contrast, Carlo Rovelli speaks of the wave function as Laplace spoke of
136 Hans Halvorson
God: Je n’avais pas besoin de cette hypothèse-là. These are just a few examples
among the many philosophers and physicists who have openly labeled themselves
as realist or antirealist about the quantum state. Self-identified state realists include
Esfeld, Goldstein, Ney, Saunders, Wallace, Zanghì, etc. Self-identified state anti-
realists include Bub, Fuchs, Healey, Peres, etc. The battle lines have been clearly
drawn, but what is at stake?
The right-wing extremists say: Quantum wave functions are things. That view is
silly. The left-wing extremists say: Quantum wave functions are just bookkeeping
devices. That view is just as silly.
heads: The wave function is a thing with a definite shape! I wager that such theories
are plausible only to the extent that it is unclear what they are really saying.
For starters, in quantum theory, the primary theoretical role of the wave function
ψ is as a state. The ψ-field theorists ask us to change our point of view. Instead of
thinking of ψ as a state, we are to think of ψ as a field configuration. There are
numerous problems with this proposal.
In classical physical theories, the word “state” is shorthand for “a maximally
consistent list of properties that could be possessed by the system simultaneously,”
or equivalently, “an assignment of properties to objects.” In that case, there are two
possible things we could mean by the sentence “the state σ exists.” First, we could
mean that the list of properties exists. But this list is an abstract mathematical
object, which would exist whether or not the corresponding theory is true. So, in
this first sense, “σ exists” is not interesting from the point of view of physics.
Second, we could use “σ exists” as an obscure shorthand for “σ is the actual state,”
which, in turn, is shorthand for saying that certain other objects have certain
properties. Thus, in this second case, “σ exists” is cashed out in terms that do
not refer to σ at all. In philosophers’ lingo, “σ exists” is grounded in facts about
other objects, and so is not really about σ at all.
Now, the defender of quantum state realism might simply say: “That was
classical physics. In quantum physics, the state takes on a new role.” I certainly
accept that quantum physics changes some of the ways we talk about the physical
world. But I am not so sure that it makes sense to reify states. According to the
normal senses of “object” and “state,” we affirm that objects can be in states. Thus,
if states are objects, then states themselves can be in states. But then, to be
consistent, we should reify the states of those states, and these new states will
have their own states, ad infinitum. In short, if you run roughshod over the
grammatical rules governing the word “state,” then you can expect some strange
results.
To continue that line of thought, we assume that things can be counted. In other
words, it makes sense to ask: How many things are there? But then, if states were
things, it would make sense to ask: How many states are there? But now I am
completely puzzled. According to quantum theory, the universe has an infinite
number of potential states, but only one actual state. What in the world would
explain the absence of all the intermediate possibilities? Why couldn’t there have
been 17 states? And what’s more, why do physicists never raise as an empirical
question: How many states are there? The reason is simple: Physicists do not treat
states as they do things, not even in the extended sense where fields also count as
things.
I hope that by this stage you are at least partially convinced that it does violence
to the logic of physical theories to talk about states as if they were things. But then
138 Hans Halvorson
you should agree that the role of a wave function is not to denote an object.
Moreover, if ψ does not denote a physical object, then the properties of ψ do not
directly represent the properties of a physical object. Granted, we should be careful
with this latter claim. Even in classical physics, the properties of a state can
represent, albeit indirectly, the properties of a physical object. For example, for a
classical particle, “being in subset Δ of statespace” is a property of states that
represents a corresponding property of the relevant particle. Nonetheless, there are
two different types of things here – the particle, which is a concrete physical object,
and its state, which is an abstract mathematical object. The latter tells us about the
former, but should not be conflated with it.
In classical theories, there is also a sharp distinction between instantaneous
configurations and states. If a configuration is represented by a point in the
manifold M, then a state is represented by a point in the cotangent bundle
T ∗ M. In many scenarios, T ∗ M looks like a Cartesian product M M, where
the first coordinate gives instantaneous configuration, and the second compon-
ent gives momentum. In every case, there is a projection mapping
π : T ∗ M ! M, and the preimage of any particular configuration q 2 M is an
infinite subset of T ∗ M. But now, if ψ is both a state and a field configuration,
then it is unclear where it lives. Does ψ live in the space M of configurations,
or does it live in the space T ∗ M of states? How can it do both jobs at the
same time?
These considerations show that the ψ-field view stretches the logic of classical
physics beyond the breaking point. To treat ψ as representing a field configuration
is to disregard its primary theoretical role as a state. Or, at the very least, to treat it
thus would obscure the difference, central to classical theories, between configur-
ations and states.
If that is not enough trouble for ψ-field views, we can also ask them to give an
account of the properties that are possessed by this thing, the ψ-field. Recall that in
a classical theory with statespace S, properties are typically represented by subsets
of statespace S. (We might require that these subsets be measurable or something
like that. But that point will not matter in this discussion.) Then, we say that the
system has property E S just in case it is in state σ 2 E.
Now, ψ-field theorists would like us to think of quantum theory on the model of
a classical field theory. In this case, the statespace would be the space C ∞ ðX Þ of
smooth complex-valued solutions to some field equation, and subsets of C ∞ ðX Þ
would represent properties that the system can possess. (Let us ignore here the fact
that classical field theories typically use the space of real-value functions.) For
example, for any field state f 2 C ∞ ðX Þ, the singleton set ff g represents the
property of being in state f , and its complement C ∞ ðX Þ∖ff g represents the property
of not being in state f .
To Be a Realist about Quantum Theory 139
Of course, ψ-field theorists are too clever to fall into the trap of carrying
classical inference rules into the quantum domain. Although they purport to view
the ψ-field classically, they stop short when it comes to reasoning about it. For the
purposes of reasoning and making predictions, they turn to the Hilbert space
formalism to guide them. Thus, we might summarize the attitude of these ψ-field
views in a phrase: You can look at the world from the god’s-eye point of view; just
don’t reason about it as god would.
Pragmatists agree with QBists [quantum Bayesians] that quantum theory should not be
thought to offer a description or representation of physical reality: in particular, to ascribe a
quantum state is not to describe physical reality.
(Healey 2016: on line, emphasis added)
implications that come along with this particular definition of “describes,” they are
forced to say that the quantum state does not describe at all. Thus, Healey and
Rovelli lay themselves open to the charge of antirealism – which, of course, carries
highly negative connotations. To be an antirealist implies a sort of failure of
courage – it implies a sort of retreat. Ergo, Healey and Rovelli are seen as making
less bold assertions about reality than their realist counterparts are making.
In fact, Wallace (2018b) locates the crucial divide between “representational” and
“non-representational” views of the quantum state. Thus the shift is signaled from
the material mode of speech (does the state exist?) to the formal mode of speech
(does the state represent?). In particular, Wallace and Timpson claim that realism
involves commitment to both literal and direct representation. Thus, Carroll utters
the shibboleth when he says, “the wave function simply represents reality directly”
(Carroll 2017: 167). But what work is the word “directly” doing here? I am led to
think that the task of representing must be a bit like getting to work, where you
have to take the right turns in order to follow the most direct route. So what are the
instructions for following the direct route to representation?
When a person says that Y represents X, then that typically signals that the
person endorses some inferences of the form
(†): If Y has property ϕ, then X has property ϕ0 ,
where ϕ ↦ ϕ0 is some particular association of properties (the details of which need
not detain us). Let us call (†) a property transfer rule. For example, if I say that a
certain map represents Buenos Aires, then I mean that some facts about Buenos
Aires can be inferred from facts about the map.
What then is the force of insisting that Y does not merely represent X, but that it
represents X directly? I suspect that the word “directly” is supposed to signal
endorsement of quite liberal use of property transfer rules. But just how liberal?
The key question to keep in mind is: Which specification of permitted property-
transfer inferences corresponds most closely to the notion of “direct representa-
tion” that is favored by realists such as Carroll, Wallace, Saunders, and Timpson?
To Be a Realist about Quantum Theory 143
When they say that “ψ directly represents reality,” what exactly are they saying
about the relation between ψ and the world?
Consider first the proposal:
(DR1) Y directly represents X just in case every property of Y is also a property of X.
This proposal is logically consistent, but also absurd. One of the properties that X
has is being identical to X. Thus, according to DR1, if Y directly represents X then
Y ¼ X. Could Wallace and Timpson possible intend this? Does Carroll mean to
say our universe is a subset of R3n C? If so, then scientific realism is truly a
radical point of view. The wave function is an abstract mathematical object. Thus,
if the universe is a wave function, then the universe is an abstract mathematical
object. Perhaps mathematicians will applaud this conclusion, because then pure
mathematics tells us everything there is to be known about the universe.
I suspect that the realists do not mean their direct representation claim in the
sense of DR1. Let us try a more reasonable proposal.
(DR2) Y directly represents X just in case each mathematical property of Y corres-
ponds to some physical property of X.
Here we need some precise account of the “mathematical properties” of Y.
According to standard set-theoretic foundations of mathematics, the mathematical
properties of Y are precisely those properties that can be described in the language
of Zermelo-Fraenkel (ZF) set theory. Thus, for example, the mathematical proper-
ties of Y would include its size (cardinality). In contrast, arbitrary predicates in
natural language do not pick out mathematical properties of Y. For example, “is an
abstract object” cannot be articulated in ZF set theory, and so would not count as a
mathematical property of Y. Thus, DR2 does not say that “anything goes” in terms
of the representationally significant properties of Y.
Even so, DR2 is still implausibly profligate in the number of representationally
significant properties it assigns to the wave function. In particular, for each
definable name c in ZF set theory, there is a definable predicate Θc given by
Θc ðSÞ $ c 2 S:
Among these definable set names, we have ∅, f∅g, and so on. Now, a wave
function is a function ψ : A ! B, with domain set d 0 f ¼ A and codomain set
d1 f ¼ B. Thus, for any definable name c, it makes sense to ask whether
Θc ðd0 ψ Þ, i.e., whether c is contained in the domain of ψ.
Imagine now the following scenario. Two physicists, Jack and Jill, are arguing
about whose wave function is a better representation of the universe. The funny
thing is, Jack and Jill’s wave functions are both Gaussians, centered on 0, and with
the same standard deviation. If you ask Jack to draw a picture of his wave function,
then he draws a Gaussian centered at 0. If you ask Jill to draw a picture of her wave
144 Hans Halvorson
function, then she also draws a Gaussian centered at 0. They agree that this picture
is a correct representation of their respective wave functions. They also agree that
their wave functions are written in the configuration space basis, and that the origin
0 represents the same point in the universe. It seems that there is nothing left for
them to disagree about.
And yet, Jack and Jill insist that their wave functions cannot both be correct.
According to Jack, the correct wave function ψ has the property that Θ∅ ðd 0 ψ Þ, that
is, the empty set is an element of the domain of the wave function. According to
Jill, the correct wave function ψ 0 does not have that property. They both believe
that Θ∅ corresponds to a genuine physical property. Jack asserts that this property
is instantiated, and Jill asserts that it is not.
Jack and Jill would fail their quantum mechanics course. They do not under-
stand how the theory works. In using the formalism of quantum theory to represent
reality, we do not care about these fine-grained set theoretic differences. If
two wave functions have the same shape, then we consider them to be the same.
If two wave functions can be described via the same equation, then we take them to
be identical. But what is this notion of same shape that we are using here? How can
we tell when two wave functions are the same, at least for the purpose of doing
physics?
At this point, we might want to lay down the ace card of recent philosophy of
science: the notion of isomorphism. Can’t we just say that two wave functions are
representationally equivalent just in case they are isomorphic? In this case, we
could then propose the following criterion for direct representation:
(DR3) Y directly represents X just in case Y and X are isomorphic.
This proposal sounds a lot more plausible than the previous two – especially
because the word “isomorphism” is simultaneously precise (within certain fixed
contexts) and flexible (since it means different things in different contexts). But
that is precisely the problem with DR3: the phrase “Y is isomorphic to X” is no
better defined than the phrase “Y directly represents X.”
In mathematics, isomorphism is a category-relative concept. If you hand me two
mathematical objects and ask, “Are they isomorphic?” then I should reply by asking
“Which category do they belong to?” For example, two mathematical objects can be
isomorphic qua groups, but nonisomorphic qua topological spaces. Thus, it makes no
sense to say that a mathematical object is isomorphic to the world tout court. In order
to make sense, we would first have to specify a relevant type (or category) of
mathematical objects. For example, one might say that the world is isomorphic to a
topological space Y, as shorthand for saying that the world has topological structure,
and is in this sense isomorphic to Y. But if you give me a concrete mathematical object
A and say that the world is isomorphic to A, then I have no idea what you are saying.
To Be a Realist about Quantum Theory 145
So, if we want to say that the world is isomorphic to a wave function ψ, then we
need to say what category of mathematical objects we take ψ to belong to. And that
is not going to be easy, for ψ is not a group, or a topological space, or a
differentiable manifold, or any other of the standard types of mathematical struc-
ture. There is no category of wave functions; and, there is no nontrivial notion of
isomorphism between wave functions. It will not help to say that ψ and ψ 0 are
isomorphic wave functions just in case there is a unitary symmetry U such that
Uψ ¼ ψ 0 , for in that case, all wave functions would be isomorphic. The closest we
come to finding a home for ψ is in the category of Hilbert spaces: ψ is an element
of a Hilbert space, which is an object in the category of Hilbert spaces. But that will
not help, because we do not want to say that the world is isomorphic to the Hilbert
space H, but that it is isomorphic to a particular wave function ψ.
There are numerous other problems with analyses of representation in terms of
isomorphism, some of which are discussed in a recent article by Frigg and Nguyen
(2016). We mention two further problems here, each of which might be taken to
deliver a fatal blow to the account. First, an isomorphism is a function between two
mathematical objects, and the world is not a mathematical object. In fact, as
pointed out long ago by Reichenbach (1965), the only grip we have on the
structure of the world is by means of our representations.
Second, our account of representational significance should mesh with our
account of theoretical equivalence, and many philosophers of science hold views
of theoretical equivalence according to which equivalent theories need not have
isomorphic models. For example, Halvorson (2012) labels this view as “the model
isomorphism criterion of theoretical equivalence,” and he argues that it must be
rejected. However, if the model isomorphism criterion of theoretical equivalence is
rejected, then we must also reject the claim that representation entails isomorphism
between the world and one of the theory’s models. We can argue as follows: If two
theories T and T 0 are equivalent, and if T is representationally adequate, then T 0 is
also representationally adequate. But if the models of T are not isomorphic to the
models of T 0 , then it cannot be the case that the world is isomorphic to a model of
T and also to a model of T 0 . Therefore, to say that T is representationally adequate
does not entail that the world is isomorphic to one of the models of T.
Clearly there are two functions ψ and ψ 0 such that ψ ψ 0 , but Θðψ Þ and ¬Θðψ 0 Þ.
Therefore, Θ is not a representationally significant property of wave functions.
Second, wave functions are not actually functions at all. In fact, the space of
square integrable functions on configuration space is not a Hilbert space. Instead,
to define a positive-definite inner product, one has to take equivalence classes of
functions relative to the equivalence relation of “agreeing except on a set of
measure zero.” But now consider the property Θ defined by:
Θðψ Þ $ jψ ð0Þj2 ¼ 1 :
Again, there are two functions ψ and ψ 0 such that ψ ψ 0 , but Θðψ Þ and ¬Θðψ 0 Þ.
Therefore, Θ is not a representationally significant property of wave functions.
This is not to say that there are no representationally significant properties of
wave functions. For example, consider the property
ð
Θðψ Þ $ j ψ ðxÞ j dμðxÞ ¼ 1:
Δ
This property Θ can be shown to be invariant under the equivalence relations mentioned
previously. Indeed, practitioners of quantum theory know exactly what this property is:
It is the property ½Q 2 Δ of being located in the region Δ. What other invariant
properties are there? Can we give some sort of systematic description of them?
As mentioned before, the Hilbert space formalism is normally taken to represent
properties by means of the subspaces of the statespace. Let us think about how this
works in the case of the space L2 ðX Þ of (equivalence classes of ) wave functions.
What does a subspace of L2 ðX Þ look like? Some subspaces correspond to proper-
ties of functions. For example, consider the property
Θðψ Þ ψ has support in the region Δ:
It is not difficult to see that the set of functions satisfying Θ forms a closed
subspace of L2 ðX Þ. But not every subspace of L2 ðX Þ has such an interpretation
in terms of straightforwardly geometric features of functions. For example, let
U : L2 ðK Þ ! L2 ðX Þ be the unitary isomorphism between the momentum-space
and position-space representation of wave functions. Now begin by defining the
same sort of subspace, but relative to the momentum-space representation. That is,
let E be the subspace of L2 ðK Þ consisting of functions with support in Δ. The
natural interpretation of E is: having momentum value in the set Δ. Then U ðE Þ is a
subspace of L2 ðX Þ, and hence, represents a quantum-theoretic property Θ. But this
property Θ does not manifest itself as a natural property of functions on the original
configuration space X. Indeed, it is not clear that it would be possible to express Θ
without making reference to the isomorphism between L2 ðK Þ and L2 ðX Þ.
148 Hans Halvorson
But that criterion is too imprecise. And, in any case, the operationalist criterion is
stricter than quantum theory’s own criterion, which countenances many natural
properties that cannot be operationally detected.
The language of quantum theory, represented via the Hilbert space formalism,
comes with a vocabulary, including a list of predicates.
(QM properties) A predicate Θ of wave functions represents a natural physical
property if and only if the set fψ 2 L2 ðX Þ j Θðψ Þg is a subspace of L2 ðX Þ.
By this result, the previous criterion can be restated as follows:
(QM properties) A predicate Θ of wave functions represents a natural physical
property if and only if there is a dynamical variable Z, and a measurable Δ R, such
that Θðψ Þ if and only if ψ lies in the subspace ½Z 2 Δ.
These predicates can then be taken as giving quantum theory’s preferred account of
natural properties. In short, the natural properties are precisely those picked out by
saying that a quantity Z has value in a certain range.
So, we return to the original question: If Θðψ Þ is the predicate “ψ is a
smooth function,” then does Θ pick out a physical property of wave functions?
Quantum theory answers this question by saying: Θ represents a physical
property only if there is some quantity Z such that that Θ picks out the subspace
½Z 2 Δ.
When we talk about giving a “physical interpretation” to a subset E of state-
space, the demand is not that E be given an operational interpretation, as, e.g.,
corresponding to some measurement operation. Instead, we are simply asking that
the mathematical object E be describable in words that have some antecedent
physical meaning. It is simply the demand that we understand what the formalism
purports to represent.
The aim of science is to give us a literally true story of what the world is like; and the
proper form of acceptance of a theory is to believe that it is true.
(van Fraassen 1976: 623, emphasis added)
The debates of the last 40 years seem not to have brought into question the
connection between realism and literalism. In a recent authoritative account of
scientific realism, Chakravartty reasserts the connection:
Semantically, realism is committed to a literal interpretation of scientific claims about
the world.
(Chakravartty 2017: on line, emphasis added)
But something fishy must be going on here. The idea that a scientific theory is a set
of claims (i.e., sentences) fell out of favor about 40 years ago. Nowadays, most
philosophers of science say that a scientific theory consists of a collection of
models, plus some claim to the effect that one of these models represents the
world. But if a theory is a collection of models, then how am I supposed to read it
literally? Nor can this problem be brushed away by adopting a different view of
scientific theories. For better or worse, the theories of mathematical physics
involve collections of mathematical models, such as Lorentzian manifolds, Hilbert
spaces, etc. So how then are we supposed to read these theories literally?
The answer, in short, seems to be: To read a theory literally is to take one of its
models M as a reliable guide to features of the world. But now we are right back to
where we were when considering analyses of “Y directly represents X.” If I am a
literalist about M, then which features of M should I take to be representationally
significant? The simple answer “all features of M” leads immediately to absurdity.
The answer “all mathematical features of M” also leads to a bizarre and untenable
picture. Thus, we are thrown back on a more piecemeal approach, where one has to
know how to interpret the model M, which means being able to distinguish its
representationally significant properties from the insignificant ones.
Indeed, learning how to use a physical theory requires that learning the art of
“reading claims off” of a model. Consider, for example, the general theory of
relativity (GTR), where a model M is a Lorentzian manifold. What might it look
like to read M literally? Well, GTR claims that at each point p 2 M, there is a four-
dimensional tangent space T p . And living on top of T p there is an infinite tower of
ðm; nÞ tensors, for all natural numbers m and n. Are these things I have just said
among the “scientific claims” of GTR? If I am a realist about GTR, then am
I committed to these claims? Should I envision an infinitely extended tangent space
T p of four dimensions sitting on the tip of my nose, and indeed, a different such
tangent space for each instant of time? Are these tangent spaces “part of the
furniture of the world”? If this is what it means to be a realist about GTR, then
Einstein himself was no realist.
To Be a Realist about Quantum Theory 151
These coordinate charts are just as much elements of a model of GTR as a wave
function is an element of a model of QM. Thus, if literalism demands commitment
to the wave function ψ, then it also demands commitment to the coordinate chart ϕ.
If quantum state realism is just a “literal reading of QM,” then coordinate chart
realism is just a “literal reading of GTR.”
If you do not think that GTR involves a commitment to an ontology of tangent
spaces, coordinate charts, etc., then I can only agree: Not every true statement,
made within the language of a theory, is one of the “scientific claims” of that
theory. To say that a model M accurately represents the physical world does not
mean that every mathematical thing in M represents a physical thing. Realism,
according to Chakravartty, Timpson, Wallace, van Fraassen, et al., requires com-
mitment to the scientific claims of a theory, interpreted literally. But you cannot
interpret a mathematical object literally. That simply does not make sense. The
demand for literal interpretation only makes sense after we have used the formal-
ism to express claims in a language that we understand.
Here we have to lay some blame at the door of the semantic view of theories.
The semantic view of theories plus realism suggests the idea that one ought to
interpret models literally – an idea that can lead to absurd consequences if not
further nuanced. A model’s elements need not all play the same representational
role. For example, suppose that I make a map of Princeton University, on which
I draw several buildings. Suppose that I also draw a picture of a compass in the
lower right hand corner of my map – to indicate its orientation. Now, I am a
realist about the geography of Princeton, and I believe that my map is a faithful
representation of it. But that does not mean that I believe there is a huge compass
lying on the ground just outside of the university. Nor would I say that the
compass on the map is “just a bookkeeping device” or that it “has no representa-
tional role.” The compass does have a representational role: It represents a claim
about how my map is related to the actual town of Princeton. And if this compass
can be said to have a representational role, then so can a wave function. (For an
illuminating investigation of the notion of “literal interpretation,” see Hirsch
2017).
Timpson 2010; Wallace 2018a). But does this technical maneuver dodge the
various philosophical problems that confront wave function realism? In order to
press the question further, we need to sketch the idea behind spacetime state
realism.
Let us begin with the simplest (and least interesting) case of spacetime state
realism – the case where spacetime consists of a single point. In this case, we
represent a quantum system by means of a C∗ -algebra A of observables (For
an account of this formalism, see Ruetsche 2011). The important point is that
A is closed under operations of addition, multiplication, and conjugation
A ↦ A∗ . Moreover, there is a preferred multiplicative unit I 2 A, the identity
operator. The prototypical case of a C ∗ -algebra is the algebra of n n complex
matrices.
We need a few definitions. An operator A 2 A is said to be self-adjoint just in
case A∗ ¼ A, and A is said to be positive just in case A ¼ B∗ B for some operator
B 2 A. A function ω : A ! C is said to be a linear functional just in case
ωðcA þ BÞ ¼ cωðAÞ þ ωðBÞ for all A, B 2 A and c 2 C. A linear functional ω is
said to be positive just in case ωðAÞ 0 for every positive operator A 2 A.
A positive linear functional ω is said to be a state just in case ωðI Þ ¼ 1. We will
use Σ ðAÞ to denote the space of states of A.
We can formulate quantum mechanics in the language of C∗ -algebras just as
well as we can in the language of Hilbert spaces. Indeed, the self-adjoint operators
in A represent observables (or more accurately, quantities), and the elements of
Σ ðAÞ represent physical states. As a particular case in point, if A is the algebra of
2 2 matrices, then the self-adjoint operators are simply the Hermitian matrices,
and the states on A correspond one-to-one with density operators on C2 via the
equation
ωðAÞ ¼ TrðW ω AÞ:
With these definitions in hand, we can state Wallace and Timpson’s proposal quite
simply:
For a system represented by the algebra A, the properties correspond one-to-one with the
states in Σ ðAÞ.
This proposal can be made more picturesque and plausible if you think of a “field
of states,” where each point p in spacetime is assigned a state ωp . And if you feel
that this is just empty mathematics, then it might help to think of the typical case,
where ωp is represented concretely by a density operator W p . Then the field
p ↦ W p of density operators starts to look more like a classical field configuration,
where some mathematical object, such as a tensor, is assigned to each point in
space. The only mathematical difference is that W p is a complex matrix instead of a
To Be a Realist about Quantum Theory 153
tensor. But as Wallace and Timpson point out, the relative unfamiliarity of
complex matrices such as W p should not rule them out as legitimate values of a
physical field.
To this point, I agree with Wallace and Timpson. What bothers me is not the
difference between tensors and complex matrices. What bothers me is the confla-
tion of the various theoretical roles of states, quantities, and properties. The typical
job of states is to assign values to quantities. So, if we ask states also to serve as
values of quantities, then the job of states will be to assign states. In order to try to
keep things straight in our heads, we might try to declare some “types.” First, the
standard way of thinking of states is that they are of type Q ! V, where Q is the
quantity type, and V is the value type. But now, Wallace and Timpson tell us that
states are also of type V. In this case, states would be both of type Q ! V and of
type V, resulting in a type confusion.
What’s more, we typically ask a physical theory to provide some sort of “state-
to-property” link. For example, the so-called orthodox interpretation of quantum
theory proposes the eigenstate-eigenvalue link:
(EE link) A property E of the system is possessed in state ψ just in case Eψ ¼ ψ.
Wallace and Timpson also propose a state-to-property link. However, their prop-
erties are of the form “being in state W,” and so their proposal reduces to:
(WT link) A system has property W when it is in state W
Or perhaps it would be better to say:
(WT link) A system has the property of being in state W just in case it is in state W.
I suppose this claim is true. But I did not need to learn any physics to draw that
conclusion. This is nothing more than a disquotational theory of truth.
Is it possible that Wallace and Timpson’s proposal only trivializes in the trivial
case – where spacetime consists of a single point? Perhaps their proposal is only
meant to give an interesting picture in the case where we associate a different
algebra of observables AðOÞ to each region O of spacetime. In that case, their
recipe would yield a much richer structure, something like a co-presheaf of states
(see Swanson 2018). But I do not see any reason to think that this additional
mathematical structure can undo the conflation of states and properties that already
occurs at the level of individual algebras.
Finally, even if you can get past these other worries, there is a worry that the
Wallace-Timpson proposal shows too much. Indeed, there is a case to be made that
any reasonable generalized probability theory can be formulated in the framework
of C ∗ -algebras. In that case, it would seem that the Wallace-Timpson proposal
yields a realistic physical ontology for any reasonable generalized probability
theory. In other words, it is realism on the cheap.
154 Hans Halvorson
8.9.1 Bohr
Analytic philosophers have been quick to categorize Bohr as an operationalist
about the wave function, citing statements like this one:
the symbolic aspect of Schrödinger’s wave functions appears immediately from the use of
a multidimensional coordinate space, essential for their representation in the case of atomic
systems with several electrons.
(Bohr 1932: 370)
Faye, for example, seems to think that Bohr’s use of “symbolic” is code for
“should not be taken literally.”
Thus [for Bohr], the state vector is symbolic. Here “symbolic” means that the state vector’s
representational function should not be taken literally but be considered a tool for the
calculation of probabilities of observables.
(Faye 2014: on line)
To Be a Realist about Quantum Theory 155
As is typical with reading Bohr, one does not feel that the situation has been greatly
clarified. However, one thing is clear: Bohr does not intend to single out the
quantum state for operational treatment. If Bohr is an antirealist about the quantum
156 Hans Halvorson
state, then he is an antirealist about all of mathematical physics. For Bohr, all
mathematical representation is “symbolic,” whether observable or unobservable
aspects of reality are being represented. Among the symbolic representations of
physics, he would include the F ab of Maxwell’s equations, the gab of general
relativity, as well as functions representing the trajectories of material bodies
through spacetime. Bohr’s point might be summed up simply by saying that
mathematical objects are not sentences, and so they cannot “be read literally.”
To understand Bohr’s use of “symbolic,” it might also help to look at a
philosopher whose career ran in parallel with his. In fact, it is well known that
Bohr interacted extensively with Ernst Cassirer when the latter was composing his
book Determinismus und Indeterminismus in der Modernen Physik, first published
in 1937. Whether there is a more substantial overlap in their usage of “symbolic”
will have to await more detailed historical investigations.
Nonetheless, it is clear that there are many common themes in the views of Bohr
and Cassirer (see e.g., Pringe 2014). One such common theme is giving careful
thought to the way that mathematical objects can be used to represent the physical
world. In putting forward his views on this issue, Cassirer is clear that “symbolic”
should not be opposed to “representational.” The interesting question is not
whether something is representational, but rather how it represents. In particular,
Cassirer believes that the development of mathematics and physics in the ninteenth
century provides a particularly clear demonstration of the need to expand the
notion of representation beyond a simplistic “similarity of content” account.
Mathematicians and physicists were first to gain a clear awareness of this symbolic character
of their basic implements. . . . In place of the vague demand for a similarity of content
between image and thing, we now find expressed a highly complex logical relation, a general
intellectual condition, which the basic concepts of physical knowledge must satisfy.
(Cassirer 1955: 75)
For the former, more narrow, use of symbols, Cassirer uses the word Darstellings-
funktion. For the latter, more general, use of symbols, Cassirer uses the word
Bedeutungsfunktion. Thus, to relate back to our earlier analysis of “Y represents
X,” we might think that Darstellungsfunktion picks out a kind of representational
relation that licenses many inferences about X from Y, especially inferences having
to do with spatiotemporal properties. The paradigm case, of course, of such
representations are the directly geometric. In contrast, Bedeutungsfunktion picks
out a more general kind of representation relation that does not imply geometric
similarity between X and Y.
Bohr does not avail himself of Cassirer’s classification of symbolic forms.
However, he often does speak of things being “unvisualizable” (uanskuelig) –
opening a door to the deep dark recesses of the Kantian tradition. Bohr’s notion of
To Be a Realist about Quantum Theory 157
8.9.2 Carnap
We began the chapter with a story about how the early interpreters of quantum
theory were operationalists. That story is often neatly combined with another story
that post-Quinean analytic philosophers love to tell: the story about how silly and
stupid the logical positivists were. According to this story, the logical positivists
viewed scientific theories as “mere calculi” for deriving predictions. Thus, the
story concludes, it is no surprise that Bohr et al. were operationalists about the
quantum state, given that operationalism had so thoroughly infected the prevailing
view of scientific theories.
If you have ever read a serious historical account of the origins of quantum
theory, you know that the first story is mostly propaganda. None of the pioneers of
quantum theory – Bohr, Heisenberg, Dirac, etc. – was a crass operationalist. And if
you have ever read a serious historical account of twentieth-century philosophy,
you also know that the second story is largely Quinean propaganda. In fact, Carnap
himself was a vocal critic of operationalism – long before he felt the pressure of
Quine’s critiques of the positivist program.
Some, especially philosophers, go so far as even to contend that these modern theories,
since they are not intuitively understandable, are not at all theories about nature but “mere
formalistic constructions”, “mere calculi”. But this is a fundamental misunderstanding of
the function of a physical theory.
(Carnap 1939: 210)
Notice how Carnap feels the same pressure that Bohr and Cassirer feel – the
pressure that the new theories of physics are not “intuitively understandable.”
Moreover, like Bohr and Cassirer, he refuses to take the breakdown of intuitive
understandability (or anskuelighed, or Darstellbarkeit) to demand a retreat to
operationalism. Instead, Carnap – like Bohr and Cassirer – asks us to think harder
about how our theories purport to represent physical reality.
Like Bohr, Carnap insists that the representational status of the quantum wave
function is not all that different from the situation of the symbols of classical
mathematical physics.
158 Hans Halvorson
If we demand from a modern physicist an answer to the question what he means by the
symbol “ψ” of his calculus, and are astonished that he cannot give an answer, we ought to
realize that the situation was already the same in classical physics. There the physicist
could not tell us what he meant by the symbol “E” in Maxwell’s equations. . . .Thus the
physicist, although he cannot give us a translation into everyday language, understands the
symbol “ψ” and the laws of quantum mechanics. He possesses the kind of understanding
which alone is essential in the field of knowledge and science.
(Carnap 1939: 210–211)
We might just add that the concept of spatial regions does not provide us with a
truly Archimedian reference point – for these regions themselves are understood in
a mediated way, via their description in physical theory.
At this point, it should be thoroughly unclear how the views of Bohr, Cassirer,
and Carnap differ from some of the more moderate and reasonable quantum state
realists. To one such view we now turn.
propositions encoded in ψ are objectively true, i.e., they correspond to reality. But
what then are these propositions that are encoded in ψ? Of course, Bohmians have
an answer ready at hand: ψ encodes propositions about the trajectories of particles.
Notice that the specific Bohmian answer is not implicit in the very idea that ψ
encodes true propositions. Even a rank operationalist will say that ψ encodes true
propositions – about the probabilities of measurement outcomes. Only we might
question whether these propositions are “objectively true,” because probabilities of
measurement outcomes are indexed by measurements, and the latter has yet to be
objectively defined.
So what makes the nomological view realist? Is it simply that ψ encodes
objectively true propositions? Or is it that ψ encodes true propositions about
particle trajectories? I would be loath to accept the second answer, because it
would make realism hostage to one idiosyncratic ontological picture, viz., a
particle ontology. Surely one can be a realist and have some sort of gunky
ontology, or a field ontology. So, it seems that realist-making feature of the
nomological view is merely its commitment to the idea that ψ represents object-
ively real features of the world. But now, if that is enough to make a view realist,
then Healey’s view is also a realist view. For Healey says that each physical
situation is correctly represented by at most one quantum state. Healey and the
nomologists agree that ψ represents objectively real features of the world.
Nor can we say that the nomological view is more realist than Healey’s because
it takes ψ to be a direct representation of reality. The representation relation
posited by the nomological view is every bit as indirect and nuanced as that
posited by Healey (or by Bohr for that matter). Indeed, the nomological view
includes an intricate translation scheme from mathematical properties of ψ to
various meaningful physical statements, some of which are about occurent states
of affairs, and some of which are about how things will change as time progresses.
Thus, in terms of how ψ represents, the nomological view is closer to the views of
Healey, Bohr, and Carnap than it is to ψ-field views. The nomologists may be
horrified to hear this, for they take great pride in being realists. But recall that
Bohm often emphasized that his point of view was not so radically different from
Bohr’s. He even offered his point of view as a clarification of Bohr’s. Perhaps then
the nomological view could be thought of as an attempt to clearly articulate some
of the things that Bohr was trying to say about the wave function.
8.11 Conclusion
The primary aim of this chapter was to investigate the meaning of realism about
quantum theory, and in particular, realism about the quantum state. We found that,
for the most part, these phrases are empty of substantive content. They are emotive
160 Hans Halvorson
catch phrases that are meant to muster the troops – and perhaps to sell books. But
please don’t get me wrong. I am not saying that there are no substantive questions
about how to interpret the quantum state. First of all, dissolving the antirealism/
realism distinction does not solve the measurement problem. There is still the
thorny issue of why it appears to us that measurements have outcomes. Second,
there are genuine disagreements about how to use quantum states – even if these
disagreements do not correlate directly with a distinction between “real” and
“not real.”
First, there is a genuine question of how to think of the relation of quantum
states to physical situations. (For simplicity, I will suppose that a physical situation
is picked out by an ordinary language description, for example, by the sorts of
instructions that one might give to an engineer or to a postdoc in the lab.) At one
extreme, we have objectivists who think that each such situation corresponds to a
unique, correct quantum state. At the opposite extreme, we have the Quantum
Bayesians who propose no correctness standards whatsoever between physical
situations and quantum states. For these QBists, a quantum state just is a person’s
point of view – it is neither correct nor incorrect, appropriate or inappropriate.
Between these two extremes, we have views like Rovelli’s, where each physical
situation can be described equally by at least two quantum states, depending on
one’s choice of a direction of time. Some people also think that Bohr was a
nonobjectivist about quantum states (see Zinkernagel 2016). However, I find that
view hard to square with Bohr’s repeated pronouncements of the “objectivity of
the quantum-mechanical description.”
I propose that we stop talking about the ill-defined notion of quantum state
realism, and that we instead start talking about these sorts of questions, e.g.,
whether quantum theory comes with objective standards for the ascription of states
to physical situations. First of all, what role do physical situations, described in
ordinary language, play in this debate? Could we replace “physical situation” with
something more neutral and description-free, such as “object” or “system”? The
problem with that suggestion is that the bare notion of an object or a system cannot
give us any sort of standard for comparison. For example, we might say:
“According to Healey, for each object X, there is a unique correct quantum state.”
But how does Healey individuate objects? If he has different standards for indi-
viduating objects than Rovelli has, then their apparently diverging views might in
fact agree. Thus, the question of appropriate use of quantum states requires a
target, or standard of reference, on which all parties antecedently agree. The notion
of a “physical situation” is supposed to offer a plausible standard of reference.
I have already suggested a shift from the ontological question: Do states exist?
to the representational question: How do states represent? Now I am suggesting
that this representational question be given a normative reading: What are the rules
To Be a Realist about Quantum Theory 161
governing the use of quantum states? That, I believe, is the real issue at stake,
although it is masked by emotionally charged words such as “ontological status.”
There is a second question, closely related to the first one. Should we apply
unitary dynamics without exception? Some people say yes (e.g., Bohm, Everett,
Wallace), and others say no (e.g., Ghirardi-Rimini-Weber [GRW], Rovelli,
Healey). But even this disagreement is not as clear-cut as it may seem. Even those
who believe in the universal validity of unitary dynamics allow themselves to use
“effective states.” The “true state,” they say, follows unitary dynamics. But for
calculational purposes, there can be great advantages to using the effective state.
I am no verificationist, and so I do not propose that we collapse the distinction
between real and effective states. Nonetheless, I am interested here in the rules for
using states, i.e., for deciding whether one ought to use the state that results from
unitary evolution or whether one is permitted to use the state that results from
application of the projection postulate. Or, to put it in explicitly representational
language: The question is whether the state that results from unitary evolution is
the only one that is “apt” to one’s situation or whether the state resulting from the
projection postulate might also be “apt” to one’s situation. Interestingly, all parties
seem to agree that the state resulting from the projection postulate is “apt” in some
sense. Even the most fervent anticollapsers will tell you that the projected state is
correct for all practical purposes. Then they will remind you that it is not the “real”
state. But I would then ask: not the real state of what? We are back again to the
question of how to identify the target X of our representation via a quantum state.
Acknowledgments
I thank Eddy Chen for guidance about wave function realism, to Catherina Juel for
help translating Bohr’s letter to Møller, and to Tom Ryckman for sending a
preprint of (Ryckman 2017), which got me interested in Cassirer’s view.
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9
Locality and Wave Function Realism
alyssa ney
9.1 Introduction
Wave function realism is a framework for interpreting quantum theories. Applied to
nonrelativistic versions of quantum mechanics, wave function realism yields a
metaphysics according to which the central, fundamental object is the quantum
wave function, understood as a field on a high-dimensional space with the structure
of a classical configuration space, perhaps supplemented with additional degrees of
freedom to capture spin and other variables. Particles and other low-dimensional
objects are understood by the wave function realist to be ontically derivative objects,
constituted ultimately out of wave function stuff. For more sophisticated relativistic
quantum theories and quantum field theories, the framework recommends a suitable
relativistic extension of this metaphysics: a field in whatever high-dimensional
space is capable of capturing the full range of pure quantum states.
The case for such high-dimensional field interpretations varies from one frame-
work proponent to another, but a recurrent theme is wave function realism’s ability
to provide ontologies for quantum theories that have some intuitively nice meta-
physical features. For example, one may note the fact that quantum entanglement
threatens to force a fundamentally nonseparable metaphysics on the interpreter or,
what is to some (Howard 1985, and, he argues, Einstein) worse, a fundamentally
nonlocal metaphysics. However, these defects may be seen to drop away in the
higher-dimensional interpretations preferred by the wave function realist. For her,
what initially appear to be distinct entities possessing primitive relations and
communicating instantaneously across distant regions of space are revealed to be
manifestations of a single object, fundamentally possessing only intrinsic features
and acting locally on a high-dimensional space. This motivation for wave function
realism is, as I shall explain later, more compelling than that suggested by others,
who have argued that one should adopt such a framework simply because it is the
sort of thing that is most naturally read off the physics.
164
Locality and Wave Function Realism 165
Although some have challenged the wave function realist’s claim to provide a
separable metaphysics for quantum theories (e.g., Myrvold 2015, Lewis 2016),
I would say it is the claim that wave function realism provides a local metaphysics
that is more difficult and less straightforward (I rebut the concerns about separabil-
ity in Ney 2019b). And so, this is what I wish to examine in the present chapter.
I will introduce and distinguish several senses in which a metaphysics for physics
may be local, starting with two notions made use of by Bell (1964, 1976). From
there, we may evaluate in which sense or senses, if any, the wave function realist’s
metaphysics are local; and what, after all, is the virtue of having interpretations for
quantum theories that are local in that (those) sense(s). I will focus on the
nonrelativistic case. Ney (2019b) examines the extension to relativistic theories.
and many worlds. The competitor primitive ontology approach is more naturally
combined with hidden variable theories like Bohmian mechanics).
As mentioned, wave function realism is a framework for the interpretation of
quantum theories in which the wave function is the central ontological item and is
interpreted as a field on a high-dimensional space that (for the nonrelativistic case)
is assumed to have the structure of a classical configuration space. For hidden
variable theories, this wave function is supplemented with additional ontology,
e.g., for Bohmian mechanics, a single particle evolving in a way determined by the
motion of the wave function. Wave function realists thus take wave function
representations literally and straightforwardly. As Albert has written
The sorts of physical objects that wave functions are, on this way of thinking, are (plainly)
fields – which is to say that they are the sorts of objects whose states one specifies by
specifying the values of some set of numbers at every point in the space where they live,
the sorts of objects whose states one specifies (in this case) by specifying the values of two
numbers (one of which is usually referred to as an amplitude, and the other as a phase) at
every point in the universe’s so-called configuration space.
(Albert 1996: 278)
As has been mentioned and will be discussed in more detail later, this simple
interpretation of the wave function has the advantage of providing a metaphysics
for quantum theories that is fundamentally separable and local. However, other
interpreters challenge this reading.
The primitive ontology approach of Dürr, Goldstein, Zanghì, Allori, and Tumulka
(Dürr, Goldstein, and Zanghì 1992, Allori et al. 2008, Goldstein and Zanghì 2013)
insists that the ontology of quantum theories consists primarily of entities in ordinary
space or spacetime, for example, particles for Bohmian mechanics and matter
density fields for collapse theories or many-worlds approaches. On the primitive
ontology approach, the wave function is interpreted as real, but not an element of a
quantum theory’s primitive ontology: It is not what any physical theory is primar-
ily about, not what constitutes the matter in the theory. (See Ney and Phillips 2013
for a detailed examination and critique of the notion of a primitive ontology).
Instead, the wave function plays some other role, to guide the behavior of the
matter, and so it is something more like a law, broadly speaking.
Those adopting the primitive ontology framework (and some of the other
approaches I describe later in the chapter) complain about the use of the name
‘wave function realism’ to apply solely to views according to which the wave
function is a physical field on a higher-dimensional space, claiming they too are
realists about the wave function, taking it to be a real, mind-independent element of
quantum ontology. In a sense, this complaint is fair, but by now the terminology
has become so entrenched, I will continue to use it. And anyway, in defense of the
terminology, one thing that distinguishes the status of the wave function on the
Locality and Wave Function Realism 167
wave function realist view from how it is viewed in primitive ontology or other psi-
ontic approaches to interpretation is that, for the wave function realist, the wave
function is real in the classical sense of being res- or thing-like; it is a substance,
re-al. This contrasts with its status on other interpretational approaches, in which it
occupies one or another distinct ontological category; rather than being res, it
viewed as law, property, or a pattern of relations.
To many, the primitive ontology framework has appeal over wave function
realism in providing metaphysics for quantum theories that are intuitive in certain
respects. According to most applications of the approach, the fundamental entities
of the theory inhabit our familiar space or spacetime, and the macroscopic objects
we observe may be built out of these basic constituents (particles or matter fields)
in straightforward ways. (For a critique, see Ney and Phillips 2013.) An exception
is the flash ontology offered as a primitive ontology interpretation of some collapse
approaches; this ontology is surprisingly sparse and unfamiliar. Moreover, these
approaches hold out the promise of a separable metaphysics. The features of
composite objects are determined by the features of their smaller constituent
particles or field values at individual spacetime points. For example, for Bohmian
mechanics, one may argue that there are no facts about joint states of the particles
that fail to be determined by the states of individual particles. Facts about quantum
entanglement do fail to be determined by facts about the states of particles taken
individually or together, but on a primitive ontology approach to Bohmian mech-
anics, entanglement is a feature of the wave function, not the particles. And so, one
could say that the matter ontology of Bohmian mechanics is perfectly separable.
The same may be said for the matter density ontology that the primitive ontology
view attaches to collapse theories.
Nonetheless, such metaphysics are not local. As Bell showed
In a theory in which parameters are added to quantum mechanics to determine the results of
individual measurements, without changing the statistical predictions, there must be a
mechanism whereby the setting of one measuring device can influence the reading of
another instrument, however remote. Moreover, the signal involved must propagate
instantaneously, so that such a theory could not be Lorentz invariant.
(1964/1987: 199)
The first clause is needed in order to rule out monistic metaphysics in which there
is only one thing or one spatial location. Separability implies that there are distinct
objects, or at the very least (if one prefers a field metaphysics), distinct field values
instantiated at distinct locations in space.
One drawback to this definition is that, as it stands, it requires a separable
metaphysics to be a Humean metaphysics. Since it speaks of all facts being
determined by facts about what occurs at individual spatial regions, it makes
separability require that all facts about dispositions, counterfactuals, causation,
and laws be determined by what occurs at individual spatial regions. Loewer
(1996) defends wave function realism explicitly for its ability to provide an
interpretation of quantum theories compatible with Humean supervenience. One
might avoid this implication by modifying the second clause of the definition to
state only that the categorical, nondispositional, or non-nomic facts are determined
by the facts about individuals at subregions. We then have the following:
A metaphysics is separable if and only if (i) it includes an ontology of objects or properties
instantiated at distinct regions, and (ii) when any objects or properties are instantiated at
distinct regions R1 and R2, all categorical facts about the composite region R1[R2 are
determined by the facts about the objects and properties instantiated at its subregions.
Some might object that the matters of concern when we discuss entanglement
relations are dispositional – this electron would be measured spin up were its z-spin
to be measured. And so we really want a definition of separability that also requires
dispositional features to reduce to localized facts about individual spatial regions.
But although these are some of the features of interest, entanglement can appear to
force on us as well, the violation of even this weaker account of separability. For,
unless one adopts the Copenhagen-ish view that we can only talk sensibly about
the results of measurements or the features of systems when they are in eigenstates,
it is the occurrent and categorical spin states of entangled pairs as well, not merely
how they would behave upon measurement, that appears to be determined only
jointly, not individually, by objects at distinct spatial regions.
170 Alyssa Ney
With this definition of separability in hand, we may see how the wave function
realist may claim to provide interpretations of quantum theories that recognize the
phenomenon of quantum entanglement without committing to fundamental non-
separability. To illustrate, consider the EPRB state, in which a pair of atoms is
entangled with respect to their z-spin. Suppose our atoms are created in the
singlet state
ψS ¼ 1=√2 j z-upiA j z-downiB 1=√2 j z-downiA j z-upiB
and are then sent in opposite directions toward two Stern-Gerlach magnets, which
will bend them up or down in accordance with their z-spin toward two respective
measurement screens. Consider four locations between the magnets and screen
with the following labels:
R1: where atom A goes at time t should it get deflected up
R2: where atom A goes at t should it get deflected down
R3: where atom B goes at t should it get deflected up
R4: where atom B goes at t should it get deflected down
At time t, the atoms will be an entangled state of position:
ψx ¼ 1=√2 j R1iA j R4iB 1=√2 j R2iA j R3iB
And so, there are facts at t about properties instantiated at the joint regions R1[R4
and R2[R3 that are not determined by any facts local to their subregions, e.g.,
there is an atom at R1 if and only if there is one at R4. There is an atom at R2 if and
only if there is one at R3. We thus have a violation of separability.
The wave function realist argues that what appears as nonseparability arises
because what we are seeing is a three-dimensional manifestation of a more
fundamental and higher-dimensional metaphysics that is entirely separable. The
individual atoms A and B are ultimately constituted out of a field – the quantum
wave function. This field is spread not over our familiar three-dimensional space in
which there are the four locations R1, R2, R3, and R4, but instead over a space
with the structure of a classical configuration space. This space instead contains
(for example) regions we may suggestively label R13, R14, R23, R24. The wave
function has amplitude at these regions corresponding to the Born Rule probabil-
ities for the quantum state. So, in the present case, given the quantum state ψx, the
wave function will have nonzero amplitude only at the two locations R14 and R23
and it will have an amplitude of ½ at each of these locations. Spin states will
correspond to additional degrees of freedom. For a system initially appearing to
have N particles then, the dimensionality of the wave function’s space is posited to
be at least 3N.
The wave function realist’s proposed higher-dimensional metaphysics is thus
entirely separable. All categorical features are determined by features of the wave
function instantiated at individual regions in its space. (The wave function has
Locality and Wave Function Realism 171
By “beable,” Bell simply means entity, something that is real. This is a stronger
principle than the early “locality” principle from 1964. It states not only that the
probabilities for the results of a measurement on one system are independent of
how we may manipulate another system at a spacelike separation from it, but also
that these probabilities are independent of the actual measurement results we find
when we measure that other system. Wiseman argues that it is local causality that
Bell took to be the primary locality principle of interest from at least 1976 on. And
it is what he argued must be violated if quantum theory is correct.
Arguably, neither of these interpretations ‘locality’ suffices to explicate the
sense in which the metaphysics of the wave function realist is claimed to be
local. For the principles invoked by Bell both concern the existence of causal
relations in spacetime – the second one especially explicitly by invoking facts
about certain events exhibiting spacelike separation. Yet there is no spacetime
interval defined on the space that the wave function is said to inhabit, nor is the
space of the wave function the space in which light propagates, and so there is
no sensible notion of spacelike separation in the wave function realist’s funda-
mental metaphysics to let us settle the issue of whether these senses of
‘locality’ obtain. To explain the way in which wave function realism may be
claimed to involve a local metaphysics, we must move to a concept of locality
that makes sense in the context of the high-dimensional space of the wave
function realist.
Sometimes, when Bell discusses his principle of local causality, he states it in
broader terms than we just saw:
What is held sacred is the principle of ‘local causality’ – or ‘no action at a distance’.
(Bell 1981: 46)
Or perhaps:
A metaphysics is local if and only if it contains no instantaneous and unmediated action
across spatial distances,
given by the Born Rule probabilities, which are associated with the amplitude
squared of the wave function at the different points in its space. But there is still no
reason to say that the amplitude of the wave function at one distant region R causes
a collapse to be localized at another region R’ of the space instantaneously. Even if
the wave function later becomes more peaked around R’, the collapse is not
something that takes place at R’, but is rather something that happens across the
entire space. So there is not really a localized effect that may be influenced by some
distant cause. The evolution of the wave function through collapse may be jerky
and discontinuous, but it does not result in nonlocal action.
Finally, in the case we are interpreting noncollapse theories with hidden
variables such as Bohmian mechanics, the wave function behaves identically to
how it does in the nonrelativistic Everettian model. However, in this case, there
will be some additional ontology, such as a particle (the so-called marvelous point)
that moves around the wave function’s space in a way described by the theory’s
guidance equation
dQk ℏ ψ ∂k ψ
¼ Im * ðQ1 ; . . . ; QN Þ
dt mk ψ*ψ
In this case, the behavior of this additional ontology, the particle, is determined by
the state of the wave function in the neighborhood of the place in the high-
dimensional space it occupies, and so, there is no threat of nonlocal action. Despite
this fact, Bohmian mechanics is a quantum theory (or solution to the measurement
problem) that combines rather poorly with the interpretational framework of wave
function realism. After all, the very motivation for adopting Bohmian mechanics
(at least as presented in Dürr et al. 1992) depends on an argument that wave
function realists should not accept, namely that quantum theories are not theories
about the behavior of the wave function but rather of something else, matter in
three-dimensional space or spacetime Regarding this point, see Section 5 of the
paper by Alyssa Ney and Ian Phillips (2013). But if one is worried about nonlocal
action, a wave function realist interpretation of Bohmian mechanics could be
of help.
locality that some would say are mainly at issue when one worries about the
incompatibility of relativity and quantum mechanics. But perhaps there is more
one can say, and the fact that wave function realism provides a metaphysics local
in its own space may help alleviate some of the concerns arising about nonlocality
in spacetime.
In his recent book Quantum Ontology, Peter Lewis (2016) states the reason why
nonlocal action is in tension with relativity in the following way. Suppose one
allows that there exists at least some instantaneous action at a distance. Then there
is some one time at which one event influences another at a spatial distance from it.
For example, something happening right here, right now depends on the simultan-
eous mass of a distant star. But according to special relativity, there are no absolute
facts about which spatially distant events are simultaneous with which others. So,
in Lewis’s example, there is no fact about the mass of the star right now. Thus, this
action at a distance is ill-defined according to relativity. Thus, it would seem,
according to relativity, there cannot be action at a distance.
However, what looks puzzling, ill-defined, or brute from the perspective of a
nonfundamental metaphysics may be revealed as expected and explained in terms
of a more fundamental metaphysics. To the extent that wave function realism
supports a derivative ontology, it will yield an account of which spacetime
configurations exist and are causally related in that derivative ontology. So, at
least Lewis’s concern about the conflict between special relativity and quantum
mechanics seems avoided if one adopts wave function realism. This is not to say
that other issues, which I would concede are more basic, are not avoided, namely a
conflict with Lorentz covariance.
Another important feature of local theories is articulated by Einstein, who, in a
famous paper from 1948, argued that local metaphysics seem to be required for the
possibility of physical theories:
For the relative independence of spatially distant things (A and B), this idea is
characteristic: an external influence on A has no immediate effect on B; this is known as
the ‘principle of local action’, which is applied consistently only in field theory. The
complete suspension of this basic principle would make impossible the idea of the
existence of (quasi-)closed systems and, thereby, the establishment of empirically
testable laws in the sense familiar to us.
(Einstein 1948: 321–322)
The point seems straightforward enough. If what is nearby and observable may be
affected by objects that are spatially distant, then without full knowledge of the
occupants of the total spacetime manifold, how are we to make predictions about
how the objects we observe will behave? Locality appears required to allow us to
formulate testable empirical theories.
Locality and Wave Function Realism 177
not the high-dimensional space of the wave function. Thus, it seems, Einstein’s
defense of locality justifies a local metaphysics in three-dimensional space or
spacetime, the framework in which we interact with objects, but not a local wave
function metaphysics.
Perhaps another case to be made for local interpretations of physical theories
may be found in the work of Allori. Allori (2013) defends another view she finds in
Einstein, that “the whole of science is nothing more than a refinement of our
everyday thinking.” She elaborates:
The scientific image typically starts close to the manifest image, gradually departing from it
if not successful to adequately reproduce the experimental findings. The scientific image is
not necessarily close to the manifest image, because with gradual departure after gradual
departure we can get pretty far away. . . The point, though, is that the scientist will typically
tend to make minimal and not very radical changes to a previously accepted theoretical
framework.
(Allori 2013: 61)
One might then say that since our prescientific thinking and subsequent physical
theories postulated local and separable metaphysics, our quantum theories should,
if possible, do so as well.
To be clear, Allori is herself not making this point to argue for the local
metaphysics of wave function realism. She is using the point to argue for her
preferred primitive ontology view, because she believes that all previous (i.e.,
nonquantum) physical theories also possess a primitive ontology. But one might
hope that her point extends to make a case for a local wave function metaphysics
as well.
Unfortunately, I do not think it does. Because wave function realism also rejects
as fundamental a three-dimensional spatial background, replacing it with an
unfamiliar, high-dimensional background, it is not really so plausible to argue that
this local metaphysics is closer to the manifest image and classical theories than
one that would jettison one or both of separability and locality, but retain the low-
dimensional spatial background of our experience. If we agree with Allori that
minimal departures should, where possible, be preferred, the move to higher
dimensions is very far from a minimal departure.
Finally, we may move to consider more purely a priori reasons in support of a
local metaphysics. Some of these were brought to bear in the eighteenth century as
natural philosophers struggled with Newton’s characterization of gravitational
forces as acting immediately across spatial distances. Newton himself sometimes
claimed that action at a distance is impossible, for example:
The cause of gravity is what I do not pretend to know and therefore would take more time
to consider of it. . . That gravity should be innate, inherent, and essential to matter, so that
Locality and Wave Function Realism 179
one body may act upon another at a distance through a vacuum, without the mediation of
anything else, by and through which their action and force may be conveyed from one to
another, is to me so great an absurdity that I believe no man who has in philosophical
matters a competent faculty of thinking can ever fall into it.
(Newton, letter to Bentley, in Bentley 1838: 202)
To claim nonlocality is absurd is not thereby to offer an argument against it. Nor to
my knowledge did Newton ever offer a clear argument for why action at a distance
is absurd; however, we do find something in the work of Clarke in his correspond-
ence with Leibniz:
That one body should attract another without any intermediate means, is not a miracle, but
a contradiction: for ‘tis supposing something to act where it is not. But the means by which
two bodies attract each other, may be invisible and intangible, and of a different nature
from mechanism; and yet, acting regularly and constantly, may well be called natural. . .
(Clarke, fourth letter to Leibniz, in Alexander 1956: 53)
Clarke, like Newton, supposes that gravity must act locally, even if the means by which
it does so may be invisible. And the reason why this must be so is for something to act,
it must be located where it acts. Otherwise, it would not be it itself that is so acting, but
something else, or nothing at all. There is something, I believe, that is sensible about
this point and it explains at least one reason why nonlocal action strikes us as deeply
unintuitive and worse, incoherent. And it is, finally, a consideration that may be
brought to be bear in support of wave function realism’s local metaphysics.
9.7 Intuitions
It is my view that the best case the wave function realist has for developing a
distinctive local metaphysics comes from such conceptual considerations and
intuitions. But one might question whether it is at all desirable to have an
interpretation of quantum theories that conforms to our intuitions. Ladyman and
Ross (2007) criticize such interpretational projects, calling them “domestications
of science.” My project is openly one of the domestication of a large part of
physics. It is my attitude that quantum theories stand very much in need of
domestication to the scientific community and greater public (this is not to deny
that the project of domestication has already been carried out to a large extent by
the work of those providing clear solutions to the measurement problem).
Following out interpretations that are compatible with our intuitions may be useful
for a number of reasons. I will now mention three benefits that such an interpret-
ation may bring. All are unabashedly pragmatic.
First, an interpretation of a physical theory, by providing one with a clear
account of what the world is like according to the theory, benefits students and
180 Alyssa Ney
scientists in allowing them a clearer handle on the theory with which they are
working. Although it is not possible to understand our best scientific theories
without having a handle on the mathematics used to state it, a clear metaphysics
to supplement the mathematics can be instrumental in seeing more clearly what the
theory says, allowing one to more easily learn and use it. As an example, the
special theory of relativity, before it is supplemented with the clear interpretation
of a four-dimensional Minkowski spacetime, can seem to lead to paradoxes in
measurements that are difficult to comprehend – like the paradox of the train and
the tunnel or the twins paradox. These are not genuine paradoxes; there is no such
inconsistency in the theory, but this is much easier to comprehend when one grasps
the theory not purely through the predictions the mathematics produces, but
supplements it with a picture of entities spread out in four-dimensional spacetime,
for which facts about elapsed time or spatial distance fail to be absolute. I believe
something similar can come to pass for quantum theories. Once supplemented with
a clear metaphysics, what looks paradoxical or surprising becomes clear and
natural and easier to use. And there is no reason why distinct interpretations cannot
produce alternative accounts that are useful in this respect.
Second, an interpretation says things that go beyond what the theory on its own
says, and in this respect, interpretations can be fruitful in generating new specula-
tions or predictions that can then extend the theoretical power of the theory. Should
one adopt the wave function metaphysics and its attendant higher dimensions, one
can begin to ask more questions about the structure and contents of this higher-
dimensional space and learn more facts about it that would simply not be discussed
without attention to this question of interpretation.
Third, for myself and many other former physics students, the reason we chose
physics as a focus of study was to learn about the fundamental nature of reality.
Without an interpretation, physics does not provide this. Under the influence of
Copenhagen, Mermin’s “Shut up and calculate!”, and Feynman’s “I think I can
safely say no one understands quantum mechanics,” students often come to
quantum theories puzzled about what they say about the world, but then they are
told not to ask such questions because the theory is impossible to understand. This
is disappointing, and it drives students out of the field. Not all physics students care
about questions of interpretation and the deep issue of the nature of reality, but for
those that do, it is worth having serious work on interpretation that can give them
what they are looking for. We need more, not fewer students of physics.
I do not want to leave the reader with the sense that anything I am saying
challenges the idea that we should not at the same time work on interpretations that
challenge our thinking. In fact, all of the interpretations of quantum theories that
are available have aspects of unintuitiveness – this is simply unavoidable in the
interpretation of quantum theories. In addition, this is what is so exhilarating about
Locality and Wave Function Realism 181
the study of these theories – how they challenge what we previously thought was
obvious. What is being suggested in this last section, however, is that there is
nothing problematic about trying to fit these startling aspects of the world into a
picture we can understand.
9.8 Conclusion
The wave function realist need not deny that there is a clear sense of locality in
which our world contains nonlocal influences. This is the sense of local causality
taken up by Bell from 1976 onwards. The question is whether one should take this
to be a brute fact about our world or should attempt to provide explanations in
terms of an underlying metaphysics. Wave function realism is such an attempt at
explanation. The virtues of having interpretative options that provide such an
explanation justify the exploration and development of this framework that should
be pursued alongside others.
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424001.
Part III
Individuality, Distinguishability, and Locality
10
Making Sense of Nonindividuals in Quantum Mechanics
jonas r. b. arenhart, otávio bueno, and décio krause
“It is only a slight exaggeration to say that good physics has at times
been spoiled by poor philosophy.”
(Heisenberg 1998: 211)
10.1 Motivation
As the epigraph by Heisenberg suggests, physics and philosophy may both benefit
from a constructive exchange in which one may enlighten the other. Physics can
illuminate philosophy, and philosophy can illuminate physics. Of course, some
may think that philosophy has nothing to contribute to physics (see Weinberg
1992), and although we shall not provide a detailed defense of why we take
philosophy to be relevant for science in general, we want to defend the relevance
of ontology, as a field of metaphysics, to physics and to what physics is about. We
stress, in this work, through a case study, the way in which ontology, as a
philosophical field, can engage with physics, particularly in clearing the ground
for the understanding of the nature of physical reality.
Ontology is concerned with what exists and with what kinds of things exist.
Although this description may sound abstract and far from the concerns of physics,
the relation between ontology and physics is a close one. Of course, we are not
claiming that physics cannot be successful without ontology. If that were the case,
ontology would be required for physics, and it is not. However, physicists work
with ontological problems all the time. For instance, when it is claimed that, in
general relativity, space and time are no longer independent and that a new kind of
entity is required, spacetime, this is a physical move with significant ontological
consequences. It directly affects how the furniture of the world looks.
Physicists need not be concerned with ontological problems raised by physics,
just as one need not be familiar with the Peano axioms in order to be able to use
arithmetical operations. Nevertheless, ontology is part of the enterprise, shared by
185
186 Jonas R. B. Arenhart, Otávio Bueno, and Décio Krause
most physicists, of obtaining information about how the world works and what it is
made of. What kinds of things are there? Particles, fields, space, time? What are
they like? Answering questions like these is part of the articulation of an under-
standing of physical reality. As a result, the furniture of the world is involved in
such understanding. Ignoring those questions and their importance may prevent
one from getting closer to the most fundamental problems.
In this chapter, we will not focus on such general questions, but rather on a very
specific case study: Assuming that quantum theories deal with “particles” of some
kind (point particles in orthodox nonrelativistic quantum mechanics, field excita-
tions in quantum field theories), what kind of entity can such particles be? One
possible answer, the one we shall examine here, is that they are not the usual kind
of object found in daily life – individuals. Rather, we follow a suggestion by Erwin
Schrödinger (among others, as will become clear later), according to which
quantum mechanics poses a revolutionary kind of entity – nonindividuals. While
physics, as a scientific field, is not concerned with whether entities posited by a
specific physical theory are individuals or not, answering this question is part of the
quest for a better understanding of physical reality. Here lies, in large measure, the
relevance of ontology.
10.2 Introduction
There is little doubt that quantum entities are difficult to categorize. Quantum
mechanics introduces so many oddities that it is easier to state what quantum entities
are not than to affirm what they are. (We use ‘entity’ here as a term that is neutral
regarding whether the things that are referred to have well-defined identity condi-
tions or not). According to some of the first creators of quantum theory, quantum
entities are nonindividuals. This view is now known as the Received View on
quantum non-individuality (henceforth, for the sake of brevity, “Received View”;
see French and Krause 2006: chapter 3, for further historical details on this view).
In a section aptly called “A particle is not an individual,” Erwin Schrödinger
(1998) advanced one of the formulations of the Received View. One passage is
worth quoting in full:
This essay deals with the elementary particle, more particularly with a certain feature that
this concept has acquired – or rather lost – in quantum mechanics. I mean this: that the
elementary particle is not an individual; it cannot be identified, it lacks “sameness” . . . In
technical language it is covered by saying that the particles “obey” a new fangled statistics,
either Bose-Einstein or Fermi-Dirac statistics. The implication, far from obvious, is that the
unsuspected epithet “this” is not quite properly applicable to, say, an electron, except with
caution, in a restricted sense, and sometimes not at all.
(Schrödinger 1998: 197)
Making Sense of Nonindividuals in Quantum Mechanics 187
Several significant points are made in this passage. It is noted that quantum
particles (i) are not individuals, (ii) cannot be identified, (iii) lack “sameness,”
and (iv) cannot be referred to by the use of “this,” at least not typically. Of course,
it is not clear, by considering this quotation alone, what Schrödinger’s conception
of identification, individuality, and sameness ultimately is, nor is it specified what
the proper relations among these concepts are. But a central feature of his view
becomes salient in another important passage. He notes:
I beg to emphasize this and I beg you to believe it: it is not a question of our being able to
ascertain the identity in some instances and not being able to do so in others. It is beyond
doubt that the question of “sameness”, of identity, really and truly has no meaning.
(Schrödinger 1996: 121–122)
Here, it is emphasized that the very question of the identity of quantum entities, the
question of their “sameness,” has no meaning. As a result, the difference between
these entities, provided their sameness is meaningless, has no meaning either. One
still needs to examine, of course, what exactly is the relation between the lack of
sameness (or identity) of quantum entities, on the one hand, and their lack of
individuality, on the other. It seems that Schrödinger takes them all to be conceptu-
ally the same: to “lose” one’s individuality just is to lose one’s identity. On his
view, the question of the identity of quantum particles in general makes no sense.
The proper understanding of the relations between these concepts, and the kind of
view that results from them in the context of quantum particles, is the topic of this
chapter.
These issues were also central to another contributor to the development of
quantum theory. In a classical passage, in which the issues of identity and individu-
ality were prominent, Hermann Weyl points out:
. . . the possibility that one of the identical twins Mike and Ike is in the quantum state E1
and the other in the quantum state E2 does not include two differentiable cases which are
permuted on permuting Mike and Ike; it is impossible for either of these individuals to
retain his identity so that one of them will always be able to say ‘I’m Mike’ and the other
‘I’m Ike’. Even in principle one cannot demand an alibi of an electron!
(Weyl 1950: 241)
As will becomes clear, to articulate this proposal it is required that identity and
indiscernibility be distinguished. In classical logic and standard mathematics,
identity is formulated in terms of indiscernibility. So, in order to keep one and
change the other, one needs to resist this identification and clearly separate the two
notions. (We will return to this later).
As these quotations illustrate, when it comes to the investigation of the nature of
quantum entities, various possibilities are open. One can examine the commonal-
ities between the conceptions underlying Schrödinger’s and Weyl’s approaches or
pursue their differences. A major feature that is common to both is that they seem
to suggest that something is lost by quantum entities: something that marks a
difference between quantum entities and classical entities.
In this chapter, we address the articulation of the Received View and the
conception of nonindividuality that it attributes to quantum entities. As we discuss
in Section 10.3, the conception can be formulated in distinct ways, some more
radical, others more conservative, at least with regard to the role of the concept of
identity as used in quantum theories. The main issue turns on the behavior of
identity and its relation with individuality. Central to the Received View is the
claim that identity makes no sense, a claim that, as just noted, Schrödinger seems
to have favored. We discuss, in Section 10.4, how to make metaphysical sense of
that idea. The bare claim that identity makes no sense should be accompanied by
an account of how this view entails that particles are not individuals. In Section
10.5, we discuss the formal consequences of the idea, and apply the Received
View to suggest a revision of classical logic. In Section 10.6, we draw some
consequences of this case study to the significance of research in the foundations of
physics.
experience. If a familiar article, say an earthenware jug, disappears from your room, you
are quite sure somebody must have taken it away. If after a time it reappears, you may
doubt whether it really is the same one – breakable objects in such circumstances are often
not. You may not be able to decide the issue, but you will have no doubt that the doubtful
sameness has an indisputable meaning – that there is an unambiguous answer to
your query.
(Schrödinger 1998: 204)
Compare the view articulated in this passage with the one Schrödinger advanced
earlier when he claimed that the notion of identity makes no sense for quantum
entities (see the quotation from Schrödinger 1996: 121–122, in the previous
section). While ordinary objects typically are supposed to have well-defined
identity conditions, which allows one to answer questions about their identity over
time (even if, in some cases, one may be unable to decide the issue), for quantum
objects such questions do not even make sense. As a result, there is simply no fact
of the matter regarding the individuality (as well as the identity or sameness) of
quantum particles. In fact, in the case of quantum particles, situations involving
distinct observations of an object through time generate problems that prevent the
individuality of the items in question from making sense. As Schrödinger notes
Even if you observe a similar particle a very short time later at a spot very near to the first,
and even if you have every reason to assume a causal connection between the first and the
second observation, there is no true, unambiguous meaning in the assertion that it is the
same particle you have observed in the two cases. The circumstances may be such that they
render it highly convenient and desirable to express oneself so, but it is only an
abbreviation of speech; for there are other cases where the “sameness” becomes entirely
meaningless . . .
(Schrödinger 1996: 121)
Schrödinger highlights the need for identity in order to claim that an entity is
observed in distinct places at distinct times. It is then just one step to add that
without the possibility that a particle observed at one instant of time t 1 is the same
as a particle observed at a later time t 2 , individuality is lost. Given that, on this
view, it makes no sense to state that those particles are the same (or different),
identity loses its meaning. As we have already noted, identity, individuality, and
sameness are taken as conceptually the same by Schrödinger.
Based on these considerations, a straightforward version of the Received View
emerges. Quantum particles are not individuals, given that they have no well-
determined trajectories in spacetime, and it is not possible to identify distinct
detections of an entity as being detections of the same entity (we will return to
this view in the next section and will provide additional details there).
However, this is not the only way to articulate the Received View via spacetime
continuity. Another form is to keep the restriction that quantum entities fail to have
Making Sense of Nonindividuals in Quantum Mechanics 191
In contrast, quantum particles have no alibi – nothing that accounts for their
individuality. Not even spatiotemporal location can be employed to this effect. Due
to the permutation symmetry, quantum particles are indiscernible by their proper-
ties, including both state-dependent and state-independent ones. Hence, the version
of PII presented earlier, according to which there is always some property that
accounts for the numerical diversity of particles, fails in quantum mechanics. The
result is clear: As Weyl noted, there is no alibi for quantum entities (see French and
Krause 2006: chapter 4, for further discussion).
It could be argued that, if properties are unable to account for the numerical
difference of quantum particles, perhaps some relations could do that, such as the
relation “to have spin opposite to” in a given spatial direction. But this proposal is
still unable to account for the particles’ individuality. After all, if x has spin
opposite to y, y also has spin opposite to x (the relation is symmetric). While no
particle has spin opposite to itself (the relation is irreflexive), there is no quantum
mechanical fact of the matter to determine which of x or y has spin up in a given
direction, and which has spin down in that same direction. Thus, those relations,
called weakly discerning relations, in principle can account for the numerical
diversity of the particles (although whether they do account for that is still
debatable; see French and Krause 2006: chapter 4). Despite that, they are unable
to provide an alibi for the particles in question, because weakly discerning relations
are unable to individuate such particles. Accounting for the particles’ numerical
diversity (if at all) is the closest one can get in quantum mechanics to discernibility
(see Muller and Saunders 2008, and the discussion in Lowe 2016).
However, if weakly discerning relations are implemented in a mathematical
context whose underlying set theory is ZFC (Zermelo-Fraenkel set theory with the
axiom of choice), as is the case of Muller and Saunders (2008), all entities become
fully discernible and identifiable in virtue of the resources of set theory alone (we
will return to this point and provide the argument later). Thus, there is a tension
between the motivation for the introduction of weakly discerning relations and the
adopted set-theoretic framework.
In principle, if the option of maintaining that identity holds for quantum
particles can be fully worked out, one could claim that they are different or
identical, without thereby implying that they are individuals. What is required, as
we have been suggesting, is that their individuality be grounded in some kind of
alibi (in Weyl’s sense) that is not formulated in terms of identity.
There are additional possibilities to articulate alibis (that is, principles of
individuality) without requiring the removal of identity (see Arenhart 2017). It is
enough that the content of identity be deflated from the metaphysical content that
would be required if identity also played the role of a principle of individuality.
(For a defense that identity should be deflated, see again Bueno 2014). As will
Making Sense of Nonindividuals in Quantum Mechanics 193
become clear later (when a formal approach to identity is discussed), identity can
be thought of as something very minimal and without much metaphysical content,
just in terms of two features: reflexivity (that is, every object is identical to itself )
and substitutivity (if x is identical to y, then if x is F, so is y). One could add some
metaphysical content to identity, so that it can be used as a principle of
individuality. But that changes identity by making it more substantive than it
needs to be. Schrödinger, of course, does not seem to follow this path since he
appears to keep identity and individuality very closely connected. As a result,
insisting on nonindividuality requires abandoning identity, at least for quantum
entities. In what follows, we investigate the prospects for a Schrödingerian
approach to nonindividuality.
The problem, however, is not to account for an item’s individuality, but rather
for its nonindividuality. How can haecceity achieve that? The answer seems to be:
through the notion of identity. As French and Krause put it
. . . the idea is apparently simple: regarded in haecceistic terms, “Transcendental
Individuality” can be understood as the identity of an object with itself; that is, ' a ¼ a' .
We shall then defend the claim that the notion of non-individuality can be captured in the
quantum context by formal systems in which self-identity is not always well-defined, so
that the reflexive law of identity, namely, 8xðx ¼ xÞ, is not valid in general.
(French and Krause 2006: 13–14)
This is only one of the possible ways to accommodate metaphysically the combin-
ation of nonindividuality and the loss of identity. This proposal allows one to make
a good case for the failure of identity, given that the relation between individuality
and identity is very clearly established in this approach. However, in addition to
burdening identity with the role of attributing individuality, there is another
disadvantage of adopting this approach to nonindividuality: It takes us very far
from the Schrödingerian ideas with which we started. Of course, it allows us to
make sense of the claim that identity and difference do not apply to quantum
entities. But the lack of haecceity arguably was not what Schrödinger had in mind
in his discussion of identity and identification of quantum particles. Rather, as
discussed previously, he seems to favor an account of individuality framed expli-
citly in terms of spatiotemporal trajectories.
Making Sense of Nonindividuals in Quantum Mechanics 195
(B.2) Closest continuers: given an individual i that satisfies condition (A), at each
moment of time the closest continuer individual to i (the one that shares most
properties with i) is taken to be i (Nozick 1981: chapter 1).
Of course, a haecceity could be an essential trait, and in this way, haecceities could
be used to account for the permanence of an individual. Given that we have already
suggested avoiding a theory of haecceities to account for individuality and to frame
an approach to nonindividuality, we favor the less metaphysically committing
option (B.2). The idea is that an individual persists through a sequence of closest
continuers, which, taken together, account for the permanence of an individual
over time despite the changes it undergoes.
Given this theory of individuality, formulated by the conjunction of conditions
(A) and (B.2), for something to be a nonindividual, three options emerge: condi-
tion (A) can be violated; condition (B.2) can be undermined, or both conditions can
fail. Quantum entities, as the discussion of Schrödinger’s view indicates, violate
both conditions. This is a Humean point: There appears to be no causal connection
that would allow one to determine that similar objects detected in different
moments in time are, in fact, the same. In the quantum case, consider some
quantum entities that have no continuous trajectory. One cannot look for a
quantum mechanical justification to connect two observations of two such entities
196 Jonas R. B. Arenhart, Otávio Bueno, and Décio Krause
through a single trajectory. Nothing in the theory allows us to do that (unless one is
a Bohmian). As a result, as we have seen, Schrödinger claimed that identity makes
no sense for those entities, given that there is no fact of the matter to determine
whether the two observations correspond to the same entity or not. The question of
the identity of the observed entities ends up being entirely ungrounded.
This accounts for both the nonindividuality of the particles and the fact that
identity does not apply to them. This metaphysical picture is closer to what
Schrödinger had in mind, it seems, and it is less inflated than the one first suggested
by French and Krause (2006), which proceeds through the concept of haecceity.
However, both approaches require a corresponding rejection of the overall validity
of identity. One of the ways to accommodate such a limitation of identity is
through logics that restrict identity, the so-called nonreflexive logics. We turn to
them now.
(Note the use of identity in the very formulation of the substitutivity rule: The
variables x and y need to be distinct, that is, not identical). From these postulates, it
follows that identity is symmetric and transitive. Thus, it is an equivalence relation
as well as a congruence relation due to the presence of substitutivity. Logicians say
that identity is the finest congruence over the domain in the sense that if ffi is
another congruence, then a ¼ b entails a ffi b, for all a and b.
However things are not so easy. Postulates (R) and (S) cannot guarantee that the
interpretation of the predicate ‘=’ is the set ID. In fact, it can be shown that a
congruence, other than identity, can be defined over the domain that also models
the predicate of identity (da Costa and Bueno 2009, Krause and Arenhart 2018). In
other words, from the point of view of L, it cannot be known whether one is
working with a structure where ‘=’ is interpreted as the identity of the domain D,
namely, the set I D , or in terms of another structure that has the defined congruence
as the interpretation of syntactic identity. These structures are elementary
equivalent.
Leaving first-order languages behind, higher-order languages should then be
considered. It suffices to consider L as a second-order language (the generalization
to other higher-order languages is immediate). In this case, identity can be
(allegedly) “defined” in terms of indiscernibility (indistinguishability) by what is
called Leibniz law, namely:
x ¼ y if, and only if, 8F ðFx $ FyÞ, (10.4)
where x and y are variables for individuals, and F is a variable for properties of
individuals. The right side of the biconditional expresses the indiscernibility of x
and y, and it states that the objects that stand for x and y have the same properties
(hence they also share all relations).
The problem now is with the semantics. Suppose that the domain is the none-
mpty set D ¼ f1; 2; 3; 4; 5g and that our second-order language has three monadic
predicate constants – P, Q, R – and two individual constants – a and b. Consider
the following interpretation: 1 is assigned to a and 2 to b. Furthermore, the
extensions of the predicates are interpreted as the following sets: A ¼ f1; 2; 3g,
B ¼ f1; 2; 4g, and C ¼ f1; 2; 5g. Thus, since 1 and 2 belong to all sets, it follows
that a and b have all properties in common. In other words, the right side of
Eq. (10.4) holds, despite the fact that 1 6¼ 2.
The only way of guaranteeing that Eq. (10.4) will have its full intuitive meaning
is to add all subsets of D to the semantics, that is, to consider what Church calls
principal interpretations (Church 1956: 307). But then, as is well known, com-
pleteness is lost.
As these considerations make clear, identity is not a simple concept when one
tries to provide a rigorous account of the intuitive idea. But from a logical point of
Making Sense of Nonindividuals in Quantum Mechanics 199
view, this is what classical logic presents us with. Based on this theory of identity,
which is called classical theory of identity (CTI), we can consider stronger
systems, such as various set theories.
As is well known, there are several nonequivalent set theories with distinct
properties and which yield significantly different, and even incompatible, the-
orems. For instance, ZFC includes the axiom of choice; Quine-Rosser’s NF
(New Foundations) system does not: It is incompatible with this axiom (Forster
2014). In ZFC, if consistent, there is no Russell set, namely R ¼ fx : x 2 = xg, but in
some paraconsistent set theories, this set is legitimate (da Costa, Krause, and
Bueno 2007). It can be proved, in ZFC, that there are sets that are not Lebesgue
measurable, but in “Solovay set theory” all sets are Lebesgue measurable (Mait-
land Wright 1973). What is remarkable is that all these set theories invoke the same
theory of identity (CTI). Thus, our considerations apply to all of them.
It is undeniable that set theory is the most widely used basis for standard
mathematics, that is, the part of mathematics that can be developed in theories
such as ZFC. This is also the mathematics that underlies quantum theories. In fact,
it is unclear what kind of quantum mechanics could be developed in a system such
as NF, given its incompatibility with the axiom of choice (AC). After all, AC is
necessary for the usual mathematical formulation of quantum mechanics, so that it
can be guaranteed, for instance, that the relevant Hilbert spaces have a basis (of
course, quantum mechanics can be developed in many different ways that need not
rely on von Neumann’s approach; see Styer et al. 2002).
It is a remarkable fact, we noted, that in all of these set-theoretic frameworks, all
objects are individuals, in the sense that all of them have identity. In other words,
given any objects (that is, any sets; the case of Urelemente will be mentioned
soon), there is always a way to distinguish them, if not effectively, at least in
principle. The proof is immediate. Given a certain object a, which is either a set or
a Urelement, the postulates of a set theory enable us to form the set {a}, the
singleton of a (as is well known, there are pure set theories, containing only sets,
and impure set theories, systems that also include atoms – the Urelemente in the
original Cantor’s terminology. These atoms are not sets but can be elements of
sets). Define the “property” Id a ðxÞ≕ x 2 fag. The only object that has such a
property is a itself, so a has at least one property distinguishing it from any other
object. Leibniz law applies and, thus, there cannot be indistinguishable but non-
identical objects.
Indiscernible entities can be accommodated in a set theory via equivalence
relations. The elements of an equivalent class can be taken as representing the
same object, but this is clearly a mathematical trick and does not work as part of a
philosophically well-motivated proposal. A trick similar to this is used in orthodox
quantum mechanics when symmetric and antisymmetric wave functions are
200 Jonas R. B. Arenhart, Otávio Bueno, and Décio Krause
chosen to stand for certain quantum systems: Functions are selected that do not
alter the probabilities when particle labels are exchanged. (This trick was called
“Weyl’s strategy” because it was used by Hermann Weyl; see French and Krause
2006: section 6.5.1).
As a result, within standard mathematics, there are no absolutely indiscernible
objects as quantum objects are said to be in certain situations. Thus, if we
use standard mathematics in our preferred formulation of quantum mechanics
(the same point applies to quantum field theories), from the simple fact that there
are two quantum objects, it results from the mathematics alone that the objects are
different (they are not identical), and by Leibniz law, there is at least one property
that one of them has and the other does not. However, if the objects in question are
indiscernible, such as two bosons in the same state, which property would that be?
The assumption of the existence of such a property amounts to the introduction of
hidden variables – even in those formulations of quantum mechanics that do not
accept them. But the fact that there is such a property follows from Leibniz law
(which, as noted, is part of the package formed by classical mathematics, which
includes a corresponding logic, and the standard theory of identity). Thus, in any
situation, given two quantum objects, there is a difference between them. Such a
difference cannot be given by a substratum (a haecceity), because the existence of
such a substratum is ruled out in quantum theories (see Teller 1998). The differ-
ence can be expressed in terms of a bundle theory of properties, which leads to the
conclusion that there is a property that only one of the quantum objects in question
have, but not the other. The problem is that, according to quantum theories,
assuming their usual interpretations, this is not a viable possibility. Otherwise,
quantum objects would be discernible. In the end, what is needed is a framework
that does not preclude the possibility of indiscernible but potentially distinct
systems of entities – a framework that makes room for nonindividuals.
An appropriate, philosophically well-motivated, strategy would then be to leave
standard set theories behind and adopt a set theory in which identity is not taken to
hold in general, namely, a quasi-set theory. This is a mathematical framework
which can be used as a metamathematics for quantum theories (see French and
Krause 2006, Domenech, Holik, and Krause 2008, Krause and Arenhart 2016). In
this theory, collections (called quasi-sets) can be formed by absolutely indiscern-
ible elements without thereby becoming identical. As a result, Leibniz law is
violated for some objects (although it remains valid for another kind of objects,
called classical). These collections of indiscernible entities can have a cardinal,
called its quasi-cardinal, even if they do not have an ordinal. The theory provides a
framework to examine collections of objects without ordering them, without
identifying or individuating them. And differently from classical set theories, the
theory offers a framework in which nonindividuals can be formulated and
Making Sense of Nonindividuals in Quantum Mechanics 201
thoroughly studied without the incoherence found in the use of classical set
theories for the formulation of the foundations of quantum mechanics (for details,
see French and Krause 2006, Krause and Arenhart 2016). We conclude this chapter
by noting the significance of foundational studies of physics, of which quasi-set
theories provide a clear case.
10.6 Conclusion
In this chapter, the metaphysical underpinnings of the idea that quantum entities
are nonindividuals have been examined. Schrödinger’s claim that identity does not
make sense for quantum entities was interpreted, and the connections between this
claim and some issues related to continuous trajectories in quantum theory were
investigated. The resulting metaphysics of nonindividuals assumes a tight connec-
tion between identity and individuality, so that if individuality has to go, so does
identity. Given that identity is a logical relation, which is part of classical math-
ematics, and since almost every system of logic has a version with identity, it is
important to provide an account of what it is like for a system of logic to have no
identity. A few more words on the importance of developing such formalisms and
their relation with physics are, thus, in order.
First, consider some reasons to look for alternative mathematics (and logic) for
quantum mechanics. Leaving aside historical considerations (an up-to-date analy-
sis, which also considers some historical facts, can be found in Maudlin 2018), the
motivation has to do with the foundations of physics. One could argue that physics
works fine with standard (Leibnizian) mathematics (and logic), as it can be seen by
considering any book on quantum mechanics. In particular, the argument goes,
questions about the foundations of physics could be regarded as “mere philosoph-
ical” problems that, on their own, contribute nothing to the clear understanding of
physics. That this view is untenable becomes clear by considering some of
the papers in Cao (1999) and the significant insights that a careful reflection on
the foundations of physics provides (for the sake of brevity, we will not revisit the
various arguments here).
It could be argued that something similar happens with current physical theor-
ies. Current physics works fine with two incompatible theories, namely, the
standard model of particle physics and general relativity. The former provides
the best way developed so far to account for the physics of the small, while the
latter offers the best physics of the big, as it were. One or the other is applied
depending on the subject matter under study. However, these two theories are
logically incompatible with one another, for gravity has not been quantized.
Should the situation be left at that, with everyone being encouraged to accept that
physics has reached its final limit, and no unification is ultimately necessary?
202 Jonas R. B. Arenhart, Otávio Bueno, and Décio Krause
Of course not. Novelties have always emerged when foundational issues have
been pursued. In mathematics, this is undeniably the case, as the development of
logic and set theory clearly illustrates. There is no reason to think that quantum
mechanics would be any different. Indeed, the relevance of string theories, loop
gravity, and any other attempt to find a more fundamental theory, in particular, the
quantization of gravity, cannot be appreciated without acknowledging the signifi-
cance of foundational research. In fact, without the latter, it would be difficult to
make sense of why physicists systematically pursue such enterprises.
To look for more suitable mathematical bases for a coherent metaphysical
conception of quantum objects as nonindividuals is reasonable and even necessary
(to prevent inconsistencies). Arguably, no one seems to know, and perhaps no one
will ever know, what quantum entities ultimately are. All one has are one’s
theories. Even the concept of particle changes from theory to theory (see Falken-
burg 2007: chapter 6). Foundational research provides some perspective and
insight to pursue the search for understanding that is integral to the attempt of
making sense of these issues as well as their significance.
This brings the second topic to be addressed here: ontology. Physicists, in
general, have a broad and intuitive idea of what ontology is, but some of them
do not find it relevant to their work in physics. Ontology, it was noted, is tradition-
ally occupied with what there is (in the world) – with existential questions, such as:
Are there winged horses? Are there electrons? Are there transcendental numbers?
Metaphysics is more general and includes ontology as a proper part. For instance,
Democritus’ claim that the world is composed by atoms (indivisibles) is a meta-
physical view. It concerns the basic structures of the world. Even in classical logic
one finds metaphysical assumptions. For example, classical propositional logic
assumes a metaphysical, semantic principle to the effect that the truth of a complex
formula depends on the truth of its component formulas, usually referred to as
Frege’s Principle of Compositionality (Szabó 2017).
Physics is not different, and it also has its share of metaphysical claims. One of
them, crucial to the entire discussion examined in this work, is that quantum
objects – whether they are particles in orthodox quantum mechanics or field
excitations in quantum field theories – are ultimately nonindividuals. This claim,
of course, does not force us to assume that nonindividuals exist. Situations can be
presented in a conditional form: If there are things like quantum entities, then they
can be interpreted as nonindividuals. This process of interpretation, itself an
integral part of foundational research, provides a possible approach to the under-
standing of the nature of such entities.
Just to be clear, we are not asserting that quantum objects are nonindividuals. It
is unclear how this claim could be established. Rather, the goal is to develop the
view as coherently as possible and indicate how it helps one to understand
Making Sense of Nonindividuals in Quantum Mechanics 203
References
Arenhart, J. R. B. (2017). “The received view on quantum non-individuality: Formal and
metaphysical analysis,” Synthese, 194: 1323–1347.
Bueno, O. (2014). “Why identity is fundamental,” American Philosophical Quarterly, 51:
325–332.
Bueno, O. (2019). “Weyl, identity, indistinguishability, realism,” in A. Cordero (ed.),
Philosophers Look at Quantum Mechanics. Dordrecht: Springer.
Cantor, G. (1915/1955). Contributions to the Founding of the Theory of Transfinite
Numbers. New York: Dover.
Cao, T. (ed.) (1999). Conceptual Foundations of Quantum Field Theories. Cambridge:
Cambridge University Press.
Church, A. (1956). Introduction to Mathematical Logic. Princeton: Princeton University
Press.
da Costa, N. C. A. and Bueno, O. (2009). “Non-reflexive logics,” Revista Brasileira de
Filosofia, 232: 181–196.
da Costa, N. C. A., Krause, D., and Bueno, O. (2007). “Paraconsistent logics and para-
consistency,” pp. 791–911 in D. Jacquette (ed.), Philosophy of Logic. Amsterdam:
North-Holland.
Domenech, G., Holik, F., and Krause, D. (2008). “Q-Spaces and the foundations of
quantum mechanics,” Foundations of Physics, 38: 969–994.
204 Jonas R. B. Arenhart, Otávio Bueno, and Décio Krause
11.1 Introduction
This chapter consists of three main sections. In Section 11.2, I review Nick
Huggett’s finding regarding the nonrelevance of permutable labeling for the
classical/quantum divide (Huggett 1999). In Section 11.3, I review the derivations
of the classical and quantum statistics and argue that a form of separability is a key
feature of the quantum-to-classical transition. In Section 11.4, I consider the
question: What allows separability to serve as a form of distinguishability in the
classical limit? First, let us review some basic considerations regarding issues of
individuality and distinguishability.
Steven French (2015) has noted that the concept of individuality is primarily a
metaphysical issue, while that of distinguishability is primarily an epistemological
issue. Nevertheless, distinguishability does have bearing on ontological questions
such as:
What is an individual?
Are there any true individuals?
Does Leibniz’ Principle of Identity of Indiscernibles apply to nature?
However, I will not enter here into the metaphysical debate concerning questions
such as: What is an individual? and Are quantum systems individuals? Rather,
I will focus on the issue of distinguishability regarding the quantum-classical
divide and attempt to identify some ontological features that may underlie the
form of distinguishability obtaining in that context.
205
206 Ruth Kastner
which transcends all its qualitative features. This concept can be identified with the
term ‘transcendent individuality’ (TI) as discussed in French and Redhead (1988).
Anti-haecceitism consists in saying that an individual’s identity is constituted by its
qualitative features and nothing more. Although the precise definition of haecceit-
ism is still a matter under discussion, for our purposes we can think of it as the
capacity of an entity to carry a label or “name,” where that label is not contingent
on any of its qualitative features. Thus, what makes a person named Fred “Fred the
individual” is his primitive this-ness, not the color of his eyes or hair or his height,
weight, etc.
Now let us consider this notion as applied to some typically classical objects,
such as a pair of coins that are assumed to be completely identical and can be either
“heads” or “tails.” Give them name-labels, say ‘Fred’ and ‘Joe’ their assumed
haecceitism is represented by their name-labels. In this context, haecceitism
implies that if we consider the case in which Fred and Joe are in different states
(one of them being “heads” and the other “tails”), then interchanging Fred and Joe
constitutes two different possible configurations. If we include the cases in which
Fred and Joe are both “heads” or both “tails,” we have four possible states of the
coins, as seen in Figure 11.1.
In contrast, for a pair of hypothetical “boson coins,” the usual story is that there
is no such thing as Fred or Joe no nameable identity that transcends the
qualitative properties of the quantum coins. So the possible configurations are just
three in number (Figure 11.2).
It should be noted that French and Redhead (1988) dissent from this usual
identification of individuality with the capacity to carry a label such that permuta-
tion of the labels establishes a different state of the total system. They argue that a
form of individuality can still be retained for quantum systems if one argues that
certain states are not accessible to the total system. For purposes of this discussion,
we work with Huggett’s formulation, but note that his interpretation of the
metaphysical bearing of the labels is not obligatory.
Huggett (1999) notes that there are two ways of describing the state-space of a
composite system such as the set of two coins. We can either use a phase space
(Γ-space) description, which specifies which component system is in which state,
or we can use a distribution space (Z-space) description, which just specifies how
many systems are in each state. The Γ-space description assumes that each
component can be meaningfully labeled and/or distinguished from the others, so
it supports haecceitism in that respect. In contrast, since Z-space specifies only the
occupancy number of each state, without identifying any particular system with
any particular state, it does not support haecceitism in the same way. Since it is
typically supposed that the key distinction between classical and quantum objects
is the ability of the former to carry a label, one would think that the two kinds of
descriptions phase space and distribution space would lead to different kinds
of statistics; i.e., classical and quantum statistics, respectively.
However, Huggett shows that if we assume that classical objects are impene-
trable, i.e., that no more than one such object can never occupy a given spacetime
point, then it turns out that the Γ- and Z-space descriptions give the same empirical
predictions. Thus, we cannot use any experimental data to decide between them.
This means that there is no empirical support for the idea that classical and
quantum objects differ fundamentally in their metaphysical nature as individuals.
The basic argument goes like this: In terms of the coin analogy, we have to pretend
that there are no other qualitative differences between the coins and forbid the two
coins, Fred and Joe (they can keep their labels), from occupying the same state. Of
course, real coins would not fulfill this criterion. For the more realistic case of classical
gas molecules, the operative condition is that no two molecules can ever occupy the
same individual phase space state, since they can never be at the same spacetime point.
208 Ruth Kastner
Figure 11.3 States of two classical coins if we forbid them from occupying the
same state.
In the case of the idealized coins, if we forbid them from occupying the same
state, there are now only two available composite Γ-states for Fred and Joe the
ones in which they are in different “heads” or “tails” states (Figure 11.3).
Additionally, since both of these correspond to the distribution “one coin in each
state,” the frequency of this distribution is 2/2 = 1. Meanwhile, the frequency of
this distribution in terms of the Z-space representation, which ignores the phase
space configurations, is just 1/1 = 1. We see that, for the idealized classical coins,
the frequency of occurrence of the distribution is exactly the same in either
representation, so there is no empirical difference between the two spaces they
both predict the same probabilities. Huggett shows that this holds in general, for an
arbitrary number of systems and states (i.e., in which the frequency of a given
distribution may differ from unity, in contrast to the trivial example shown earlier).
Thus, it turns out that there is no empirical support for the Γ-space description
over the Z-space description for classical systems, if they are correctly character-
ized as impenetrable, and thus no empirical support for haecceitism as applying to
classical objects—if one identifies that as a criterion for haecceitism, as does
Huggett. Based on the dissent of French and Redhead (1988), from the criterion
described earlier for transcendental individuality, these authors could, of course,
still argue that both quantum and classical systems possess metaphysical individu-
ality. What is off the table, in view of Huggett’s argument, is the idea there is any
empirical support for a fundamental difference between quantum and classical
systems regarding their status as individuals. While somewhat bewildering for our
intuitions about the difference between classical and quantum objects, we actually
need this result. Why? Because, in keeping with the correspondence principle, the
classical (Maxwell-Boltzmann) distribution must (and does) emerge as a limit from
From Quantum to Classical Physics: The Role of Distinguishability 209
ðMBÞ eβεi
ni ¼ N P βε (11.1a)
e j
j
ðBEÞ 1
ni ¼ (11.1b)
eβðεi μÞ 1
ðFDÞ 1
ni ¼ (11.1c)
eβðεi μÞ þ1
In the quantum distributions Eq. (11.1b) and Eq. (11.1c), the chemical potential μ
(related to the number of degrees of freedom N) necessarily enters for systems with
a fixed N. This will turn out to be significant, as we shall see.
Now, recall that classical distributions can only be wavelike or particle-like. The
particle-like classical distribution (applying to systems such as ideal gases) is just
the Maxwell-Boltzmann distribution Eq. (11.1a). Meanwhile, the classical wave
distribution is what was applied to blackbody radiation prior to the advent of
quantum theory, resulting in the Rayleigh-Jeans distribution and the “ultraviolet
catastrophe”:
2ε2 kT
PðεÞRJ ¼ (11.2)
ðhcÞ2
In view of “wave-particle duality,” it is well known that the quantum distri-
butions interpolate between these two extremes, as follows. Consider the
210 Ruth Kastner
and
X
N¼ n
r r
(11.7)
At this point, it may already be noted that Eq. (11.6) represents a distribution over
the possible energy states, and in that sense is the “Z-space” representation. Since this is
a general derivation (leading to both classical and quantum statistics), it is clear that the
Z-space representation is applicable for both cases, reinforcing Huggett’s observation.
Now, for the case in which the total system of N degrees of freedom is taken as
capable of exchanging energy with its environment at temperature T (the “canon-
ical ensemble”), the probability that the total system is in the state R is given by:
PR ¼ C eβER (11.8)
where β ¼ 1=kT.
The constant of proportionality C is 1/Z, where Z is the total system partition
function:
X
Z¼ j
eβEj (11.9)
So, analogously with Eq. (11.10), the probability that a single system is in state εi is
eβεi eβεi
Pðεi Þ ¼ P βε ¼ (11.14)
je
j ζ
212 Ruth Kastner
We make note of this because for a classical gas of N degrees of freedom, one finds
the average number simply by taking Z(N) for the entire gas as the product of the
individual partition functions:
Z ðN Þ ¼ ζ N (11.15)
So that, using Eq. (11.12), the distribution for ni becomes:
1 ∂ ln Z ðN Þ 1 ∂ ln ζ eβεi
ni ¼ ¼ N ¼ N P βε (11.16)
β ∂εi β ∂εi je
j
Then we can use Eq. (11.12) to get the distribution for average occupation
number ns :
" #
^
1 ∂ ln Z 1 ∂ X∞
ns ¼ ¼ N ln 1 eβðεr μÞ
β ∂εs β ∂εi r¼0
eβðεs μÞ 1
¼ β ð ε μÞ
¼ βðε μÞ (11.20)
1e s e s 1
which is the Bose-Einstein distribution.
The first thing to notice here (besides the fact that we could not use Eq. (11.15)
to obtain this quantum distribution) is that the total number of degrees of freedom,
N, seems to have “disappeared.” It got “dissolved” into infinite sums over all the
possible values of the ns ðεs Þ. Thus, ironically, N has to become a variable in order
to be able to “fix” N for a gas of quantum systems. We recover N as the sum over
the average occupation numbers ns :
X
NN ¼ ns (11.21)
s
The situation is similar for fermions, except that they obey the Pauli Exclusion
principle which limits state occupancy to zero or one. Without going through the
derivation here, we note that, given the restriction described earlier on occupancy,
the inability to express the partition function Z(N) as a direct product of N
individual degrees of freedom yields for the mean occupancy number:
1
nðsFDÞ ¼ (11.22)
eβðεs μÞ þ1
which is the Fermi-Dirac distribution.
Thus, for both bosons and fermions, the chemical potential μ is involved in a
crucial, nonseparable way. Its relation to N is fixed by (11.21), i.e.,
X 1
¼
N¼N (11.23)
s eβðεs μÞ 1
Does the chemical potential play any role in the classical case? Yes, but only
trivially, as a normalizing factor. In the “dilute” (low-occupancy) limit yielding the
classical case, the exponential factor involving μ approaches the particle number N
divided by the single-particle partition function, i.e.,
N N
eβμ ! ¼ P βε (11.24)
ζ e j
j
214 Ruth Kastner
component systems, in that the collective partition function can be obtained from
individual partition functions ζ i in-principle capable of carrying the permutable
label i. This indicates that in the classical limit, the component systems acquire a
form of distinguishability. In the next section, we investigate the nature of this
emergence, in the classical limit, of the capacity to carry a label.
So, from Eq. (11.26) and Eq. (11.27), the quantity ΔpRMS is taken to be:
pffiffiffiffiffiffiffiffiffiffiffi
ΔpRMS ¼ 3mkT (11.28)
and this (despite the fact that it is not a real momentum uncertainty but rather a
root-mean-square error) is plugged into the uncertainty relation to obtain a corres-
ponding thermal wavelength λth :
h
λth ¼ pffiffiffiffiffiffiffiffiffiffiffi (11.29)
3mkT
Clearly, in this context, λth is assumed to be a kind of position uncertainty. It is then
demanded that this be much smaller than the average interparticle spacing d, where:
1=3
V
d¼ (11.30)
N
216 Ruth Kastner
Figure 11.4 One way of picturing the thermal wavelength condition for the
classical limit of a quantum distribution.
the idea being that this condition makes the gas ‘dilute’ (i.e., no particles ever
occupying same position x; and many positions unoccupied, see Figure 11.4.) So
the thermal wavelength condition for classicality becomes:
1=3
h V
pffiffiffiffiffiffiffiffiffiffiffi (11.31)
3mkT N
the gas are not necessarily described by wave packets (i.e., in the quantum limit,
they are plane waves).
Of course, it is well known that classical behavior emerges in the small-
wavelength limit, so rather than try to pretend that a wavelength is a position
uncertainty, one can simply work with the de Broglie wavelength of the average
momentum (still using the Equipartition Theorem) to obtain the same condition
Eq. (11.31). But in this case, one cannot explain the classical behavior in this limit
by saying that the gas is dilute, as pictured in Figure 11.4, because it retains a
nonlocal character arising from the presumed exact wavelengths of its degrees of
freedom. So the question is: Why does condition Eq. (11.31) seem to work so well
as a criterion for the classical limit?
It turns out that if we reexpress Eq. (11.31) in terms of thermal energy kT, we
find the condition (neglecting numerical constants of order unity):
2=
h2 N 3
kT (11.32)
m V
But the quantity on the right hand side is then recognized as the Fermi energy,
which is the chemical potential μ for fermions at T = 0:
2=
h2 N 3
EF ¼ ¼ μðT ¼ 0Þ (11.33)
m V
And in fact, this condition kT E F is also the well-known condition for the
classical limit of the Fermi-Dirac distribution. So what happens in this limit that
could justify a classical description? Once again, the chemical potential μðT Þ has
much to teach us.
Specifically, μ is crucially related to the Helmholtz free energy F, defined as
F ¼ U TS (11.34)
The chemical potential μ is the change in F when adding a degree of freedom (at a
given T and V), i.e.,
ΔF
μ¼ (11.35)
ΔN T , V
Now, μ is of large magnitude and negative in the classical limit of large T, small
N/V, and small λth , as can be seen from the well-known relation (see, e.g., Kelly
2002 for details):
!
V
μ ! kT ln (11.36)
clas Nλ3th
218 Ruth Kastner
11.5 Conclusions
By examining the derivations of the quantum and classical distributions, we have
found that separability of the individual probability spaces fails in the quantum
domain. In contrast, in the classical limit, the probability spaces of the component
degrees of freedom are fully separable. In addition, by examining the role of the
chemical potential μ, we find a clear manifestation of the highly nonclassical
constraints that quantum degrees of freedom impose on one another via the so-
From Quantum to Classical Physics: The Role of Distinguishability 219
Of course, when he made this statement, Einstein was resisting the quantum
nonlocality and/or nonseparability that was evident in the context of the famous
EPR experiment (Einstein, Podolsky, Rosen 1935). It has since become clear that it
is indeed possible, and necessary, to formulate and test the laws of physics without
relying on this sort of classical picture at all levels.
We can trace the emergence of Einstein’s “being thus” in the classical limit by
noting that the latter obtains for high thermal energies kT Eq. (11.32). What can be
deduced from that depends on one’s interpretation of the quantum formalism. In a
unitary-only account, high thermal energies enable decoherence arguments to
proceed (Joos and Zeh 1985), although that account has been criticized on the
basis of entanglement relativity and circularity (e.g., Fields 2010, Dugić and
Jeknić-Dugić 2012, Kastner 2014, 2016a).
Another interpretive approach is to take the projection postulate of von
Neumann as a real physical process (i.e., a “collapse” interpretation). One
such approach is actually a different theory from quantum mechanics: The
220 Ruth Kastner
In contrast, under RTI, the previously discussed form of global interference does
vanish upon the nonunitary transition in which a transaction is actualized. Since
transactions are very frequent in the conditions defining the classical limit, this can
be seen as directly supporting the independence, or being thus, of systems in the
classical limit.
Whichever interpretation one adopts, in the domain of high thermal energies kT,
one gets at least effective determinacy of position over time, and thus a unique
spacetime trajectory for each degree of freedom. Such a trajectory confers the capacity
for a unique label and therefore supports distinguishability of the degree of freedom to
which it corresponds. This does not amount to a haecceitistic label, because it is
From Quantum to Classical Physics: The Role of Distinguishability 221
conferred based on qualitative features of the degrees of freedom (i.e., their trajector-
ies). Nevertheless, as noted by French and Redhead, one may still regard all systems
(classical and quantum) as haecceitistic under a suitable interpretation of individuality.
This chapter takes no position on that metaphysical issue.
Acknowledgments
The author is grateful for valuable correspondence from Jeffrey Bub and Steven
French.
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Modern Physics, 58: 647–688.
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systems,” Pramana, 79: 199–209.
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(trans.). The Born-Einstein Letters. London: Walker and Co.
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reality be considered complete?”, Physical Review, 47: 777–780.
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Stanford Encyclopedia of Philosophy (Fall 2015 Edition), https://plato.stanford.edu/
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macroscopic systems,” Physical Review D, 34: 470–491.
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12
Individuality and the Account of Nonlocality:
The Case for the Particle Ontology in Quantum Physics
michael esfeld
222
Individuality and the Account of Nonlocality 223
emits one particle every morning at 8 a.m. so that one gets an outcome recorded in
the form of exactly one dot on a screen every morning. The question then is what
happens between the emission of the particle from the source and the measurement
record on the screen. Is there a particle that goes through one of the two slits?
A wave that goes through both slits and that contracts afterwards to be recorded as
a dot on a screen? Or something else? The constraint on the ontology and dynamics
that are to answer this question is that they have to account for the characteristic
distribution of the dots on the screen that shows up if one runs this experiment
many times. In other words, the ontology and the dynamics have to explain the
measurement outcome distribution.
It is not possible to infer the law that describes the individual processes that
occur in nature from the rule to calculate the measurement outcome statistics. The
characteristic pattern of the measurement outcome distribution in the double-slit
experiment by no means reveals what happens between the source and the screen.
Trying to make such inferences runs into the famous measurement problem of
quantum physics. The measurement outcomes show a jψ j2 distribution, the law for
the evolution of the wave function ψ is the Schrödinger equation, but the Schrö-
dinger evolution does, in general, not lead to measurement outcomes. More
precisely, the by now standard formulation of the measurement problem is the
one of Maudlin (1995: 7):
2.A The wave-function always evolves in accord with a linear dynamical equation
(e.g., the Schrödinger equation).
3.A Measurements of, e.g., the spin of an electron always (or at least usually) have
determinate outcomes, i.e., at the end of the measurement the measuring device is
either in a state which indicates spin up (and not down) or spin down (and not up).
Any two of these propositions are consistent with one another, but the conjunction
of all three of them is inconsistent. This can be easily illustrated by Schrödinger’s
cat paradox (Schrödinger 1935: 812): If the entire system is completely described
by the wave function, and if the wave function always evolves according to the
Schrödinger equation, then, due to the linearity of this wave equation, superpos-
itions and entangled states will, in general, be preserved. Consequently, a meas-
urement of the cat will, in general, not have a determinate outcome: At the end of
the measurement, the cat will not be in the state of either being alive or being dead.
Hence, the measurement problem is not just a philosophical problem of the
interpretation of a given formalism. It concerns the very formulation of a consistent
quantum theory. Even if one takes (1.A) and (2.A) to define the core formalism of
224 Michael Esfeld
quantum mechanics and abandons (3.A), one has to put forward a formulation of
quantum physics that establishes a link with at least the appearance of determinate
measurement outcomes. If one retains (3.A), one has to develop a formulation of a
quantum theory that goes beyond a theory in which only a wave function and a
linear dynamical equation for the evolution of the wave function figure. Accord-
ingly, the solution space for the formulation of a consistent quantum theory can be
divided into many-worlds theories, rejecting (3.A); collapse theories, rejecting
(2.A); and additional variable theories, rejecting (1.A).
However, research in the last decade has made clear that we do not face three
equally distinct possibilities to solve the measurement problem, but just two: The
main dividing line is between endorsing (3.A) and rejecting it. If one endorses
(3.A), the consequence is not that one has to abandon either (1.A) or (2.A), but that
one has to amend both (1.A) and (2.A). Determinate measurement outcomes as
described in (3.A) are outcomes occurring in ordinary physical space, that is, in
three-dimensional space or four-dimensional spacetime. Hence, endorsing (3.A)
entails being committed to the existence of a determinate configuration of matter in
physical space that constitutes measurement outcomes (such as a live cat or an
apparatus configuration that indicates spin up, etc.). If one does so, one cannot stop
at amending (2.A). The central issue then is not whether or not a collapse term for
the wave function has to be added to the Schrödinger equation, because even with
the addition of such a term, this equation still is an equation for the evolution of the
wave function, by contrast to an equation for the evolution of a configuration of
matter in physical space. Consequently, over and above the Schrödinger equation –
however amended – a law or rule is called for that establishes an explicit link
between the wave function and the configuration of matter in physical space. By
the same token, (1.A) has to be changed in such a way that reference is made to the
configuration of matter in physical space and not just the quantum state as encoded
in the wave function (see Allori et al. 2008). This fact underlines the point made
earlier: We need both a dynamics, filling in proposition (2.A), and an ontology,
filling in proposition (1.A), that specifies the entities to which the dynamics
refers and whose evolution it describes. The way in which these two interplay
then has to account for the measurement outcomes and their distribution, filling in
proposition (3.A).
This point can be further illustrated by a second formulation of the measurement
problem that Maudlin (1995: 11) provides:
1.B The wave function of a system is complete, i.e., the wave function specifies
(directly or indirectly) all of the physical properties of a system.
2.B The wave function always evolves in accord with a deterministic dynamical
equation (e.g., the Schrödinger equation).
Individuality and the Account of Nonlocality 225
3.B Measurement situations which are described by identical initial wave functions
sometimes have different outcomes, and the probability of each possible outcome is
given (at least approximately) by the Born Rule.
Again, any two of these propositions are consistent with one another, but the
conjunction of all three of them is inconsistent. Again, the issue is what the law is
and what the physical entities are to which the law refers such that, if one takes
(3.B) for granted, configurations of matter in physical space that constitute definite
measurement outcomes are accounted for.
All mathematical formulations of nonrelativistic quantum mechanics work with a
formalism in terms of a definite, finite number of point particles and a wave function
that is attributed to these particles, with the basic law for the evolution of the wave
function being the Schrödinger equation. The wave function is defined on configur-
ation space, thereby taking for granted that the particles have a position in three-
dimensional space: For N particles, the configuration space has 3N dimensions so
that each point of configuration space represents a possible configuration of the N
particles in three-dimensional space. This fact speaks also in quantum physics
against regarding configuration space as the physical space, because its dimension
depends on a definite number of particles admitted in three-dimensional space.
Even if one pursues an ontology of configuration space being the physical space
in quantum physics, one can add further stuff than the wave function to configur-
ation space, such as, e.g., the position of a world-particle in configuration space, or
take the wave function to collapse occasionally in configuration space in order to
solve the measurement problem (see Albert 2015: chapters 6–8). However, the
problem remains how to connect what there is in configuration space and its
evolution with our experience of three-dimensional physical objects, their relative
positions, and motions. That experience is the main reason to retain (3.A and 3.B).
This chapter is situated in the framework that envisages abandoning (3.A and 3.B)
only as a last resort, that is, only in case it turned out that the consequences of the options
that endorse (3.A and 3.B) were even more unpalatable than the ones of rejecting (3.A
and 3.B). It seeks to make a contribution to assessing the options that are available in this
framework, namely, the option that starts from amending the Schrödinger dynamics by
admitting a dynamics of the collapse of the wave function – (not 2.A and 2.B); see next
section – and the option that starts from admitting particles, although the information
about their positions is not contained in the wave function – (not 1.A and 1.B); Section
12.3. The chapter closes with a few remarks on quantum field theory (Section 12.4).
collapse rate. Between two successive jumps, the wave function Ψt evolves
according to the usual Schrödinger equation. At the time of a jump the
kth component of the wave function Ψt undergoes an instantaneous collapse
according to
1=2
Lxxk Ψt ðx1 ; . . . ; xk ; . . . ; xN Þ
Ψt ðx1 ; . . . ; xk ; . . . ; xN Þ↦ 1= (12.1)
x 2
L
xk Ψt
where the localization operator Lxxk is given as a multiplication operator of the form
1 2
e2σ2 ðxk xÞ
1
Lxxk ≔ (12.2)
ð2πσ2 Þ3=2
and x, the center of the collapse, is a random position distributed according to the
x 1=2 2
probability density pðxÞ ¼ Lxk Ψt
. This modified Schrödinger evolution
captures in a mathematically precise way what the collapse postulate in the
textbooks introduces by a fiat, namely the collapse of the wave function so that
it can represent localized objects in physical space, including in particular meas-
urement outcomes. GRW thereby introduces two additional parameters, the mean
rate λ as well as the width σ of the localization operator. An accepted value of λ is
of the order of 1015s1. This value implies that the spontaneous localization
process for a single particle occurs only at astronomical time scales of the order
of 1015s, while for a macroscopic system of N~1023 particles, the collapse happens
so fast that possible superpositions are resolved long before they would be experi-
mentally observable. Moreover, the value of σ can be regarded as localization
width; an accepted value is of the order of 107m.
A further law then is needed to link this modified Schrödinger equation with a
wave ontology in the guise of an ontology of a continuous matter density field
mt ðxÞ in space:
XN ð
mt ðxÞ ¼ mk dx1 . . . dxN δ3 ðx xk Þ jΨt ðx1 ; . . . ; xN Þj2 (12.3)
k¼1
of fundamental physical systems. There is just one object in the universe, namely a
matter density field that stretches out throughout space and that has varying
degrees of density at different points of space, with these degrees changing in
time. Hence, there are no individual systems in nature according to this theory so
that the issue of identity and distinguishability of individual quantum systems does
not arise. That notwithstanding, this theory accounts for measurement outcomes
that appear as individual particle outcomes in terms of a spontaneous contraction of
the matter density field at certain locations. Position thus is distinguished in the
form of degrees of matter density at points of space. It is the only fundamental
physical property, as Allori et al. (2014) point out:
Moreover, the matter that we postulate in GRWm and whose density is given by the m
function does not ipso facto have any such properties as mass or charge; it can only assume
various levels of density.
(Allori et al. 2014: 331–332)
of the tails of the wave function. This problem arises from the fact that the GRW
formalism mathematically implements the collapse postulate by multiplying the
wave function with a Gaussian, such that the collapsed wave function, although
being sharply peaked in a small region of configuration space, does not actually
vanish outside that region; it has tails spreading to infinity. On this basis, one can
object that GRWm does not achieve its aim, namely to describe measurement
outcomes in the form of macrophysical objects having a definite position (see e.g.,
Maudlin 2010: 135–138). However, one can also make a case for the view that this
mathematical fact does not prevent GRWm from accounting for definite measure-
ment outcomes in physical space (see e.g., Wallace 2014, Egg and Esfeld 2015:
section 3).
The main drawback of GRWm, arguably, is its account of quantum nonlocality,
which occurs when the wave function collapses all over space. Consider a simple
example, namely the thought experiment of one particle in a box that Einstein
presented at the Solvay conference in 1927 (the following presentation is based on
de Broglie’s version of the thought experiment in de Broglie 1964: 28–29, and on
Norsen 2005). The box is split in two halves that are sent in opposite directions,
say from Brussels to Paris and Tokyo. When the half-box arriving in Tokyo is
opened and found to be empty, there is on all accounts of quantum mechanics that
acknowledge that measurements have outcomes a fact that the particle is in the
half-box in Paris.
On GRWm, the particle is a matter density field that stretches over the whole
box and that is split in two halves of equal density when the box is split, these
matter densities travelling in opposite directions. Upon interaction with a meas-
urement device, one of these matter densities (the one in Tokyo in the example
given earlier) vanishes, while the matter density in the other half-box (the one in
Paris) increases so that the whole matter is concentrated in one of the half-boxes.
One might be tempted to say that some matter travels from Tokyo to Paris;
however, because it is impossible to assign any finite velocity to this travel, the
use of the term ‘travel’ is inappropriate. For lack of a better expression, let us say
that some matter is delocated from Tokyo to Paris (this expression was proposed
by Matthias Egg; see Egg and Esfeld 2014: 193). For even if the spontaneous
localization of the wave function is conceived as a continuous process, as in
Ghirardi, Pearle, and Rimini (1990), the time it takes for the matter density to
disappear in one place and to reappear in another place does not depend on the
distance between the two places. This delocation of matter, which is not a travel
with any finite velocity, is a quite mysterious process that the GRWm ontology
asks us to countenance.
Apart from the matter density ontology, there is another ontology available
for the GRW collapse formalism. This ontology goes back to Bell (2004:
230 Michael Esfeld
dxk ℏ rk ψ
¼ Im ðx1 ; . . . ; xN Þ (12.4)
dt mk ψ
This equation yields the evolution of the kth particle at a time t as depending on,
via the wave function, the position of all the other particles at that time.
3. Schrödinger equation: The wave function always evolves according to the
usual Schrödinger equation.
4. Typicality measure: On the basis of the universal wave function Ψ, a typicality
measure can be defined in terms of the jΨj2 -density. Given that typicality
measure, it can then be shown that for almost all initial conditions, the distribu-
tion of particle configurations in an ensemble of subsystems of the universe that
admit of a wave function ψ of their own (known as effective wave function) is a
jψ j2 -distribution. A universe in which this distribution of the particles in
subconfigurations obtains is considered to be in quantum equilibrium.
systems within the universe, with ψ being the effective wave function of the
particular systems in question (see Dürr et al. 2013: chapter 2).
Axiom (1) defines the ontology of the theory. The universe is a configuration of
point particles that, consequently, always have a precise position relative to one
another: They stand in determinate distance relations to each other. Indeed, BM
does not require the commitment to an absolute space in which the particles are
embedded and an absolute time in which their configuration evolves. The geom-
etry and the time with its metric can be conceived as a mere means to represent the
particle configuration and its change, that is, the change in the relative distances of
the particles (see Esfeld and Deckert 2017: chapter 3.2 for the philosophical
argument, and Vassallo and Ip 2016 for a relationalist formulation of BM).
Consequently, the particles are individuals that are absolutely discernible by their
position in a configuration. They have an identity in time that is provided by the
continuous trajectory that their motion, i.e., the change in their relative positions,
traces out. In the framework of relationalism about space and time, one can employ
the distance to individuate the particles, so that the particles in BM are not bare
particulars: The distance relations make them the objects that they are and account
for their numerical plurality (see Esfeld and Deckert 2017: chapter 2.1).
This ontology contains the core of the solution to the measurement problem that
BM provides: There are always point particles with definite positions, and these
particles compose the macroscopic objects. Hence, Schrödinger’s cat is always
either alive or dead, a radioactive atom is always either decayed or not decayed, an
electron in the double-slit experiment with both slides open always goes either
through the upper or the lower slit, etc. That is to say: There are no superpositions
of anything in nature. Superpositions concern only the wave function and its
dynamics according to the Schrödinger equation – Axiom (3) – but not the matter –
the objects – that exist in the world, although, of course, the superpositions in the
wave function and its dynamics can be relevant for the explanation of trajectories
of the matter in physical space.
Axiom (2) then provides the particle dynamics. As it is evident from the guiding
equation, the evolution of the position of any particle depends, strictly speaking, on
the position of all the other particles in the universe via the wave function. This is
the manner in which BM implements quantum nonlocality: It is correlated particle
motion, with the correlation being established by the wave function and being
independent of the distance of the particles. However, it is only correlated motion.
By contrast to the collapse dynamics with the wave function representing an
ontology of a wave in physical space in the guise of a matter density field
(GRWm), there is never a delocation of matter in physical space. There are always
only particles, moving on continuous trajectories and thus with a finite velocity in
physical space (in the relativistic setting, a velocity that is not greater than the
Individuality and the Account of Nonlocality 233
velocity of light), with their motions being correlated with one another as a
primitive matter of fact. The particle ontology and this dynamics provide a
considerable advantage of the Bohmian account of measurement results over the
collapse one: The particles are always there in space, instead of spontaneously
localizing upon the collapse of the wave function – either from nowhere, as the
flashes do in GRWf, or being delocated all over space, as the matter density does in
GRWm.
The ontology of BM – known as the primitive ontology, that is, the referent of
the formalism – is given by the particles and their relative distances as well as the
change of these distances (i.e., the particle motion). The wave function is defined
by its dynamical role of providing the velocity field along which the particles
move. The wave function is set out on configuration space. It can be conceived as a
wave or a field; but, then, it is a wave or field on configuration space by contrast to
an entity in physical space over and above the particles: The wave function does
not have values at the points of physical space. It is therefore misleading to
consider BM as an ontology of a dualism of particles and a wave and to imagine
the wave function as a pilot wave that guides or pilots the particles in physical
space. These are metaphorical ways of speaking, since the wave function cannot be
or represent a wave in physical space.
The wave function is nomological in the sense that it is introduced and defined
through its dynamical role for the particle motion. Consequent upon its being
nomological in that sense, all the stances in the metaphysics of laws of nature are
applicable to the wave function. In particular, in recent years, a Bohmian
Humeanism has been developed according to which the universal wave function
is fixed by or supervenes on the history of the particle positions, being a variable
that figures in the Humean best system (see Miller 2014, Esfeld 2014, Callender
2015, Bhogal and Perry 2017). This stance is applicable to all the theories that
introduce the wave function through its dynamical role for the evolution of the
configuration of matter in physical space, including GRWm and GRWf (see
Dowker and Herbauts 2005 for a precise physical model based on GRWf).
This Humean stance makes clear that one can be a scientific realist without
subscribing to an ontological commitment to the wave function and without falling
into instrumentalism about the wave function: The role of the wave function is, in the
first place, a dynamical one through its position in the law of motion – Axiom (2) –
with its role for the calculation of measurement outcome statistics deriving from that
nomological role – Axiom (4). But the laws, including the dynamical parameters
that figure in them, can be the axioms of the system that achieves the best balance
between simplicity and information in representing the particle motion, as on
Humeanism, instead of being entities that exist in the physical world over and above
the configuration of matter (see Esfeld and Deckert 2017: chapter 2.3).
234 Michael Esfeld
The particle positions are an additional parameter in BM in the sense that the
wave function and its evolution according to the Schrödinger equation do not
contain the information about the exact particle positions and their evolution. All
there is to the Bohmian particles are their relative positions, that is, the distances
among them. Although the parameter of particle mass figures in the guiding
equation – Axiom (4) – mass cannot be considered as an intrinsic property of the
particles in BM. It is not situated where the particles are, but rather in the
superposed wave packets. The same holds for the charge (see e.g., Brown,
Dewdney, and Horton 1995, Brown, Elby, and Weingard 1996, and references
therein; see also, most recently, Pylkkänen, Hiley, and Pättiniemi 2015 and Esfeld
et al. 2017).
The particle positions are the only additional parameter. Theorems like the one
of Kochen and Specker (1967) prove that it is not possible to take the operators or
observables of quantum mechanics to have definite values independently of
measurement contexts, on pain of violating the predictions of quantum mechanics
for measurement outcome statistics. Nonetheless, these theorems leave the possi-
bility open to admit one additional parameter of the quantum objects that has a
definite value, without the precise information about that value figuring in the wave
function. The natural choice for the additional parameter then is position, since all
measurement outcomes consist in macroscopically recorded positions of discrete
objects. In making this choice, one therefore lays the ground for solving the
measurement problem. By contrast, pursuing a strategy that accords definite values
to different parameters over time – as done in so-called modal interpretations apart
from BM – falls victim to the measurement problem, as has been proven by
Maudlin (1995: 13–14).
It is often maintained that, apart from position, all the other operators or
observables are treated as contextual properties in BM, in the sense that they
acquire a definite value, signifying that they are realized as properties of quantum
systems, only in the context of measurements. But this is wrong-headed. The
operators or observables defined on a Hilbert space are never properties of
anything. Suggesting that measurements somehow bring into existence properties
that are contextual, in the sense that they do not exist independently of measure-
ment situations, is a confused manner of talking. The operators or observables
defined on a Hilbert space are instruments that provide information about how the
quantum objects behave in certain situations, and that finally comes down to
information about how their positions develop. Thus, there is no property of spin
that quantum objects possess, and there is no contextual property of a definite
value of spin in a certain direction that quantum objects acquire in the context of
measurement situations. What these measurements do, essentially, is to provide
information about particle positions. For instance, the measurement result “spin up”
Individuality and the Account of Nonlocality 235
themselves laws – not even the universal wave function in BM is a law, because the
theory admits of models with different universal wave functions). It then depends on
the stance that one takes with respect to laws whether or not one accords the
dynamical structure a place in the ontology over and above the basic or primitive
ontology or takes it merely to be a means of representation (as on Humeanism).
Not only the operators in quantum mechanics, but also the classical parameters
of mass and charge belong to the dynamical structure. Mass and charge are also
introduced in classical mechanics through their functional role for the particle
motion (see Mach 1919: 241 on mass in Newtonian mechanics). Consequently,
also in classical mechanics, the distinction is available between, on the one hand,
the primitive ontology of the theory in the guise of particle positions and particle
motion and, on the other hand, the dynamical structure in terms of mass and
charge, forces and fields, and energy and potentials introduced through their causal
role for particle motion. Thus, also in classical mechanics, one can suspend or
refuse an ontological commitment to the dynamical structure, being committed
only to the particle positions and their change (see Hall 2009: section 5.2 and
Esfeld and Deckert 2017: chapter 2.3). In short, quantum physics simply makes
evident that it is mandatory to draw a distinction that was already there from the
very beginning.
Although the particle positions make up the ontology in BM, there are limits
to their accessibility. These limits are given in Axiom (4), which implies that
BM cannot make more precise predictions about measurement outcomes on
subsystems of the universe than those generated by using the Born Rule. The
link between the dynamical laws of BM and the jΨj2 -density on the level of the
universal wave function as typicality measure is at least as tight as the link
between the dynamical laws of classical mechanics in the Hamiltonian formu-
lation and the Lebesgue measure (see Goldstein and Struyve 2007). Indeed, in
BM, the quantum probabilities have the same status as the probabilities in
classical statistical mechanics: They enter into the theory as the answer to the
question of what evolution of a given system we can typically expect in
situations in which the evolution of the system is highly sensitive to slight
variations of its initial conditions, and we do not know the exact initial condi-
tions. In such situations, the deterministic laws of motion cannot be employed
to generate deterministic predictions. Nonetheless, the probabilities are object-
ive: They capture patterns in the evolution of the objects in the universe that
show up when one considers many situations of the same type, such as many
coin tosses in classical physics or running the double-slit experiment with many
particles in quantum physics. To put it in a nutshell, the Bohmian universe is
like a classical universe in which not the motion of the planets, but the coin toss
is the standard situation.
Individuality and the Account of Nonlocality 237
However, one may wonder whether, if one lived in a classical universe of coin
tosses, one would endorse the Hamiltonian laws as providing the dynamics of that
universe; Hamiltonian mechanics could then well be a minority position like BM
today. The reason for endorsing Hamiltonian mechanics in that classical case
would be the same as the one for endorsing BM in the quantum case: We need
not only statistical predictions, but also a dynamics that describes the individual
processes occurring in nature, on pain of falling into a measurement problem in the
guise of the inability to account for the occurrence of determinate measurement
outcomes. That is to say: In a classical universe of coin tosses, the Hamiltonian
laws would be useless for predictions, as the calculation of Bohmian particle
trajectories is useless for predictions. But both are indispensable for physics as a
theory of nature. This is not only philosophical ontology, it is the business of
physics to provide dynamical laws that apply to the individual processes in nature.
Nonetheless, in contrast to the relationship between classical statistical
mechanics and classical mechanics, there is a principled limit to the accessibility
of initial conditions of physical systems in the quantum case. That limit becomes
evident, for instance, in the famous Heisenberg uncertainty relations. It is trivial
that measurement is an interaction, so that the measurement changes the measured
system, and can thus not simply reveal the position and velocity that it had
independently of the measurement interaction. Some limit to the accessibility of
physical systems may follow from the triviality that any measurement is an
interaction. Thus, it is well known in classical physics that no observer within
the universe could obtain the data that Laplace’s demon would need for its
predictions. However, the quantum case is not simply an illustration of that
triviality, since there is a precise principled limit of the epistemic accessibility of
quantum systems (as illustrated, for instance, by the Heisenberg uncertainty
relations). In BM, this principled limit follows from applying the theory as defined
earlier to measurement interactions (see Dürr et al. 2013: chapter 2).
In the last resort, of course, it is the particle motion in the world that makes
possible stable particle correlations such that one particle configuration (say a
measurement device or a brain) records the position and traces the motion of other
particles and particle configurations, and it is also the particle motion in the world
that puts a limit on such correlations. The laws of BM, including the typicality
measure and the assumption that the actual universe is a typical Bohmian universe,
bring out these facts about the actual particle motion. Again, the wave function in
configuration space represents that particle motion; it is not the wave function that
puts the limit on the epistemic accessibility of the particle positions, although we
understand that limit by representing the particle motion through the wave func-
tion. Instead of taking this limit to be a drawback and hoping for a physical theory
like classical mechanics in which there are paradigmatic cases of deterministic
238 Michael Esfeld
laws of motion enabling deterministic predictions (e.g., the motion of the planets),
it is fortunate and by no means trivial that there are such stable particle correlations
in the universe at all so that we can represent actual particle positions and motions
and make reliable predictions.
As these considerations make clear, posing a limit to the accessibility of the
objects in physical space is by no means a feature that is peculiar to BM. Such a
limit applies not only to the particles in BM, but, for instance, also to the flash
distribution in GRWf and the matter density field in GRWm (see Cowan and
Tumulka 2016). In any case of a quantum ontology of objects in physical space, if
these objects were fully accessible, we could employ the thus gained information
to exploit quantum nonlocality for superluminal signaling. This limited accessibil-
ity of the particle configuration, the flash distribution or the matter density field
confirms that if one endorses proposition (3.A and 3.B) of the measurement
problem, i.e., determinate measurement outcomes whose statistical distributions
are given by the Born Rule, the situation is not that one has to reject either
proposition (1.A and 1.B) or proposition (2.A and 2.B). One has in this case to
modify both proposition (1.A and 1.B) and proposition (2.A and 2.B). If one starts
from admitting particle positions that are not revealed by the wave function –
rejection of proposition (1. A and 1.B) – one can retain the Schrödinger dynamics
for the wave function – proposition (2.A and 2.B) – but then this is not the
complete dynamics: The central dynamical law is the law of the evolution of the
additional variables, namely, the guiding equation that tells us how the particle
positions evolve in physical space. If one starts from amending the Schrödinger
equation by collapse parameters – rejection of proposition (2.A and 2.B) – one can
retain the wave function and its dynamics as describing the evolution of the objects
in physical space – flashes, matter density field; proposition (1.A and 1.B) – but
that distribution then nevertheless is “hidden” in the sense that it follows from the
theory that it is not fully accessible.
At least three conclusions can be drawn from this situation:
1. Any theory that admits definite measurement outcomes distinguishes position –
be it particle positions, positions of flashes, or values of matter density at points
of space. All the other observables are accounted for on this basis.
2. If one is not prepared to accept a principled limit to the epistemic accessibility
of the objects in physical space, one remains trapped by the measurement
problem, because one then does not have a dynamics at one’s disposal that
accounts for determinate measurement outcomes.
3. The solution space to the measurement problem reduces to this one: Either one
abandons determinate measurement outcomes in physical space, in which case
one can retain the propositions of the wave function being complete and its
Individuality and the Account of Nonlocality 239
evolving always according to the Schrödinger equation; one then has to come
up with an Everett-style account of why it appears to us as if there were
determinate measurement outcomes in physical space. Or, one retains determin-
ate measurement outcomes in physical space, and the account of these meas-
urement outcomes then commits one to endorsing a distribution of objects in
physical space whose evolution cannot be given by the Schrödinger equation
and is not fully accessible.
BM can be seen as the answer to the following question: What is the minimal
deviation from classical mechanics that is necessary in order to obtain quantum
mechanics? BM shows that the physical ontology can remain the same – point
particles moving on continuous trajectories – and that the status of probabilities can
remain unchanged. What has to change is the dynamics, that is, a wave function
parameter has to be introduced with the wave function binding the evolution of the
particle positions together independently of their distance in physical space.
That notwithstanding, this conceptualization of quantum nonlocality is like
Newtonian gravitation, in that there is never any matter instantaneously delocated
in space. As in Newtonian gravitation, the distribution of the particle positions,
velocities, and masses all over space at any time t fixes the acceleration of the
particles at that t, so in BM the distribution of the particle positions and the
universal wave function at any time t fix the velocity of the particles at that t. Of
course, Newtonian gravitation concerns all particles indiscriminately and depends
on the square of their distance, whereas quantum nonlocality is de facto highly
selective, i.e., concerns de facto only specific particles, and is independent of their
distance. Nonetheless, in both cases, nonlocality means that there are correlations
in the particle motion without these correlations being mediated by the instantan-
eous transport of anything all over space. There is no reason to change more.
Doing so only leads to unpalatable consequences beyond the quantum nonlocality
with which one has to come to terms anyway.
to say: Bohmian QFT has to rely on cut-offs, as does the standard model of particle
physics when it comes to dynamical laws of interactions (by contrast to scattering
theory). Given appropriate cut-offs, one can formulate a Bohmian theory for QFT
in the same way as for quantum mechanics: On what is known as Bohmian Dirac
sea QFT, the ontology is one of a very large, but finite and fixed number of
permanent point particles that move on continuous trajectories as given by a
deterministic dynamical law (guiding equation) by means of the universal wave
function.
More precisely, one can define a ground state for these particles that is a state of
equilibrium. This state is one of a homogeneous particle motion, in the sense that
the particle interactions cancel each other out. Consequently, the particle motion is
not accessible. This state corresponds to what is known as the vacuum state.
However, on this view, it is not at all a vacuum, but a sea full of particles (known
as the Dirac sea) in which the particles are not accessible. What is accessible, and
what is effectively modeled by the Fock space formalism of calculating measure-
ment outcome statistics, are the excitations of this ground state that show up in
what appears to be particle creation and annihilation events. Again, by defining a
typicality measure on the level of the universal wave function, one can derive the
predictions of measurement outcome statistics in the guise of, in this case, statistics
of excitation events from the ground state. Thus, again, the quantum probabilities
are due to a – principled – limit to the accessibility of the particle motion (see Colin
and Struyve 2007, Esfeld and Deckert 2017: chapter 4.2).
When pursuing a solution to the measurement problem in terms of an ontology
of particles, it is worthwhile to go down all the Bohmian way also in QFT. What is
known as Bell-type Bohmian QFT (Bell 2004: chapter 19, and further elaborated
on in Dürr et al. 2005) goes only half the way down: On this theory, the particles
come into and go out of existence with statistical jumps between sectors of
different particle numbers in the dynamics. However, this proposal amounts to
elevating what is known as quasi-particles that are dependent on the contingent
choice of a reference frame to the status of particles in the ontology. Furthermore, it
is committed to absolute space as the substance in which the particles come into
and go out of existence. The Bell-type (quasi-) particles are much like the GRWf
flashes, apart from the fact that they can persist for a limited time, instead of being
ephemeral. By contrast, on the Bohmian Dirac sea theory, the particles are
permanent so that they can be conceived of as being individuated by the distances
among them in any given configuration and as having an identity in change
through the continuous trajectories that their motion trace out. Consequently, there
is no need for a commitment to a surplus structure in the guise of absolute space
and time in the ontology; there is no need for a medium in which the particles exist
(viz. come into and go out of existence). Probabilities then come in through linking
242 Michael Esfeld
the deterministic dynamics with a typicality measure. Filling negative energy states
with particles is no problem on this ontology, because the only property of the
particles is their position; energy is not a property of anything, but a variable in the
formalism to track particle motion.
In sum, on the Bohmian Dirac sea ontology, the account of measurement
outcomes is of the same type as in Bohmian quantum mechanics, with the
difference that there are many more particles in the sea than one would expect in
an ontology of particle positions that are only given by the distances among the
particles (corresponding to empty space in the representation in terms of a space in
which the particles are embedded). The account again has two stages: The ontol-
ogy of particles – in this case, the excitations of particles against the background of
the particle motion in the Dirac sea – accounts for the presence of the measured
quantum objects as well as the one of the macroscopic systems; the latter are
constituted by these particle excitations, with the particle dynamics that yields
these excitations explaining their stability. The measurement outcome statistics
then are accounted for in terms of the limited accessibility of the quantum particles
by means of defining a typicality measure from which one then deduces the
formalism to calculate these statistics.
In any case, the objects that one poses in an ontology of quantum physics are
theoretical entities. They are admitted to explain the phenomena as given by the
measurement outcome statistics. That is why the solution to the measurement
problem is the standard for assessing these proposals. In any case, if one admits
quantum objects in physical space beyond the wave function, there is a limit to
their accessibility; the wave function has, in this case, an exclusively dynamical
status, namely, yielding the dynamics for these objects. The Bohmian solution to
the measurement problem provides the least deviation from the ontology of
classical mechanics that is necessary to accommodate quantum physics, both in
the case of quantum mechanics and in the case of quantum field theory. There is no
cogent reason to go beyond that minimum.
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13
Beyond Loophole-Free Experiments:
A Search for Nonergodicity
alejandro a. hnilo
13.1 Introduction
Quantum mechanics (QM) has been controversial since its very inception. In the
climactic point of a famous debate with Niels Bohr, Albert Einstein argued that the
correlations between measurements performed on distant entangled particles dem-
onstrated that the description of physical reality provided by QM was incomplete
(note the subtle difference: There was no objection on the completeness of the
theory, it was the description of physical reality that was claimed to be incom-
plete). Bohr answered that the idea of a “physical reality” independent of an
observer was meaningless. In my opinion, Einstein’s argument was impeccable
and Bohr’s reply was dangerously close to Saint Bellarmino’s refutation to
Galileo’s observation of mountains in the moon. Yet, quite incomprehensibly to
me, physicists’ public opinion gave the reason to Bohr. Using a football analogy, it
was a beautiful Einstein’s goal disallowed by off-side (the most arcane of foot-
ball’s rules, see Figure 13.1). The history of QM would have been very different if
Bohr had replied in 1935: “Wow Albert, you have a good point. I don’t know. But
let’s use this new theory. There is a lot of exciting work to do. Perhaps, by
applying the theory to new problems, this issue of completeness will become
clearer.”
Bohr’s actual answer or point of view (the so-called Copenhagen interpretation
of QM) opens the door to paradoxes that even have negative social consequences.
I quote E. T. Jaynes’ opinion (Jaynes 1980):
Defenders of the (quantum) theory say that this notion (“real physical situation”) is
philosophically naïve, a throwback to outmoded ways of thinking, and that recognition
of this constitutes deep new wisdom about the nature of human knowledge. I say that it
constitutes a violent irrationality, that somewhere in this theory the distinction between
reality and our knowledge of reality has become lost, and the result has more the character
of medieval necromancy than science.
(Jaynes 1980: 42)
245
246 Alejandro A. Hnilo
Figure 13.1 Einstein’s goal in the “EPR paradox” argument was disallowed by the
referee of public opinion. By the way, Bohr did play as goalkeeper (and a very
good one) in his youth.
The analysis of how measured numbers are inserted into Bell’s inequalities shows
that the actual logical relationship is:
Locality + Realism + “ergodicity” ) Bell’s inequalities are valid.
That naturally leads to speculating that it is “ergodicity,” and not LR, that has been
disproved by the recent loophole-free experiments. It is convenient to recall here
that the ergodic hypothesis means that the average of the dynamical variables (of
the system being considered) calculated over the phase space, which is called
ensemble average, is equal to the average obtained over the actual evolution of the
system, or time average. The interest of the ergodic hypothesis is that ensemble
averages are far easier to calculate than time averages. The former can be deduced
from the system’s symmetry or from laws of conservation. The latter requires the
complete solution of the equations of motion of the system. But the ergodic
hypothesis is not always valid. The Fermi-Pasta-Ulam system (a series of coupled
nonlinear oscillators) is the best known example of a nonergodic system. The idea
that ergodicity may be involved in the QM vs. LR controversy is not new. It was
indicated long ago by V. Buonomano (1978). It has probably passed mostly
unnoticed because he linked the nonergodic possibility (which is a general argu-
ment) to specific mechanisms, or loopholes, which always have a conspiratorial
flavor. The issue has been updated by Khrennikov (2017 and references therein).
Beyond Loophole-Free Experiments: A Search for Nonergodicity 249
I stress that I use “ergodicity” here as a shorthand for naming a set of hypotheses
that retrieve the validity of the usual form of the Bell’s inequalities. In this set,
some hypotheses are weaker and others stronger than strictly speaking ergodicity.
The origin and meaning of the hypotheses in the set have been discussed in detail
in Hnilo (2013). In that paper, an example of a LR model relevant to the Bell’s
experiment, which is able to violate ergodicity (in its broadest sense), is also
presented. The main conclusion is that nonergodicity (in its broadest sense) is a
condition necessary, but not sufficient, to violate Bell’s inequalities without violat-
ing LR. Be aware that that model was devised as an example of how ergodicity can
be reasonably violated, but that it does not survive (it was not intended to survive)
the performed loophole-free Bell’s experiments.
The pertinent question now is: How to determine whether ergodicity is violated
or not in a Bell’s experiment? Pragmatically speaking, randomness ) ergodicity,
hence nonergodicity ) nonrandomness (I leave aside the precise definition of
“randomness,” which is a difficult issue). In consequence, the study of the devi-
ations from randomness in a time series of measurements in a Bell’s experiment
may reveal that it is ergodicity (instead of LR) that has been refuted in the recent
loophole-free experiments. It is worth noting here that a Bell’s experiment is
equivalent to the “quantum link” of a quantum key distribution (QKD) setup.
Therefore, such deviation from randomness may imply a QKD vulnerability of a
fundamental origin. This possibility has practical consequences.
In summary, the key to test this way to save the validity of LR in nature is the
search for evidence of nonergodic dynamics in Bell’s experiments. This evidence
cannot be found by measuring average magnitudes, as it was done in almost all
Bell’s experiments performed until now, but by analyzing the time evolution of
these magnitudes. This requires time-resolved acquisition of data, a procedure also
known as “stamping” or “tagging” the time values when the entangled particles are
detected. If the analysis of the time series obtained in this way showed the
existence of nonergodic dynamics, QM could be then interpreted as the steady-
state approximation of a more general theory, still unknown. Recall that ergodicity
) steady-state and that the states of QM belong to a Hilbert space, where the
ergodic hypothesis is valid. In any case, such revelation would entail rewriting
the basis of microscopic physics. It would also give the reason to Einstein in the
famous debate with Bohr.
In the next section, the reason why the ergodic hypothesis is necessary to insert
measured data into the Bell’s inequalities is reviewed. A brief analysis of the case
of an experiment using Greenberger-Horne-Zeilinger (GHZ) states is presented. In
Section 13.3, the basic idea of the experiment to detect nonergodic behavior is
presented, together with the discussion of some existing antecedents. In Section
13.4, I indulge myself in some free considerations.
250 Alejandro A. Hnilo
α β
L
A B
(+) (+)
(–) (–)
Locality also implies that {α,β,λ} are statistically independent variables: PA(α,λ) =
PA(α)PA(λ), and that ρ(λ) is independent of {α,β}. The set of all these properties
often receive the specific name of measurement independence. To enforce it in the
practice, outstanding experiments have been performed. I point out the ones by
Giustina et al. (2015), Shalm et al. (2015), and Weihs et al. (1998). In these
experiments, the angle setting {α,β} is randomly changed in a time shorter than
L/c. In what follows, measurement independence is taken for granted.
Given {x,y 0, X x’, Y y’} the following equality holds: 1 xy – xy’ +
x’y + x’y’ – Xy – Yx’ 0. Choosing x = PA(α,λ), x’ = PA(α’,λ), y = PB(β,λ), y’ =
PB(β’,λ) and X = Y = 1, where {α,β,α’,β’} are different analyzers’ orientations,
Ð and
after integration over the space of the hidden variables applying dλ.ρ(λ) and
Eq. (13.2), we get:
1 PAB ðα; βÞ PAB α; β’ þ PAB ðα’ ; βÞ þ PAB α’ ; β’ PB ðβÞ PA ðα’ Þ
J0
(13.3)
which is the Clauser-Horne (CH) inequality. The QM predictions violate it. That is,
for the entangled state jφ+i = (1/√2){jxa,xbi + jya,ybi}, PAB(α,β) = ½.cos2(α,β), and
choosing {α,β,α’,β’} = {0, π/8, π/4, 3π/8}, we get: J = 0.427 – 0.073 + 0.427 +
0.427 – ½ – ½ = 0.208, violating the right-hand side (r.h.s.) of the inequality. It is
concluded that QM is incompatible with at least one of the assumptions (i.e.,
locality and/or realism).
This equation represents the result of the following real process: Set A = α during
the time interval [θ, θ + Δt], sum up the number of photons detected after the
analyzer A, and obtain PA(α) as the ratio of detected over incident photons. But the
integrals in Eq. (13.1) and Eq. (13.4) are different: The former is an average over
the possible states of the hidden variables; the latter is an average over time. They
are not necessarily equal. Assuming they are equal means assuming the ergodic
hypothesis valid. In all Bell’s experiments to date, this assumption has been
(implicitly) made to insert measured numbers into derived inequalities. If this
assumption is not made, the insertion of measured data into the derived Bell’s
252 Alejandro A. Hnilo
measurements are performed during time and that it is impossible to measure with
two different angle settings simultaneously. It is impossible to travel in time to
measure again, at the same value of time, with a different angle setting.
where x,y are the planes of linear polarization and 1, 2, 3 label the entangled
photons. The Pauli operator σl, which acts: σljxi = jyi, σljyi = jxi, represents
a polarization analyzer that fully transmits a photon linearly polarized at 45o of the
x,y axes. The Pauli operator σr, which acts: σrjxi = ijyi, σrjyi = ijxi, does the
same with circularly polarized photons (rightwards: transmitted, leftwards:
reflected). These operators are applied to each particle of the state and are chosen
in a random and independent way in three remote stations. The configuration of the
whole setup can be lll (the three photons find linear polarization analyzers), llr
(photons 1 and 2 find linear analyzers, the third one a circular one), etc. with equal
probability. There are eight possible configurations. A transmitted photon means a
result +1, a reflected one 1, in each station. The result of a complete measurement
is the product of the results obtained at the three stations. That is, if the photons are
transmitted at the first two stations and reflected at the third one, the result of the
measurement is: (+1) (+1) (1) = 1. The state jϕ(3)i is an eigenstate with
eigenvalue +1 for the configurations rll, lrl, and llr, and with eigenvalue 1 for the
configuration rrr. For the other four possible configurations (the ones having an
even number of r) jϕ(3)i is not an eigenstate, and the result of a measurement may
be 1 or +1 with equal probability. In a long run, they average to zero. Let call the
four elements of the set {rll, lrl, llr, rrr} “words,” and the other four possible
configurations “strings.”
A simple hidden variables theory can be constructed in the form of a set of 2 3
matrices. The matrix determines the result of detection for each photon of the trio,
depending what type of analyzer it finds. For example, the following matrix yields
254 Alejandro A. Hnilo
Table 13.1: Example of a matrix that yields the result (+1) if the configuration is lrl or llr,
and (1) if it is rrr, all in agreement with QM predictions; but, it yields (1) for rll,
contrarily to QM predictions.
the result (+1) if the configuration is lrl or llr and (1) if it is rrr, all in agreement
with QM predictions. However, it yields (1) for rll, contrarily to QM prediction
(see Table 13.1).
There are 26 = 64 of these matrices. Each matrix reproduces the QM predictions
for three of the four “words” and for all the “strings” (in the average). Let’s call
“bad word” the configuration for which a given matrix cannot reproduce the QM
prediction (in the example stated earlier, the “bad word” is rll). There are eight
matrices sharing the same “bad word,” and each configuration is the “bad word” of
eight matrices. The probability that a (randomly chosen) configuration is the “bad
word” of the (randomly chosen) matrix carried by the entangled trio is hence:
8/64 = 1/8. This is the probability of the matrix theory to not reproduce the QM
predictions. In other words, the probability of the matrix model to reproduce the
QM predictions for a single trio is 7/8. In order to disprove LR with a reliability
>99%, 35 trios (all of them showing results coincident with the QM predictions)
must be detected in an ideal setup. There are actually 70 trios, because half of the
configurations are “strings,” in a setup where the bases are randomly and independ-
ently chosen in each station. There is, in consequence, no single-shot disproval of
LR with GHZ states, but a statistical one, as in the usual two-particles Bell’s case.
What can be single-shot disproved is QM. It suffices to observe (+1) for rrr, or
(1) for any of the other four “words,” to get a result that refutes QM. Of course,
this is for an ideal setup. In a real setup the imperfections must be taken into
account, and the conditions to discriminate QM from LR become much like in the
two-particles case. The issue is discussed in detail in Hnilo (1994) for GHZ states
with arbitrary numbers of particles. The main conclusion of that paper is, in short,
that experiments with GHZ states do not provide any significant advantage (for the
tests of LR) over the two-particles Bell states, even if the difficulty in the
preparation of the states is not taken into account.
Now that we have seen that a single-shot disproval of LR is not provided by
GHZ states, let go back to the main question. Mermin (1990) has demonstrated that
classical theories are limited by the inequality:
ð
Fn ¼ Im dλ:ρðλÞ:Π El j þ iEr j 2n=2 ðn evenÞ, or 2ðn-1Þ=2 ðn oddÞ: (13.6)
Beyond Loophole-Free Experiments: A Search for Nonergodicity 255
where n is the number of particles in the GHZ state, λ is the hidden variable, the
product goes from j = 1 to n, and Elj (Nl+ Nl)/(Nl+ + Nl), where Nl+ (Nl) is
the number of detections that produced a result “+1” (1) in the j-station when the
setting was l (analogously for r). In the case of jϕ(3)i Mermin’s inequality takes the
form:
ð
F3 ¼ dλ:ρðλÞ: El 1 :El 2 :Er 3 þ El 1 :Er 2 :El 3 þ Er 1 :El 2 :El 3 Er 1 :Er 2 :Er 3 2
(13.7)
Note that the settings are randomly and independently chosen, so that in any
experiment (even in an ideal one) the “string” configurations (say, lrr) also appear
in the set of measured data. However, they average to zero, so that they are
dropped in Eq. (13.7). The QM prediction in the case of jϕ(3)i is F3 = 4, which
violates the inequality. In the case of the matrix model, the integral over λ is
replaced by a sum over the 64 matrices. The matrix model saturates the inequality
(F3 = 2), because all the matrices fail to reproduce at least one of the “words” (e.g.,
“bad word” of each matrix).
The definition of Fn involves an integral over the hidden variables, so that it is
clear that the ergodic hypothesis is necessary in an experiment using GHZ states,
too. To be specific, what is actually measured for n = 3 is:
ð ð ð
Fmeas: ¼ dt:ρðtÞ El :El :Er þ dt:ρðtÞ El :Er :El þ dt:ρðtÞ Er 1 :El 2 :El 3
1 2 3 1 2 3
ð
dt:ρðtÞ Er 1 :Er 2 :Er 3 (13.8)
where each integral spans over a different period of time (regardless of whether
they are continuous or divided in many randomly chosen, separate small intervals).
If we make λ = t and integrate over the whole measuring time, we are faced again
with the problem of assigning values to the counterfactual integrals, as before.
Therefore, F3 6¼ Fmeas., unless the ergodic hypothesis (or something like it) is
assumed. The involved inequality is different (Mermin’s instead of Bell’s), but the
situation is the same, as in the case of the experiment with two entangled particles.
violated in the experiments. The direct way cannot be taken, for it is impossible to
measure an average over the space of the hidden variables (to check whether it is
equal to the time average or not). Therefore, one must follow an indirect approach.
As was discussed in the introduction, the key may be deviations from randomness
in time series of data obtained in a Bell’s experiment. This means measuring not
only the average values of magnitudes (say, J, SCHSH, or concurrence), as in almost
all experiments performed to date, but also the time evolution of these or other
magnitudes.
Weinfurter 2001). This number cannot be much improved nowadays, for the
fastest currently available single-photon detectors (avalanche photodiodes) cannot
be used reliably if the rate approaches 106 s1 because of the high number of
secondary (false) counts. In consequence, to detect 38 pairs with 3105 s1 coinci-
dences in a time L/c, one must have L > 38 km. Bell’s setups with L = 13 km (Peng
et al. 2005) and even 144 km (Scheidl et al. 2010) have been performed, but the
achieved coincidence rate was much lower than necessary: 50 and 8 s–1 (scaled
values are 2 10–3 c/L and 4 10–3 c/L). The recent loophole-free experiment in
Vienna (Giustina et al. 2015) reached 200 s–1 for L = 58 m, or 4 10–5 c/L; the one
in Boulder (Shalm et al. 2015) 5 s–1 for L = 185 m, or 3 10–6 c/L. The rate of
detected pairs should be thus increased by several orders of magnitude to enter the
range where the basic oscillations are expected to be detectable. A direct search for
nonergodic dynamics seems beyond the current technical capacity.
Yet, the task is reachable under a “stroboscopic” approximation. That is, by
supposing that the system decays to a “ground state” in a time unknown, but finite
τdecay after the source of entangled states is turned off. The hypothesized picture is
then as follows: Once the source of entangled states is turned on, the nonclassical
correlation between measurements in the remote stations starts to evolve in a
nonergodic way. After the source is turned off, the correlation decays with time
τdecay (unknown, but finite). We have studied a general form that the dynamics
may take in this picture in Hnilo (2012) and Hnilo and Agüero (2015) if locality is
imposed. Noteworthy, oscillations with period 4L/c are predicted for a broad
region in parameters’ space.
If the hypothesized picture is correct, a stroboscopic reconstruction of the
system’s evolution is possible by using a pulsed source of entangled states. The
time between pulses is adjusted longer than τdecay. The pulse duration is sliced in
periods shorter than L/2c, and the number of photons detected in each time slice is
recorded. Due to technical reasons (see Agüero, Hnilo, and Kovalsky 2014), less
than one photon per pulse must be recorded in the average but, after millions of
pulses are detected, the time slices are gradually “filled” with data and the evolu-
tion of the correlation during the pulse duration can be reconstructed with arbitrary
precision. The value of τdecay is unknown, but the pulse repetition rate can be
lowered as much as necessary, at the only cost of increasing the total duration of
the experimental run. In summary, the stroboscopic approach allows the search for
nonergodic dynamics with accessible means. There is, however, a risk of failure: If
the system did not decay to the same ground state after each pump pulse, then the
initial condition before each new pulse would not always be the same, jamming the
reconstruction. This is an unavoidable risk in any stroboscopic observation.
Anyway, even an imperfect reconstruction of the dynamics may provide a valuable
antecedent to consider the realization (or not) of the “complete,” and much more
258 Alejandro A. Hnilo
Pump laser
(pulsed)
Time stamper A Time stamper B
Trigger C+ C– C– C+ Trigger
Frequency down
conversion crystals
SPCM SPCM
“Bat-ears” “Bat-ears”
Photodiode
Photodiode L
difficult, L > 38 km experiment. Note that the “complete” experiment would be, in
any case, a simpler setup than, e.g., a laser interferometer gravitational-wave
observatory (LIGO) or a supercollider.
The specific experiment proposed (Figure 13.3) uses a pulsed laser to pump the
nonlinear crystals to generate entangled (in polarization) states of photons. Both
the pump repetition rate and the pulse length are adjustable. This is to explore the
unknown value of τdecay and of the time of evolution of the dynamics. The
entangled photons are inserted into single-mode optical fibers and transmitted to
remote stations. The time-stamped files allow the calculation of the variables of
interest (say, concurrence, efficiency, etc.) after the experimental run has ended.
The distance between the stations is adjustable. Varying the value of L allows
discarding artifacts in the case some dynamics are actually observed, for L is
supposed to define the timescale of the problem. It is relevant to mention here that
the effect of polarization mode dispersion limits the use of optical fibers to about
L 1 km. Photons are detected at the stations with avalanche photodiodes, and
detections’ time values stored in time-stamped files with a resolution of 1 or 2 ns.
A better resolution is meaningless, because of the intrinsic time jitter of the
photodiodes. A sample of the pump pulse is sent to each station to synchronize
the time-stamping devices, to set a uniform and stable starting point for the
stroboscopic reconstruction, and to avoid the drift of the clocks that apparently
occurred in some previous experiments (see the next section).
Beyond Loophole-Free Experiments: A Search for Nonergodicity 259
Entanglement
Efficiency
Pump pulse
Time
The stroboscopically obtained time series (Figure 13.4) are then analyzed,
looking for the existence of a low-dimension object in phase space.
the data files, except one. The exception was the longest run in real time (6 minutes,
rather than 10 to 30 seconds of most files), and the result was dE = 10 with four
positive Lyapunov exponents, which meant that the series was hyperchaotic. Based
on the reconstructed attractor, we were able to “predict” the future outcomes in the
series, with satisfactory precision, up to an average of five to six pairs, that roughly
corresponded to the inverse of the largest positive Lyapunov exponent. As there
were four possible settings in the series, about 20 bits of the key were predictable.
The importance of this finding for the security of QKD was remarked. When we
found this result, the Innsbruck experiment had been dismantled, so that the cause
of the detected dynamics is impossible to know for sure. It is believed to be a drift
between the clocks in each station. This belief is supported by the fact that a time-
stamped experiment performed by our group some years later (Agüero et al. 2009),
with an even longer session of data recording (>30 min) but a single clock,
produced no measurable value of dE. Regardless whether the cause of the chaotic
dynamics was instrumental or fundamental, the nonlinear analysis approach was
able to reveal it in one file of the Innsbruck experiment. This result proves the
capacity and power of the approach.
Another antecedent is the stroboscopic reconstruction of the evolution of
entanglement achieved by our group (Agüero, Hnilo, and Kovalsky 2012). The
experiment’s aim was not to test ergodicity, but to close the time-coincidence
loophole. No evolution of entanglement was observed during the pulse duration.
Entanglement was constant during the pulse, and was born and dead “instantan-
eously” (strictly speaking, in a time shorter than the time resolution of the device,
12.5 ns). There was only an increase of statistical errors at the pulse’s edges,
naturally caused by scarcer data. Nevertheless, the time resolution of photon
detectors and time-stamping devices was insufficient to detect f for the small value
of L used which was, to make things even worse, fixed. After the setup was
dismantled, we realized there was a linear increase of efficiency with time during
the pulse duration. This result is consistent with the predictions of a simple LR
hidden variables theory, but we consider it in no way conclusive (see Hnilo and
Agüero 2015). A repetition of the experiment is in order to discard possible
artifacts.
Finally, in my opinion, the proposed stroboscopic experiment is worth doing for
the following reasons:
1) It is new. Instead of measuring Bell’s inequalities over and over again with
different random number generators (as in the recent “Big Bell Test,” or by
using light from remote stars) and/or improved detectors, this proposal involves
analyzing the time evolution of the system, not only the average values. I do not
mean those experiments are worthless. On the contrary, they are formidable
Beyond Loophole-Free Experiments: A Search for Nonergodicity 261
Now I would like to remark that arrows and vectors are really strange things. If
one thinks in the familiar terms of sets, the properties common to different sets are
found by their intersection. The same for arrows is found by projecting one arrow
into the other. Orthogonal arrows have “nothing” in common. The intersection of
sets is associative and commutative. The projection of arrows is not. In a popular
children’s game, one has to find a person in a set by asking whether the person is a
man or a woman, if he or she is blond or not, wears glasses or not, etc. The result is
the same regardless of the order in which the questions are posed. If played with
arrows, however, the game may have a different result depending the order of the
questions. Some centuries-old mysteries are impossible to grasp by thinking in
terms of sets; e.g., Christian Trinity – three different persons, but only one God.
Considered in terms of sets, it has no solution other than faith. In terms of arrows,
God can be thought as the sum of three orthogonal (i.e., completely different)
entities: Son, Father, and Holy Spirit. This picture also explains the Jesuits’
commandments to study nature (to get close to the Father), to do charitable actions
(to get close to the Son), and to exercise introspection (to get close to the Holy
Spirit). Failing to complete any of these three commandments means to fall short
of reaching God by a distance 1/√3 (in God’s space, assumed to be Euclidean).
The QM description of Bell’s experiments requires interference of waves lying in
remote positions in real space. It is interference in an abstract, nonlocal space. This is
not only anti-intuitive (interference in real space has already led us far from intuition,
so that this is not too serious) but, much more important in my opinion, structurally
unstable. For nonlocal interference and nonlinearity (even if infinitesimally small)
lead to the possibility of transmission of information at infinite speed and quantum
field theory collapses. Some way to include nonlinear terms without leading to faster-
than-light signaling (or something even stranger, see Polchinski 1991 and references
therein) would be most welcome, for nonlinearities exist everywhere. They exist at
the large scale of the universe; the equations of general relativity are nonlinear. They
also exist at our scale – we can describe the evolution of the surrounding world with
linear equations only as an approximation. Yet, QM claims nonlinearities to do not
exist at the microscopic scale, not even infinitesimal ones. I prefer thinking that
nonlinearities also exist at the microscopic scale, but that they cannot lead to
instantaneous transmission of information, because the strong correlations character-
istic of entanglement appear only after a time L/c has elapsed. This is an effect that is
not predicted by QM. In other words, I believe entanglement to be an average
property of the setup’s symmetries and, for spatially spread systems, it requires a
time longer than L/c to grow. In shorter times, transient deviations from the high
correlations predicted by QM should be observed. I find this the most economical
way to save quantum field theory, the statistical interpretation of QM, and LR.
QM claims that only probabilities can be predicted, that information is physical,
and that an independent physical reality does not exist. This has to me the smell of an
Beyond Loophole-Free Experiments: A Search for Nonergodicity 263
Acknowledgments
I would like to offer many thanks to the participants on the workshop Identity,
indistinguishability and non-locality in quantum physics (Buenos Aires, June
2017), especially to Federico Holik, Nino Zanghì, Olimpia Lombardi, Lev Vaid-
man, and Sebastian Fortin, for so many exciting and fruitful discussions on the
subject of this contribution. This research was partially supported by the grant
PIP11–077 of CONICET, Argentina.
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Part IV
Symmetries and Structure in Quantum Mechanics
14
Spacetime Symmetries in Quantum Mechanics
cristian lópez and olimpia lombardi
14.1 Introduction
In the last decades, the philosophy of physics has begun to pay attention to the
meaning and the role of symmetries, an issue that has, however, had a great
relevance in physics since, at least, the middle of the twentieth century. Notwith-
standing this fact, this increasing interest in symmetries has not yet been trans-
ferred to the field of the interpretation of quantum mechanics. Although it is
usually accepted that the Galilean group is the group of invariance of the theory,
discussions about interpretations of quantum mechanics, with very few exceptions,
have not taken into account symmetry considerations. But the invariance of a
theory under a group does not guarantee the invariance of its interpretations, as
they usually add interpretive assumptions to the formal structure of the theory.
Symmetry considerations should thus be seriously taken into account in the field of
the interpretation of quantum mechanics.
For this reason, in this chapter we shall focus on the spacetime symmetries of
quantum mechanics. After briefly introducing certain terminological clarifications,
we shall focus on two aspects of spacetime transformations. First, we shall
consider the behavior of nonrelativistic quantum mechanics under the Galilean
group, aiming at assessing its Galilean invariance in relation to interpretive con-
cerns. Second, we shall analyze the widely-accepted view about the invariance of
the Schrödinger equation under time reversal, in order to unveil some implicit
assumptions underlying such a claim.
269
270 Cristian López and Olimpia Lombardi
invariant under the transformation gα if, for that transformation, a~i ¼ ai . In turn, the
object ai 2 A is invariant under the group G if it is invariant under all the trans-
formations gα 2 G.
In physics, the objects to which transformations apply are usually those represent-
ing states s, observables O, and differential operators D, and each transformation
acts upon them in a particular way. In turn, those objects are combined in equations
representing the laws of a theory. Then,
Def. 2 Let L be a law represented by an equation E s; Oi ; Dj ¼ 0, where s
represents a state, the Oi represent observables, and the Dj represent differential
operators, and let G be a group of transformations gα 2 G acting upon the objects
~ ~
involved in the equation as s !~s , Oi ! O i , and Dj ! D j . L is covariant under the
~ ~
transformation
gα if E ~s ; O i ; D j ¼ 0, and L is invariant under the transformation gα
if E ~s ; Oi ; Dj ¼ 0. Moreover, L is covariant invariant under the group G if it is
covariant invariant under all the transformations gα 2 G.
On this basis, it is usually said that a certain group is the symmetry group of a theory:
Def. 3 A group G of transformations is said to be the symmetry group of a theory if
the laws of the theory are covariant under the group G.
This means that the laws preserve their validity even when the transformations of
the group are applied to the involved objects.
Some authors prefer to talk about symmetry instead of covariance. This is the case
of John Earman (2004), who defines symmetry in the language of model theory:
Def. 4 Let M be the set of the models of a certain mathematical structure, and let
ML M be the subset of the models satisfying the law L. A symmetry of the law
L is a map S : M ! M that preserves ML , that is, for any m 2 ML , SðmÞ 2 ML .
When L is represented by a differential equation E s; Oi ; Dj ¼ 0, each model
m 2 ML is represented by a solution s ¼ F ðOi ; s0 Þ of the equation, corresponding
to a possible evolution of the system.
Then, the covariance of L under a transform-
~ ~
ation g – that is, the fact that E ~s ; O i ;D j ¼ 0 – implies that if s ¼ F ðOi ; s0 Þ is a
solution of the equation, ~s ¼ F~ O ~ i ; s0 is also a solution and, as a consequence, it
represents a model SðmÞ 2 ML . This means that the definition of covariance given
by Def. 2 and the definition of symmetry given by Def. 4 are equivalent.
In turn, the covariance of a dynamical law – represented by a differential
equation – does not imply the invariance of the possible evolutions – represented
by the solutions
of the equation.
In fact, the covariance of the law L, represented
by
the equation E s; Oi ; Dj ¼ 0, implies that s ¼ F ðOi ; s0 Þ and ~s ¼ F~ O ~ i ; s0 are both
solutions of the equation, but does not imply that s ¼ ~s . In the model-theory
272 Cristian López and Olimpia Lombardi
Def. 5 Let M be the set of the models of a certain mathematical structure, and let
ML M be the subset of the models satisfying the law L. Let a transformation be a
map S : M ! M that preserves ML . The law L is invariant under the transform-
ation S if, for any m 2 ML , SðmÞ ¼ m.
The general definitions just described can be applied to the Schrödinger equation
so as to explicitly state the conditions of covariance and invariance for quantum
mechanics. Here we will focus on the evolution equation of the theory, leaving
aside the collapse postulate, since it is an interpretive postulate in orthodox
quantum mechanics. Given a transformation g acting as jφi ! jφ ~
~ i, O ! O,
~ ~
d=dt ! d=dt, and i ! i (considering i as the shorthand for the operator i I), by
making ℏ ¼ 1 the Schrödinger equation is covariant under g when
d~ jφ
~i
¼ ~i H
~ jφ
~i (14.1)
dt
and it is invariant under g when
~i
djφ
~ i:
¼ i H jφ (14.2)
dt
where εijk is the Levi-Civita tensor. Strictly speaking, in the case of quantum
mechanics the symmetry group is the group corresponding to the central extension
of the Galilean algebra, obtained as a semi-direct product between the Galilean
algebra and the algebra generated by a central charge, which in this case is the
mass operator M ¼ mI, where I is the identity operator and m is the mass. The
mass operator as a central charge is a consequence of the projective representation
of the Galilean group (see Bose 1995, Weinberg 1995). However, in order to
simplify the presentation, we will use the expression “Galilean group” from now
on to refer to the corresponding central extension.
In a closed, constant-energy system free from external fields, the generators K α
are given by the basic magnitudes of the theory: the energy H ¼ ℏK τ , the three
momentum components Pi ¼ ℏK ρi , the three angular momentum components
J i ¼ ℏK θi , and the three boost components Gi ¼ ℏK ui . Then, in this case the
commutation relations turn out to be
ðaÞ Pi ; Pj ¼ 0 ðf Þ Gi ; Pj ¼ iδij M
ðbÞ Gi ; Gj ¼ 0 ðgÞ ½Pi ; H ¼ 0
ðcÞ J i ; J j ¼ iεijk J k ðhÞ ½J i ; H ¼ 0 (14.4)
ðdÞ J i ; Pj ¼ iεijk Pk ðiÞ ½Gi ; H ¼ iPi
ðeÞ J i ; Gj ¼ iεijk Gk
The rest of the physical magnitudes can be. defined in terms of these basic ones. For
instance, the three position components are Qi ¼ Gi =m, the three orbital angular
momentum components are Li ¼ εijk Qj Pk , and the three spin components are
274 Cristian López and Olimpia Lombardi
irreducible representation.
d d~ D d dU 1 d~ jφ
~i
! ¼ ¼ U ) ¼ ~i H
~ jφ
~i (14.10)
dt dt Dt dt dt dt
~
This means that the transformed differential operator d=dt is a covariant time-
derivative D=Dt, which makes the Schrödinger equation to be Galilean-covariant
in the sense of Eq. (14.1).
In a closed, constant-energy system free from external fields, H is time-
independent and the Pi and the J i are constants of motion (see Eq. (14.4g) and
Spacetime Symmetries in Quantum Mechanics 275
~ ¼ eiGx ux HeiGx ux 6¼ H
H (14.12)
Since Gx is not time-independent, dU=dt ¼ deiGx ux =dt 6¼ 0, and Eq. (14.9) yields
~i
d jφ ~ deiGx ux iGx ux
¼ i H þ i e ~i
jφ (14.13)
dt dt
276 Cristian López and Olimpia Lombardi
In order to know the value of the bracket in the right-hand side (r.h.s.) of Eq.
(14.13), the two terms in the bracket must be computed. When the task is
performed, it can be proved that the terms added to H in H~ cancel out with those
coming from the term containing the time-derivative (see Lombardi, Castagnino,
and Ardenghi 2010: appendices). Therefore, Eq. (14.2) is again obtained and the
invariance of the Schrödinger equation is proved to hold also for boost-
transformations.
The case of boost-transformations illustrates a claim previously mentioned in
Section 14.2: Even though a law is invariant under a transformation when it is
covariant and all the involved objects are invariant, this is not the only way to
obtain invariance. When the quantum system is free from external fields, the
Schrödinger equation is invariant under boost-transformations, in spite of the fact
that the Hamiltonian and the differential operator d=dt are not boost-invariant
objects.
ðP AðQÞÞ2
H¼ þ V ð QÞ (14.14)
2M
where AðQÞ is a vector potential and V ðQÞ is a scalar potential. The covariance of
the Schrödinger equation, as expressed in Eq. (14.9), fixes the way in which the
potentials AðQÞ and V ðQÞ must transform under the Galilean group. The electro-
magnetic field may be derived from a vector potential and a scalar potential; thus, a
fully Galilean-covariant quantum theory of the Schrödinger field interacting with
Spacetime Symmetries in Quantum Mechanics 277
The intuition about a strong link between invariance and objectivity is rooted in
a natural idea: What is objective should not depend on the particular perspective
used for the description. When this intuition is translated into the group-theoretical
language, it can be said that what is objective according to a theory is what is
invariant under the symmetry group of the theory. This idea appeared in the
domain of formal sciences in Felix Klein’s “Erlangen Program” of 1872, with
the attempt to characterize all known geometries by their invariants (see Kramer
1970). This idea passed to physics with the advent of relativity, regarding the
ontological status of space and time (Minkowski 1923). The claim that objectivity
means invariance becomes a main thesis of Hemann Weyl’s book Symmetry
(1952). Max Born also very clearly expressed his conviction about the strong link
between invariance and objectivity: “I think the idea of invariance is the clue to a
rational concept of reality” (Born 1953: 144). In recent times, the idea has strongly
reappeared in several works. For instance, in her deep analysis of quantum field
theory, Sunny Auyang (1995) makes her general concept of “object” to be founded
on its invariance under transformations among all representations. The assumption
of invariance as the root of objectivity is also the central theme of Robert Nozick’s
book Invariances: The Structure of the Objective World (2001). In the same vein,
David Baker (2010) has argued that symmetries are a guide to finding out which
quantities represent fundamental natural properties in a physical theory.
If the ontological meaning of symmetries is accepted, it is easy to see that
symmetries must play an active role in the understanding of a physical theory. In
the particular case of quantum mechanics, the consideration of its Galilean
covariance cannot be overlooked in the discussions about interpretation.
As it is well known, the Kochen-Specker theorem (Kochen and Specker 1967)
establishes a barrier to any realist classical-like interpretation of quantum mechan-
ics: It proves the impossibility of ascribing definite values to all the physical
quantities (observables) of a quantum system simultaneously, while preserving the
functional relations between commuting observables. This result is a manifestation
of the contextuality of quantum mechanics – the ascription of definite values to the
observables of a quantum system is always contextual. As a consequence of the
Kochen-Specker theorem, any realist interpretation of quantum mechanics is com-
mitted to selecting a subset of definite-valued observables from the set of all the
observables of the system (or a preferred basis from all the formally equivalent bases
of the Hilbert space). The observables of that subset will be those that acquire
definite values without violating quantum contextuality. It is at this point that the
symmetry group of the theory becomes a leading character. As noticed by Harvey
Brown, Mauricio Suárez, and Guido Bacciagaluppi (1998), any interpretation that
selects the set of the definite-valued observables of a quantum system in a given state
is committed to considering how that set is transformed under the Galilean group.
280 Cristian López and Olimpia Lombardi
However, now the link between invariance and objectivity comes into play. The
study of the role of symmetries is particularly pressing in the case of realist
interpretations of quantum mechanics, which conceive a definite-valued observ-
able as a physical magnitude that objectively acquires an actual definite value
among all its possible values: The fact that a certain observable acquires a definite
value should be an objective fact that should not depend on the descriptive
perspective. Therefore, the set of the definite-valued observables of a system
picked out by the interpretation should be left invariant by the Galilean
transformations. From a realist viewpoint, it would be unacceptable that such a
set changed as result of a mere change in the perspective from which the system is
described.
In his article “Aspects of objectivity in quantum mechanics,” Harvey Brown
(1999) explicitly tackles the problem in discussing the objectivity of “sharp
values.” In particular, he focuses on interpretations that specify state-dependent
rules for assigning sharp values to some of the self-adjoint operators representing
quantum magnitudes, such as the interpretations whose value-assignment rules
coincide with the eigenstate-eigenvalue link, or the modal interpretations that make
the set of definite-valued observables to depend on the instantaneous state of the
system. Brown clearly explains the difference between the classical and the
quantum case. In classical mechanics, Galilean noninvariant magnitudes modify
their values with the change of reference frame; for this reason, if their objectivity
is to be retained, they must be regarded not as intrinsic properties but as relational
properties. For instance, the values of classical position and momentum can be
conceived as relational properties that link the system and the reference frame. In
quantum mechanics, by contrast, the relational nature acquires a further degree;
whereas, in the classical case the sharp value of a magnitude depends on the
reference frame, in the quantum case the very sharpness of an observable’s value
must be relational in order to preserve its objectivity. For instance, the fact that the
position of a system has a sharp (definite) value in a certain reference frame, and,
as a consequence, that the system can be conceived as a localized particle, is itself
relational. In a different reference frame, the system may have an unsharp value of
position and, then, may behave as a delocalized particle. Brown also correctly
stresses that it is not just boosts that produce this kind of situation; passive spatial
translations can cause that some sharp-valued observables to become unsharp. On
the basis of this analysis, he concludes that
If, in the hope of providing an ontological interpretation of quantum mechanics, we
introduce state-dependent rules for assigning sharp values to magnitudes associated with
a specific quantum system, we should recognise that the objective status of such sharp
values is relational, not absolute.
(Brown 1999: 66–67)
Spacetime Symmetries in Quantum Mechanics 281
The idea is that, by knowing how dynamical equations (standing for physical laws)
behave under time reversal, one can learn about the nature of time according to a
Spacetime Symmetries in Quantum Mechanics 283
theory. This kind of principle is well-seated in the literature (see also Earman 1974,
Sklar 1974, Arntzenius 1997), and it is the key element that links time’s properties,
physical laws, and the time-reversal transformation.
However, here nothing has yet been said about the time-reversal transformation
in itself, other than that it minimally performs the transformation t ! t. As
typically noted in the literature, the topic is somewhat tricky as there is no shared
understanding of what time reversal is exactly supposed to do nor of what proper-
ties it should instantiate (see Savitt 1996 for a careful analysis of varied time-
reversal operators; see Peterson 2015 for an updated approach). Furthermore, time
reversal’s properties seem to change from theory to theory, to the extent that
dynamics also changes. This remark already assumes a strong premise – that the
structure of time, in particular, regarding its time-reversal symmetry property, is
closely tied up to the theory’s dynamics.
Quantum mechanics textbooks rarely offer a thorough justification for such a claim
and commonly go on by formally introducing the “proper” way to reverse time in
quantum mechanics. In some cases, the only justification is based on a classical
analogy: The transformation t ! t does not lead to the transformation of
momentum as P ! P, which is expected because this is the way in which
momentum transforms under time reversal in classical mechanics. But mere
analogy does not seem to be a sufficiently good argument; for this reason, Bryan
Roberts (2017) has very recently brought up an updated and purely quantum-
mechanic-based reasoning for defending the standard procedure. Here we will not
analyze in detail those arguments; rather, we will consider the problem in the light
of the symmetries of the Schrödinger equation related to the reversal of time.
284 Cristian López and Olimpia Lombardi
Let us use θ to call a generic time-reversal operator, which performs at least the
transformation t ! t but whose precise form is not defined yet. As explained in
Section 14.2, this operator acts as jφi ! jφ~ i, O ! O, ~
~ d=dt ! d=dt, and i ! ~i; in
~ i ¼ θjφi, O
particular, jφ ~
~ ¼ θO θ , d=dt ¼ θ d=dt θ , and ~i ¼ θ i θ . The Schrö-
1 1 1
d~ jφ
~i
¼ ~i H
~ jφ
~i (14.16)
dt
Now, in order to know if the Schrödinger equation is also invariant under the
application of θ, it is necessary to define the precise form of θ to see how it acts
upon d=dt, i, and H. As in the case of the Galilean group, the situation of a closed,
constant-energy system will be considered.
Case (i): If θ ¼ T only performs the transformation t ! t, then
d~ d d ~i ¼ TiT 1 ¼ i ~ ¼ THT 1 ¼ H
¼T T 1 ¼ H (14.17)
dt dt dt
Introducing these equations into Eq. (14.16) leads to the conclusion that the
Schrödinger equation is not T-invariant, since
~i
d jφ
~i
¼ iH jφ (14.18)
dt
Summing up, independently of any interpretation, the results given by Eq. (14.18)
and Eq. (14.20) show that the Schrödinger equation is not invariant under the unitary
operator T, and it is invariant under the antiunitary operator T ∗ . The conceptual question
is which of the two operators, T or T ∗ , represents the operation of time reversal.
Why does the unitary operator T not meet that requirement? It is not unusual to read
that the reason is that the unitary operator T transforms the Hamiltonian as
THT 1 ¼ H. This sounds very strange because, by performing only the transform-
ation t ! t, the operator T should leave the time-independent Hamiltonian invari-
ant. So, now the question is: Why does the Hamiltonian transform as H ! H?
Although not always explicit, a typical answer is that offered by Stephen Gasiorowicz:
we find that [the equation of motion] can be invariant only if
THT 1 ¼ H
This, however, is an unacceptable condition, because time reversal cannot change the spectrum
of H, which consists of positive energies only. If T is taken to be anti-unitary [our T ∗ ], the
* operator changes the i to i [in the equation of motion] and the trouble does not occur.
(Gasiorowicz 1966: 27; italics added)
A symmetry can be a priori, i.e., the physical law is built in such a way that it respects that
particular symmetry by construction. This is exemplified by spacetime symmetries,
because spacetime is the theater in which the physical law acts . . . and must therefore
respect the rules of the theater.
(Dürr and Teufel 2009: 43–44)
From this perspective, the invariance under the Galilean group must be built into
the Schrödinger equation due to the homogeneity and the isotropy of space and the
homogeneity of time. This view may sound reasonable to the extent that those are
features of space and time that we, in a certain sense, can experience. But, why
should we impose time-reversal invariance? We have no experience of the isotropy
of time since we cannot travel backwards in time. Despite this, time-reversal
invariance must be introduced as a postulate:
One should ask why we are so keen to have this feature in the fundamental laws when we
experience the contrary. Indeed, we typically experience thermodynamic changes which
are irreversible, i.e., which are not time reversible. The simple answer is that our platonic
idea (or mathematical idea) of time and space is that they are without preferred direction,
and that the “directed” experience we have is to be explained from the underlying time
symmetric law.
(Dürr and Teufel 2009: 47)
Challenging the most widely-held position about time reversal in the field
of quantum mechanics, a few authors have raised their voices against it by
appealing to philosophical reasons: The antiunitary operator T ∗ would fail to
offer a conceptually sound and clear-cut representation of time reversal. On
the one hand, a far-reaching tradition, which tracks back to the work of Giulio
Racah (1937) and Satosi Watanabe (1955), pleads for a unitary time-reversal
operator in quantum theories. Oliver Costa de Beauregard (1980) has argued
for such a view by claiming that a unitary time-reversal operator that merely
reverses the direction of time by flipping the sign of the variable t goes more
Spacetime Symmetries in Quantum Mechanics 287
naturally along with relativistic contexts and is more naturally akin to the
Feynmann zig-zag philosophy. On the other hand, philosophers such as Craig
Callender (2000) and David Albert (2000) have claimed that the Schrödinger
equation should actually be considered as nontime-reversal invariant, since
it is not invariant under T, which more fairly represents what one means
by “reversing the direction of time.” Without any further ado, Jill North
claims:
What is a time-reversal transformation? Just a flipping of the direction of time! That is all
there is to a transformation that changes how things are with respect to time: change the
direction of time itself.
(North 2009: 212)
In other words,
time reversal displacement by Δt time reversal displacement by Δt = identity
This requirement is commonly interpreted in formal terms as follows:
U Δt θ1 U Δt θ s ¼ s (14.21)
where s is the arbitrary state and U Δt is the evolution operator for Δt. This
requirement is precisely Sakurai’s starting point in the argument that led him to
288 Cristian López and Olimpia Lombardi
the conclusion stated previously. Under the assumption that the evolution oper-
ators form a group (an assumption that is not always satisfied, see Bohm and
Gadella [1989], where the time evolution is represented by a semigroup), a U Δt
exists such that U Δt U Δt ¼ I. In this case, Eq. (14.21) becomes
H ¼ T H T 1 (14.25)
which is unacceptable because it leads to values of energy that are unbounded from
below. Since Wigner (1931/1959) also proved that any symmetry transformation is
represented by a unitary or an antiunitary operator, the argument concludes that the
right time-reversal operator is the antiunitary operator T ∗ .
This argument in favor of T ∗ is certainly much better than the previous one,
which takes the time-reversal invariance of the Schrödinger equation as one of the
premises. Nevertheless, as we will see, this second argument imposes the time-
reversal invariance of the dynamical equation beforehand as well, even if in a more
subtle way.
Let us begin by noticing that, when Eq. (14.21) is used to formalize Wigner’s
requirement, the “displacement by Δt” is represented by the time evolution of the
system by Δt according to the dynamical law of the theory – the Schrödinger
equation, here expressed as s ¼ U Δt s0 . However, as stressed in Section 14.3.4,
when considering the Galilean group, time displacement is not time evolution.
Time evolution is ruled by the dynamical law of the theory, in this case quantum
mechanics: According to the Schrödinger equation, the Hamiltonian is the gener-
ator of the dynamical evolution. By contrast, spacetime transformations are
purely geometric operations of displacing or rotating the system self-congruently.
In particular, time-displacement is a purely geometric operation that displaces the
system self-congruently to another time, and may agree or not with time evolu-
tion. In fact, as Hans Laue (1996) and Leslie Ballentine (1998) stress, in a generic
case, the Hamiltonian is not the generator of time displacements and only retains
its role as the generator of the dynamical evolution. This clearly shows that
time displacement and time evolution are different concepts. Hence, insofar as
Spacetime Symmetries in Quantum Mechanics 289
Wigner’s definition involves time displacement and not time evolution, it must be
formally expressed as (recall how Galilean transformations act upon states,
Eq. (14.5))
where the difference in sign between Eq. (14.27) and Eq. (14.22) is due to the
inverse relation between transformations on function space and transformations on
coordinates (see Ballentine 1998: 67). Eq. (14.27) expresses what time reversal
means: It should be a transformation such that
time reversal displacement by Δt time reversal = displacement by Δt
toward the past too, then we are introducing the time-reversal invariance of the
dynamical law by hand.
In other words, the question about the time-reversal invariance of a law is
precisely the question of whether the time displacement of the system toward the
past is also ruled by the dynamical law, that is, whether it is also a time evolution. If
the answer is positive, the law is time-reversal invariant, if the answer is negative, the
law is not time-reversal invariant. Therefore, supposing from the very beginning that
any time displacement toward the past is a dynamical evolution amounts to putting
the cart before the horse.
But, then, what do Eq. (14.21) and Eq. (14.22) mean? Actually, those equations
express the conditions that define what can be called motion reversal:
motion reversal evolution by Δt motion reversal evolution by Δt = identity
To put it precisely, the motion-reversal operator is the operator that reverses the
direction of a lawful motion of the system so as to obtain another lawful motion.
Then, the argument that, starting by Eq. (14.21), concludes with discarding the
unitary operator T is a proof of the fact that the antiunitary operator T ∗ is the right
motion-reversal operator for quantum mechanics.
Even though the difference between motion reversal and time reversal has not
been sufficiently stressed, it is acknowledged by some authors. For example,
Ballentine clearly states:
In the first place, the term “time reversal” is misleading, and the operation that is the
subject of this section would be more accurately described as motion reversal. We shall
continue to use the traditional but less accurate expression “time reversal”, because it is so
firmly entrenched.
(Ballentine 1998: 377; italics in original)
Sakurai also emphasizes the point just at the beginning of the section devoted to
time reversal:
In this section we study another discrete symmetry operator, called time reversal. This is a
difficult topic for the novice, partly because the term time reversal is a misnomer; it reminds
us of science fiction. Actually what we do in this section can be more appropriately
characterized by the term reversal of motion. Indeed, that is the terminology used by
E. Wigner, who formulated time reversal in a very fundamental paper written in 1932.
(Sakurai 1994: 266; bold and italics in original)
Summing up, it is quite clear that the antiunitary operator T ∗ is the motion-reversal
operator in quantum mechanics. But the initial question still remains: which is the
right quantum time-reversal operator?
Spacetime Symmetries in Quantum Mechanics 291
14.5 Conclusions
In this chapter we have focused on the spacetime symmetries of quantum mechan-
ics under the assumption that exploring the meaning of those symmetries is
relevant to the interpretation of the theory.
In the first part, we have considered the behavior of nonrelativistic quantum
mechanics under the Galilean group. We have shown that the Schrödinger equa-
tion is always covariant under the Galilean group, but its Galilean invariance can
only be guaranteed when it is applied to a closed system free from external fields.
We have also discussed the relevance of symmetries to interpretation; in particular,
any realist interpretation that intends to select a Galilean-invariant set of definite-
valued observables should make that set to depend on the Casimir operators of the
Galilean group, since they are invariant under all the transformations of the group.
In future works, these conclusions can be extended in two senses. On the one hand,
they can be transferred to quantum field theory by changing the symmetry group
accordingly: The definite-valued observables of a system in quantum field theory
would be those represented by the Casimir operators of the Poincaré group. Since
the mass operator M and the squared-spin operator S2 are the only Casimir
operators of the Poincaré group, they would always represent definite-valued
observables, a view that stands in agreement with a usual physical assumption in
quantum field theory. On the other hand, if invariance is a mark of objectivity,
there is no reason to focus only on spacetime global symmetries. Internal or gauge
symmetries should also be considered as relevant in the definition of objectivity
and, as a consequence, in the identification of the definite-valued observables of
the system.
In the second part of the chapter, we have carefully disentangled the different
notions involved in the issue of the time-reversal invariance of the Schrödinger
equation. We have assessed the usual claim about the matter, according to which
the Schrödinger equation is time-reversal invariant and the quantum time-reversal
operator is antiunitary. We have argued that the antiunitary operator is actually a
motion-reversal operator and that the question about the right time-reversal oper-
ator in quantum mechanics is still an open question. Those who think that time is
ontologically independent of and prior to the processes in it will stress the
difference between time reversal and motion reversal and, consequently, may tend
to prefer a time-reversal operator that only flips the direction of time. Others, by
contrast, may claim that the very concept of time as independent of motion has no
meaning. From this relationalist-like view, distinguishing between time reversal
and motion reversal as different operations makes no sense and, as a consequence,
the right time-reversal operator is necessarily a motion-reversal operator. This
shows that the question about the time-reversal invariance of quantum mechanics
292 Cristian López and Olimpia Lombardi
involves deep issues about the very nature of time. But the further development of
this aspect of the problem will be the subject of future work.
Acknowledgments
We want to thank the participants of the workshop Identity, indistinguishability
and non-locality in quantum physics (Buenos Aires, June 2017) for their contribu-
tion to a philosophically exciting and fruitful time. This work was made possible
through the support of Grant 57919 from the John Templeton Foundation and
Grant PICT-2014–2812 from the National Agency of Scientific and Technological
Promotion of Argentina.
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Spacetime Symmetries in Quantum Mechanics 293
15.1 Introduction
One of the most evocative results in the whole history of mathematical physics is
that there are exactly five polyhedra with perfect symmetry, i.e., all faces, edges,
and vertices are congruent. Starting from the intuitive and reasonable definitions
and axioms of Euclidean geometry and applying the constraint of symmetry, these
five structures are logically inevitable. Book 13, the climax of Euclid’s Elements,
constructs these solids and proves they exhaust the possibilities. While the beauty
of symmetric geometry provokes attention, I believe it is as much their “fiveness”
that inspires. Plato, Kepler, and others sought five-fold explanations for the
physical structures of the world (see Wilczek 2015). Those chimerical hopes
notwithstanding, Platonic solids do appear throughout science and technology
for natural and practical reasons: They are fundamental structures.
Starting in the nineteenth century, combining generalized notions of geometry
and symmetry yielded a bounty of new structures. In turn, these structures pro-
vided a framework for interpreting, classifying, and generating mathematical
models of physical reality. Glossing over the technical details (upon which hang
critical distinctions and academic careers), the same structural system that classifies
Platonic solids and their generalizations in any dimension also classifies Lie
groups, dynamical catastrophes, symmetric manifolds, random matrices, topo-
logical insulators, gauge quantum field theories, and so on. The imposition of
symmetry on vector spaces is surprisingly rigid, often giving finite or countable
possibilities. And then, more often than not, it seems we find these possibilities
manifest in our models of systems and dynamics. This coincidence of logically
inevitable mathematical structures with elements of physical reality remains as
seductive now as it did to Plato and Kepler. Some theoretical physicists spend their
career chasing these beautifully symmetric models and enumerating their qualities,
hoping one day the model will play an important role in the next “paradigm shift.”
294
Symmetry, Structure, and Emergent Subsystems 295
My proposal is more modest. Symmetry is not the answer to every question, and
the universe may have contingent features that no model can predict or explain.
However, symmetries seem to exist in reality and certainly exist in many effective
and productive models of reality. Within a framework like quantum mechanics, the
presence of symmetry entails the existence of mathematical structures that are
privileged by their relation to the symmetry. The focus of this chapter is the
particular structures called irreducible representations of symmetry groups. Similar
to how a Hamiltonian gives a spectrum in quantum mechanics, the irreducible
representations (irreps) of a group of symmetry transformations form a kind of
basis for possible manifestations of a symmetry on vector spaces. Like the Platonic
solids, these irreps are the denumerable “atoms” of the Hilbert spaces of quantum
mechanical models with symmetry.
For example, any quantum model with special unitary, or SU(2) symmetry has a
description in terms of units that are often called spins, even when they have no
rotational origin. These spin units are Hilbert subspaces that carry irreps of SU(2),
and the total Hilbert space is reducible into products and sums of these “atoms”
of spin.
Similarly, quantum field theory is built by associating free particles to irreps of
the Poincaré group of symmetry transformations of Minkowski spacetime (see, for
example, Streater and Wightman 1964, Haag 1992, Weinberg 1995). As Eugene
Wigner (1939) discovered, the irreps of the Poincaré group are conveniently
labeled by three invariants: mass, internal energy, and spin. Irreps of the Poincaré
group are equivalent to relativistic particles for most technical and interpretational
purposes. A similar construction with the Galilean group and nonrelativistic
particles is discussed later.
Let us recall that, technically, relativistic particles correspond to irreps of the
universal covering group of the Poincaré group. This is necessary to account for
the fact that, in quantum mechanics, we are typically interested in projective
representations, i.e., representations up to a phase. Projective representations are
more general than precise representations, and they preserve the interpretation of
pure states as rays in the Hilbert space. This technicality is mostly swept under the
rug for the rest of this chapter.
These examples make it clear that using irreps to analyze symmetry in quantum
mechanics is an old story, full of many bold successes and productive technical
details. Why rehash it here? The purpose of this contributed chapter is to explore
the connections between the mathematical units of symmetry embodied by irreps,
arguably the “most inevitable” symmetric structures of quantum mechanics, and
the conceptual units of reality that form the basis for interpretation of quantum
theories. Since irreps are symmetric structures that have the appealing properties of
being denumerable, they hold the same appeal as Platonic “fiveness.” Their logical
296 Nathan Harshman
energy levels of a more complicated system. It could also be derived from the
smallest nontrivial projective representation of an underlying three-dimensional
rotation group, i.e., a true spin. In any case, the model as a mathematical structure
can be studied without reference to any physical embodiment, ontological com-
mitment, or larger theoretical framework.
Nonrelativistic few-body models often do start with an ontological commitment,
such as N particles on an underlying d-dimensional
Euclidean space X Rd . The
Hilbert space is realized as H L R , Lebesgue square-integrable functions
2 Nd
with a finite number of particles. However, there are many-body systems and
quantum field theories where this kind of separability cannot be assumed (again,
see Streater and Wightman 1964).
mechanics often take these basic models as starting units and build more compli-
cated models out of their irreps spaces and algebras of observables.
In practice, identifying the symmetries of a nonrelativistic few-body model
takes place using a hybrid of formal, top-down and reductionist, bottom-up
approaches. Symmetries in physical space are important, but so are symmetries
in the spaces of configuration space and phase space. Although these auxiliary
spaces can be derived from the bottom-up approach, starting with free particle
models, the symmetries of these derived spaces may not be easily reducible into
free-particle symmetries and their corresponding irreps can describe collective or
emergent degrees of freedom. Additionally, few-body models can have symmet-
ries that are defined by unitary operators on the Hilbert space itself without
reference to a symmetry of an underlying space. For example, these kind of
symmetries are present when there are accidental degeneracies (Harshman 2017a).
On the surface, this looks like Eq. (15.1), but here, instead of a decomposition into
one-dimensional spaces, each energy eigenspace H E is realized by a finite-
dimensional space CdðEÞ with dimension equal to the degeneracy d ðE Þ of the
energy E.
If the Hamiltonian under consideration is time-independent and describes a
closed system, then the system has time-translation symmetry and the
302 Nathan Harshman
This may seem all a bit abstract, so here are a few examples. Perhaps the
simplest and most familiar is the case of parity. Parity is realized by a finite group
of two elements Z 2 . This abelian group has two irreps denoted + for even states
304 Nathan Harshman
under parity and for odd states. So the Hilbert space can be divided into sectors
of even and odd states H ¼ H þ ⊕ H . Because the group Z 2 has only one-
dimensional irreps, nothing more can be said except that H þ and H are built
out of one-dimensional subspaces that are invariant under parity. If parity is a
kinematic symmetry of a model, then all energy eigenstates are in one of those two
sectors and the expectation value of parity is a dynamical invariant. If parity is a
kinematic symmetry of a family of Hamiltonians, then varying the parameters of
the family mixes states within a sector, but not across sectors, i.e., changing the
control parameter does not change the parity of a state.
Another familiar example is rotational symmetry in three dimensions. The
eigenvalue of the operator representing angular momentum squared ℏ2 sðs þ 1Þ
can be used to characterize irreps and irrep spaces. The spin s comes in two infinite
series: nonnegative integers and nonnegative half-integers. A startling fact, called a
superselection rule, is that a decomposition into rotation group irreps only consist
of one of those two types, either integer or half-integer irreps. There can never be a
superposition of states with integer and half-integer total angular momentum. That
means in any quantum model, the Hilbert space only has sectors of integer or half-
integer irrep spaces.
A final example is the symmetry of particle permutations for N identical
particles, realized by the symmetric group SN . Here the irrep decomposition
provides a conceptual unit for analyzing the meaning of identical particles. Irreps
of SN are labeled by positive integer partitions of N. For example, there are four
partitions of N = 4: [4], [31], [22][22], [211][212], and [1111][14]. That
means the Hilbert space for a model with four identical particles can be decom-
posed into five sectors:
H ¼ H ½4 ⊕H ½31 ⊕H ½22 ⊕H ½212 ⊕H ½14 (15.7)
A familiar example is the reduction of hydrogen energy levels into subspaces with
fixed orbital angular momentum, doubled by the presence of spin. In principle, the
direct sum in Eq. (15.8) extends over all irreps of the kinematic group and there can
be multiple copies of the same irrep. However, when there are multiple irreps with
multiplicity, that usually signifies the presence of additional kinematic symmetries.
In the hydrogen atom example, there is the larger kinematic symmetry group SO(4)
that explains why subspaces with different orbital angular momentum have the
same energy.
Define the maximal kinematic symmetry group GH of the Hamiltonian H as the
group such that every H E corresponds to a single irrep ðμÞ of GH . When this group
can be found, energy levels are irreps of the maximal kinematic symmetry group,
and this is a powerful tool for the analysis of the model. It allows the physics of
degeneracy to be handled in a systematic, algebraic fashion because the symmetry
group provides all invariant observables necessary to diagonalize degeneracies.
Other observables in the model can be characterized by their transformation
properties under the group, simplifying calculations of expectation values, transi-
tion rates, and perturbation theory. Further, if H is part of a family of Hamiltonians,
then how GH changes with varying parameters determines how the energy levels
(irreps) split and merge and how invariants are broken and reformed.
To close this section on decomposition, consider a dynamic symmetry group G
with irrep labeled by ðμÞ representations. Since the symmetry group does not
commute with the Hamiltonian, there is no necessary relationship between irreps
of G and H E . However, one possibility is that each irrep of G is decomposable in a
sum of energy eigenspaces, i.e., the reversal of Eq. (15.8):
ðμÞ
Vi ¼ ⊕ HE (15.9)
ðμÞ
E2σi ðH Þ
ðμÞ
where σi ðH Þ is a purely symbolic shorthand for the spectrum of energies E
ðμÞ
corresponding to the irrep V i and depends on the Hamiltonian H and the
306 Nathan Harshman
Pure states in H can be classified as to whether they are separable or not separable
with respect to this factorization. In this context, separable is used in the algebraic
sense that a separable pure state jψi can be written at the tensor product of states
jψi i 2 H i as
jψi ¼ jψ1 i⊗jψ2 i⊗ . . . ⊗jψk i (15.12)
An entangled pure state is not separable and does not admit a factorization like
Eq. (15.12). To be clear, this is separable in a totally different sense than the
Symmetry, Structure, and Emergent Subsystems 307
More generally, the set A includes at least some observables that are not specific to
the tensor product structure induced by the subsystems.
For an interacting system, the Hamiltonian must be nonspecific, i.e., an operator
that cannot be constructed by Kronecker products of sub-Hamiltonians, like Eq.
(15.14). As a result, time evolution no longer factors into a specific unitary
operator like Eq. (15.15). Entanglement of a state evolving in time is typically
no longer a dynamical invariant with respect to the tensor product structure Eq.
(15.11). However, there may be other observables that are dynamical invariants.
There may even be other tensor product structures besides the original construction
that are dynamically invariant. The question becomes: Can one exploit these
observables to find an alternate factorization and an alternate notion of specificity?
Symmetry, Structure, and Emergent Subsystems 309
In this statement, the first two requirements on the subalgebras of observables are
mathematical in nature, and they could be assessed from within the model as true
or false for any particular partition of the observables. However, the third require-
ment is a physical criterion about empirical accessibility of measurement and
control. There could be partitions of the observables that satisfy the first two, but
are inadmissible based on a physical limitation of reality or some other constraint
from outside the model.
An extension of Zanardi’s theorem, called the “tailored observables theorem,”
demonstrates the flexibility provided by the first two requirements in constructing
subalgebras that factor the Hilbert space into “virtual” subsystems. For a finite-
dimensional Hilbert space H Cd , one can construct subalgebras of observables
that induce a tensor product structure from a finite basis of operators, such that any
known pure state can have any entanglement that is possible for any factorization of d
(Harshman and Ranade 2011). The proof relies on the unitary equivalence of Hilbert
spaces with the same dimension, and it is constructive in the sense that a procedure is
given to construct the generators for the subalgebras in a finite number of steps
(depending on the factorization of d). A consequence of this theorem is that, for any
pure state, observables can be found that will detect as much or as little entanglement
as is possible in a Hilbert space with dimension d. The only hitch is that entanglement
is completely relative only when the control of the system is absolute, and therefore the
third criterion of Zanardi’s theorem is unrestrictive. For example, in a system of linear
quantum optics (i.e., using only mirrors, phase shifters, and beam splitters) any finite-
dimensional unitary operator can be implemented (Reck et al. 1994). Combined with
Mach-Zehnder interferometers, this system has enough control to extract any
observer-relevant entanglement from any pure state.
310 Nathan Harshman
where σðλi Þ is the spectrum of the differential operator Λi . This is a more robust
separability than “silver” separability described in Eq. (15.5), where only the
spectrum was separable, not the Hilbert space. For “bronze” separability, the
spectrum of each differential operator depends on the values of other separation
constants, and so spectral separability is lost. Only for “gold” separability does
differential separability correspond to a tensor product structure and therewith to
an algebraic notion of separability.
A final method for identifying top-down tensor product structure is requiring
that the tensor product structure be invariant with respect to a symmetry group of
the model. A general theory of when this is possible has not been developed, but
two examples from nonrelativistic physics illustrate the idea.
Consider a model whose Hilbert space is an irrep space of the Galilean group
and whose algebra of observables is the Galilean algebra extended by a central
Symmetry, Structure, and Emergent Subsystems 311
1 ∂2
H1 ¼ þ V ðxÞ (15.20)
2m ∂x2
Here is the first place the restriction to one-dimensional systems pays dividends. First,
one-dimensional systems are always integrable. An integrable system has as many
algebraically independent, globally defined constants of the motion as the number of
degrees of freedom. (The classical definition of integrability is usually formulated in
term of operators generating flows on phase space. In a classical one-dimensional
system, the constraint provided by this integral of motion reduces the two-dimensional
phase space to a one-dimensional manifold, i.e., the trajectory of the particle. There is
some ambiguity in the quantum case. See Caux and Mossel 2011 for a review of the
difficulties). For a one-dimensional system, the Hamiltonian itself is the single
conserved integral of motion necessary for integrability. This does not mean that the
system is necessarily solvable, in the sense that the eigenvalues and eigenstates of the
Hamiltonian can be expressed in closed-form analytic expressions. However, for
moderately well-behaved trapping potentials (Harshman 2017a), the Sturm-Liouville
theory guarantees that the energy spectrum of Eq. (15.20) is a denumerable tower of
singly-degenerate states bounded from below. The wave function with lowest energy
ε0 has no nodes, and each successive state with energy εn has n nodes.
Again assuming a reasonable trapping potential, the Hilbert space of the one-particle
system H 1 is the space of Lebesgue-square-integrable functions L2 ðRÞ on the real line.
This space carries an irrep of the one-dimensional Galilean group, although the
trapping potential breaks that symmetry. Each energy eigenstate jεn i spans a one-
dimensional subspace H 1εn 2 H 1 , leading to the decomposition of H 1 into a direct sum
of energy eigenspaces (or equivalently, time translation symmetry irrep spaces):
∞
H 1 ¼ ⊕ H 1εn (15.21)
n¼0
Symmetry, Structure, and Emergent Subsystems 313
Note that in two and more dimensions, integrability is not guaranteed without
more knowledge of the potential and its symmetries. Although states can still be
labeled by a spectrum of energies and a decomposition in energy subspaces like
Eq. (15.21) is still possible, the subspaces are not necessarily one-dimensional and
much less can be inferred about the properties of the wave functions.
An energy eigenstate basis for the total system is formed by all tensor products
of single-particle basis vectors like jni jεn1 i⊗jεn2 i⊗ . . . ⊗jεnN i. The energy En
of a basis state is the sum of the single-particle energies. Most energies are no
longer singly degenerate, but they are still denumerable and provide a decom-
position of H N into a tower of energy eigenspaces
H N ¼ ⊕ H En (15.24)
En
where the direct sum is over all possible energies constructed as sums of N single-
particle energies εn . Note that each space H En has a complete basis that is
unentangled with respect to the tensor product structure Eq. (15.22).
The dimension of H En is determined by the number of ways the set of single-
particle energies that sum to E n can be permuted. For example, for three particles
the spaces H En can have one, three, or six dimensions. Irreps of S3 either have one
or two dimensions and that signals the presence of additional kinematic symmet-
ries beyond S3 (Leyvraz et al. 1997, Fernández 2013).
In fact, the decomposition Eq. (15.24) is a reduction into the irrep spaces of the
kinematic symmetry group SN o T t , there T t is the time translation group of a
single-particle system and o is the wreath product (Harshman 2016b). There could
be an even larger kinematic symmetry group of the same form incorporating
additional single-particle symmetries, like parity.
Note that I have not made the claim that all the spaces H En correspond to
different energies. That depends one whether each energy can be uniquely associ-
ated to a set of N single-particle energies. If several energy sums coincide, then
there must be an even larger kinematic symmetry group. I call this an emergent
kinematic symmetry, because it cannot be generated by single-particle symmetries
and particle permutations. One example is when the trapping potential is a har-
monic trap and then the maximal kinematic symmetry is U ðN Þ and can be realized
as symmetry transformations on phase space (Baker 1956, Louck 1965). For this
system, the degeneracies of the energy eigenspaces grow like a factorial in the
energy but can be reduced into spaces like H En . Another example of an emergent
kinematic symmetry is when the trapping potential is an infinite square well. Then
there are “pythagorean degeneracies” that do not appear to have a description as a
group of transformations realized on configuration space or phase space (Shaw
1974).
Both of these limiting cases can be understood as examples when the Yang-Baxter
equation holds and there is diffractionless scattering (Sutherland 2004).
One more special case is when the external trap is quadratic in position. The
noninteracting system H N0 is equivalent to an isotropic harmonic oscillator in N
dimensions and, as mentioned earlier, has kinematic symmetry group U ðN Þ. That
system is maximally superintegrable, meaning there are 2N 1 integrals of
motion, and exactly solvable, meaning that the energy is an algebraic function of
the quantum numbers, and all excited states are products of the ground state with
polynomials (Post, Tsujimoto, and Vinet 2012). At finite interaction strength, most
of this additional analytical tractability is lost, but there is one extra integral of the
motion corresponding to the separable center-of-mass degree of freedom (Harsh-
man 2014). This separability survives symmetrization of indistinguishable
particles and, therefore, entanglement between center-of-mass and relative degrees
of freedom remains a dynamical invariant.
For general traps and arbitrary g, the model Eq. (15.26) is not integrable, nor is it
solvable except numerically. Then two questions become: How far from integra-
bility and deep into chaos and is it? How difficult is it to achieve convergent
numerical solutions? The second question has been investigated exhaustively, by
this author and many others, because of the relevance to current experiments
(Serwane et al. 2011, Zürn et al. 2012, 2013, Wenz et al. 2013; for a partial list,
see the references of Harshman 2016a). However, I would claim a productive
metatheory of when particular approximation methods work well has not arrived.
One way to answer the first question about chaos is by comparing the spectrum
to the Wigner-Dyson distribution of eigenvalues of a random matrix (Gutzwiller
1990). According to the Bohigas-Giannoni-Schmit conjecture, this is a universal
feature of systems with quantum chaos. Work on closely related systems suggests
that chaos is present in these systems (Bohigas, Giannoni, and Schmit 1984), but
the world currently waits for a more detailed analysis, especially one that situates
the model in the hierarchy of chaos from ergodic, mixing, Kolmogorov, and
Bernoulli (Ullmo 2016, Gomez, Losada, and Lombardi 2017).
In summary, depending on the trap shape and interaction strength, the model
Eq. (15.26) can manifest the full range of possible dynamic behaviors, from (super)
318 Nathan Harshman
integrability to (conjectured) hard chaos. For integrable cases, there are observ-
ables privileged by dynamical conservation laws that fully characterize the system,
i.e., a complete set of commuting observables. The specific nature of the integrals
of motion depends on the trap and interaction strength. For the noninteracting case,
the integrals of motion are single-particle observables. In integrable interacting
cases like the hard-core limit or Bethe-ansatz solvable cases, the conserved quan-
tities are collective observables built from symmetrized polynomials of single-
particle observables. In contrast, for chaotic cases, the unique conserved observ-
able is the energy itself, and the spectrum matches with a relevant form of
randomness. A goal of this avenue of research is to see if the approach to chaos
can be understood as the dissolution of structures based on irreps of
symmetry group.
the system can be decomposed or factorized into subspaces. The structural unit that
unifies these two features is the irreducible representation.
Irreps appear in many guises – as invariant subspaces in direct sums and tensor
products, as the building blocks of towers for describing identical particles that
generate unusable entanglement and frustrate algebraic separability, as the concept
of energy levels that vary across families of Hamiltonians in the same model, and
more. No matter the underlying ontological commitment of an interpretation, any
formulation of quantum mechanics must account for the prevalence and utility of
these structures. Unlike the Platonic solids, these are not metaphors. They are
mathematical building blocks of quantum mechanics and will remain so even when
new or reformed ontologies emerge.
The most egregious oversight of this chapter is that I have not discussed how
symmetry groups partition the set of observables into irreps. This is more technic-
ally challenging that the Hilbert space arithmetic I have presented, but I think the
potential rewards are a deeper understanding of the connections among the observ-
ables, separability (in all three senses), and integrability.
At this stage, the investigation is still incomplete, but I argue that one notion
emerges – the importance of solvability. Solvability is a concept lying in that
awkward place of being a technical term with multiple overlapping and connecting
definitions in different contexts. One of the unifying themes across these contexts
is that solvable systems play a central role in the interpretation of physical
phenomena. Solvable models in mechanics, like coupled harmonic oscillators
and hydrogenic atoms, are not just touchstones for mathematical analysis. They
are ubiquitous as direct and approximate models in nature, and they provide the
cognitive framework for understanding other physical systems. Is it a coincidence
that solvable models are so useful? Is it just attention bias, i.e., we pay more
attention to things we understand more thoroughly? Or, is there something more
deeply “real” about solvable systems, either in an epistemic or ontic sense?
Acknowledgments
I would like to thank Olimpia Lombardi and the other organizers of the workshop
Identity, indistinguishability and non-locality in quantum physics (Buenos Aires,
June 2017) for assembling such a stimulating group of physicists and philosophers.
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16
Majorization, across the (Quantum) Universe
guido bellomo and gustavo m. bosyk
16.1 Introduction
In how many ways can one represent a given quantum mixed state as a mixture of
pure states? Why (and in which sense) are separable states more disordered
globally than locally? Is it possible to transform a given pure state into another
by means of local operations and classical communication? How should an
adequate formulation of the uncertainty principle be? All these questions, as
dissimilar as they may seem, share one element in common: They can be answered
by appealing to the notion of majorization partial order. Majorization is nowadays
a well-established and powerful mathematical tool with many and different appli-
cations in several disciplines, such as economics, biology, and physics, among
others. Indeed, the seminal idea of this concept had already been glimpsed by
Lorenz (1905) while studying the inequality of wealth distribution and developing
the representation of the (nowadays called) Lorenz curves. Moreover, the famous
Gini coefficient (Gini 1912), widely accepted as a legitimate quantifier of income
distribution inequality, is merely a ratio between graphical areas defined by a
Lorenz curve. Other key contributions to the subject were those by Muirhead
(1903), Dalton (1920), Schur (1923), and Hardy, Littlewood, and Pólya (1929).
The name “majorization,” though, appears first in the prominent book by Hardy,
Littlewood, and Pólya (1934).
At present, it is clear that anyone who is interested in the field would find it
appropriate to begin by the celebrated book by Marshall and Olkin (1979), whose
second edition was coauthored by Arnold (2010). Recently, Arnold (2007) pub-
lished an article entitled “Majorization: Here, there and everywhere,” in which he
presents a sampling of diverse areas in which majorization has been found useful
in the last few years, such as geometry, probability, statistical mechanics, and
graph theory. However, those contributions do not explore its quantum theoretical
implications, which are much more extensively covered, for example, in Nielsen’s
323
324 Guido Bellomo and Gustavo M. Bosyk
lecture notes (Nielsen 2002, see also Nielsen and Vidal 2001). In this chapter, we
attempt to make a brief review of the subject and then to highlight the most
important results of this research line in the quantum realm, in order to offer a
kind of quantum counterpart of Arnold’s work.
A natural question for our current work regards the roots beneath the wide
applicability of majorization in quantum mechanics. Some authors argue that the
connection arises as a result of two important theorems that link majorization to
unitary matrices, Horn’s lemma, and Uhlmann’s theorem, together with the
ubiquity of unitary matrices in quantum mechanics (Nielsen 2002: 5). Here, we
present and discuss a variety of situations to show that the spread applicability of
majorization in the quantum realm emerges as a consequence of deep connections
among majorization, partially ordered probability vectors, unitary matrices, and
the probabilistic structure of quantum mechanics. To this end, we review basic
aspects of majorization, focusing on its connections with some quantum
information problems. In particular, we organize our study here into three
different facets. The first one consists in the role played by majorization as a
disorder signature on quantum states, how quantum operations affect this, and the
connection with quantum entropies. The second one is the study of entanglement
transformations of bipartite pure states, applying local operations, and classical
communication. Third, the problem of how to formulate the uncertainty principle
is posed, and different proposals of majorization uncertainty relations are
reviewed.
In the following section, in order to make the ensuing study more self-
contained, we present some elementary definitions and mathematical properties
related to majorization theory.
We take for granted that any vector that appears below lives in Δd , that is, has its
elements sorted in a nonincreasing manner. In order to define the concept of
majorization between probability vectors, let x and y be two members of Δd . It is
said that x is majorized by y, denoted as x ≺ y, if and only if the entire set of d 1
inequalities
Majorization, across the (Quantum) Universe 325
X
n X
n
xi yi for all n 2 f1; . . . ; d 1g (16.2)
i¼1 i¼1
P P
is satisfied. (Notice that the condition di¼1 xi ¼ di¼1 yi is trivially satisfied; for
that reason we discard this condition from the definition of majorization). The
intuitive idea is that a probability distribution majorizes another if the former is
more concentrated than the latter. In this sense, majorization provides a quantifica-
tion of the notion of disorder for given probability vectors. To fix ideas, let us
observe that any probability vector x 2 Δd trivially satisfies the relation
1 1
. . . ≺ x ≺½1 0 . . . 0 (16.3)
d d
where the left-hand side (l.h.s.) corresponds to the most uncertain case (uniform
probability vector), whereas the right-hand side (r.h.s.) represents complete
certainty.
It is essential to note that majorization defines a partial order over Δd , meaning
that there exist x, y 2 Δd , such that neither x ≺ y nor y ≺ x. That is the case, for
example, for x ¼ ½0:6; 0:2; 0:2 and y ¼ ½0:5; 0:4; 0:1.
There is a way to easily visualize whether, given two probability vectors, there
is a majorization relation between them. This is done through the notion of Lorenz
curve (Lorenz 1905). A Lorenz curve of a probability
vector x 2 Δd consists of the
Pn
linear segments joining the points n; i¼1 xi for n 2 f0; . . . ; dg. Therefore, a
Lorenz curve is always concave and has extreme points ð0; 0Þ and ð1; 0Þ. In this
way, there is a majorization relation between x, y 2 Δd , if and only if their
corresponding Lorenz curves do not intersect each other, except in the extreme
points. In Figure 16.1, we illustrate these situations.
Figure 16.2 Set of vectors majorized by a fixed vector y 2 Δ3 , fx : x≺yg. The set
is given by the points inside the convex hull of the orbit of y under the group of
permutation matrices.
• x ≺ y if and only if x lies in the convex hull of the orbit of y under the group of
permutation matrices (see Figure 16.2)
The second equivalent condition, linking majorization with concave functions,
relates further to the notions of Schur-concavity and entropy. Functions that
preserve the majorization order form a large class, which was formerly studied
by Schur (1923). In his honor, we say that a function Φ : Δd ↦ R is Schur-concave
if it (anti)preserves the majorization relation, that is,
if x ≺ y ) ΦðxÞ ΦðyÞ for all x, y 2 Δd : (16.4)
For example, the very general family of ðh; ϕÞ-entropies satisfies the Schur-
concavity (Bosyk, Zozor, Holik, et al. 2016). They are defined as
!
Xd
H ðh;ϕÞ ðxÞ ¼ h ϕðxi Þ , (16.5)
i¼1
where the entropic functionals h : R ↦ R and ϕ : ½0; 1 ↦ R are such that either: (i)
h is increasing and ϕ is concave, or (ii) h is decreasing and ϕ is convex, together
with the conventions ϕð0Þ ¼ 0 and hðϕð1ÞÞ ¼ 0. The ðh; ϕÞ-entropies can be
classified according to whether they satisfy the additivity relation
Majorization, across the (Quantum) Universe 327
or not. Particular instances of Eq. (16.6) that are additive are the well-known
Shannon (1948), Rényi (1961) and Burg (1967) entropies, given respectively by
X
d
H 1 ðxÞ ¼ xi log xi , (16.7)
i¼1
sign α Xd
H α ðxÞ ¼ log xαi , for all α 2 R∖f0; 1g, (16.8)
1α i¼1
1X d
H Burg ðxÞ ¼ log xi : (16.9)
d i¼1
• reflexivity: x ≺ x,
• symmetry: if x ≺ y and y ≺ x, then x ¼ y,
• transitivity: if x ≺ z and z ≺ y, then x ≺ y.
Remarkably enough, Cicalese and Vaccaro (2002) have shown that majorization
over Δd defines an even more complex structure: a lattice. This means that there
always exist the infimum (join), x∧y, and the supremum (meet), x∨y. By defin-
ition, x∧y means that x∧y ≺ x, x∧y ≺ y, and z ≺ x∧y for all z such that z ≺ x and
z ≺ y. In a similar way, x∨y means that x ≺ x∨y, y ≺ x∨y, and x∨y ≺ z0 for all z0
such that x ≺ z0 and y ≺ z0 (see Figure 16.3). The algorithms to obtain both the
infimum and the supremum between arbitrary vectors in terms of its elements have
also been given by Cicalese and Vaccaro (2002).
Furthermore, by appealing to the subadditivity and supermodularity of Shannon
entropy, the authors introduced a proper distance on the lattice (Cicalese, Gargano,
and Vaccaro 2013): given x, y 2 Δd , the distance D : Δd Δd ↦ R is defined as
328 Guido Bellomo and Gustavo M. Bosyk
Figure 16.3 Hasse diagram for probability vectors x and y, its infimum (join) x∧y,
and its supremum (meet) x∨y.
Figure 16.4 Illustration of the convex set of quantum states. The pure states are
the extreme points of the set (the border). Notice that a given mixed state can be
written in infinitely many ways as convex combinations of pure states. In the
example in this figure, ρ can be expressed by mixing either ρ1 and ρ2 , or ρ3 and ρ4 .
vectors, we are going to work over a set of finite dimensional (positive semidefinite
and trace-class) density matrices or operators over a Hilbert space H ffi Cd ,
D ¼ ρ 2 Cdd : ρ 0 and Trρ ¼ 1 (16.13)
These matrices are also selfadjoint, ρ† ¼ ρ. The set D is convex, meaning that
given ρ1 , ρ2 2 D, then any mixture of the form p ρ1 þ ð1 pÞρ2 2 D, when
p 2 ½0; 1. The extreme states of D are the pure states that satisfy
ρ2 ¼ ρ ¼j ψihψ j, with jψ i 2 H . States which are not pure are called “mixed.”
Notice that there exist infinite ways to decompose a mixed state in terms of convex
combinations of pure states, but they are not arbitrary (we will go more deeply into
this when we discuss the Schrödinger theorem; see Figure 16.4).
Moreover, any state ρ has a spectral decomposition
X
d
ρ¼ λi jei ihei j, (16.14)
i¼1
P
with λi 0, i λi ¼ 1, and ei jej ¼ δij .
Since we plan to later talk about quantum correlations, we should remember
how to describe composite systems. The states of multipartite systems act over
the tensor product of the individual Hilbert spaces. For example, for a bipartite
A [ B system, where H A and H B are the respective individual Hilbert spaces, the
joint state acts over H AB ¼ H A ⊗H B . If ρAB is the state of the bipartite system,
we can obtain the (reduced) states of each subsystem by performing the partial
traces: ρAðBÞ ¼ Tr BðAÞ ρAB . When the joint state is pure, ρAB ¼j ψ AB i ψ AB j, we
Acan
use the Schmidt technic to decompose the same in terms of local basis, ji i
and fjiB ig, as
AB Xn
pffiffiffiffiffi
B X
ψ ¼ ψ i i i , with ψ i 0, ψ i ¼ 1 and n ¼ min d A ; dB :
i¼1 i
(16.15)
330 Guido Bellomo and Gustavo M. Bosyk
¼ ψ ⊗ ψ . In other words,
the system is in a pure separable state. When more than one Schmidt coefficient is
positive, the system is in an entangled state. The extreme case is that of a
maximally entangled
pffiffiffiffiffiffiffiffi state, characterized by the equality of the Schmidt coeffi-
cients, ψ i ¼ 1=n for all i. Hereafter, let σ ðψ Þ denote the probability
vector
formed by the squared Schmidt coefficients of a bipartite pure state
ψ AB i.
Mixed joint states demand a more complex hierarchy of correlations. A mixed
state ρAB is a product state whenever it is expressible as a product of individual
states as ρA ⊗ρB . In turn, a mixed separable state can be written as a convex
combination of product states,
X X
ρAB ¼ pi ρAi ⊗ρBi , with pi 0 and pi ¼ 1: (16.16)
i i
Mixed product states are particular cases of separable states. Any state that is
nonseparable is called “entangled.”
We are also going to consider linear, completely positive, trace-preserving maps
from the set of density operators into itself ℰ : D ↦ D, also called “quantum
operations.” All these operations have a Kraus representation of the form
X X
ℰðρÞ ¼ Pi ρP†i with P†i Pi ¼ I (16.17)
i i
ity vectors entails complementary information about the quantum state. Addition-
ally, there is still a third probability vector that one can obtain from the state ρ, if
one considers a preferred observable quantity X, namely, the one defined by the
components of the expectation value in this way. Let X be an observable with
P
discrete and nondegenerate spectrum, that is, X ¼ di¼1 xi jiX ihiX j. The correspond-
ing probability vector, pðX; ρÞ, has i component pi ðX; ρÞ ¼ Tr ðρjiX ihiX jÞ. In what
follows, we are going to present some paradigmatic problems in quantum mech-
anics in which the notion of majorization, together with one of those descriptions,
gives the right starting point toward their solutions (see Figure 16.5).
1 X d pffiffiffiffi
jψ i i ¼ pffiffiffiffi U ik λk jek i: (16.20)
pi k¼1
It follows that the proposed expression is possible whenever the vector p, with
components pi , satisfies the relation p ¼ BλðρÞ, where B is some doubly stochastic
matrix. Hence, recalling Schur’s theorem, we have (Nielsen 2000)
X
M
9fpi ; jψ i ig : ρ ¼ pi jψ i ihψ i j , p ≺ λðρÞ: (16.21)
i¼1
This last equation gives a conclusive requisite that any ensemble of pure states
must fulfill to represent a given mixed state.
332 Guido Bellomo and Gustavo M. Bosyk
Figure 16.5 Our roadmap. Given a quantum state, ρ, there are several associated
probability vectors, which involve different informational aspects of the same.
Each probability vector allows complementary descriptions that are related to
different quantum information problems, which are discussed throughout this
chapter in the sections indicated between brackets.
From the example just shown, we can also have some insight on how to
naturally define a notion of majorization between quantum states by appealing to
its spectral decomposition. Given ρ, σ over H , we say that ρ is majorized by σ as
follows:
ρ ≺ σ , λðρÞ ≺ λðσ Þ: (16.22)
Now, a reasonable question regards the connection between two given states that
satisfy a majorization relation from the viewpoint of quantum operations. In other
words, whether there is any operation linking ρ and σ which satisfies ρ ≺ σ. What
Uhlmann proved is that ρ ≺ σ if and only if the majorized state can be obtained
from the latter by means of convex combinations of unitary maps (Uhlmann
1970)
X X
ρ¼ pi U i σU †i , with U i U †i ¼ I ¼ U †i U i , pi 0 and pi ¼ 1: (16.23)
i i
That is, ρ lies in the convex hull of the unitary orbit to which σ belongs. It is
interesting to observe that, contrary to the classical case, convex combinations of
unitary operations give a more general class of bistochastic channels, also known
as random external fields (Alicki and Lendi 1987). Moreover, one can also prove
Majorization, across the (Quantum) Universe 333
where the entropic functions h and ϕ satisfy the same requirements as that in the
classical case. Notice that the quantum entropy of a given state ρ coincides with the
classical one for the probability vector formed by the eigenvalues of ρ, that is,
Sðh;ϕÞ ðρÞ ¼ H ðh;ϕÞ ðλðρÞÞ. In this way, we find, in the quantum realm, the same close
connection between majorization, bistochastic maps, and entropies.
Several properties of Eq. (16.25) have been studied by Bosyk, Zozor, Holik,
et al. (2016). In addition to the Schur-concavity of Eq. (16.25), two specific links to
majorization are the following. On the one hand, as a consequence of Schur-
concavity and Schrödinger theorem, one has that the quantum ðh; ϕÞ-entropy of
P
an arbitrary statistical mixture of pure states ρ ¼ M i¼1 pi jψ i ihψ i j, is upper-
bounded by the classical ðh; ϕÞ-entropy of the probability vector formed by the
mixture weights, that is,
Sðh;ϕÞ ðρÞ H ðh;ϕÞ ðpÞ: (16.26)
On the other hand, when dealing with quantum systems, it is of interest to estimate
the effect of a given quantum operation on them. In particular, one may guess that
a measurement can only perturb the state and, thus, that the entropy will increase.
This is also true for more general quantum operations. Indeed, as a consequence of
the Schur-concavity and a result by Chefles (2002), one has that any bistochastic
map ℰ has a nondecreasing effect on the entropy, that is,
334 Guido Bellomo and Gustavo M. Bosyk
where the equality holds if and only if the map is unitary ℰðρÞ ¼ UρU † : Notice that
in the case of a nonbistochastic map, the entropy can decrease. For instance, let ρ
be the density operator of an arbitrary mixed qubit system, with nonvanishing
quantum ðh; ϕÞ-entropy, and ε a completely positive and trace-preserving (but not
unital) map characterized by Kraus operators P1 ¼ j0ih0 j and P2 ¼ j0ih1 j. Then,
the system, after the action of this map, is on the pure state ℰðρÞ ¼ j0ih0 j; thus, it
has zero quantum ðh; ϕÞ-entropy.
Projective measurements are particular cases of bistochastic maps (see Eq. (16.18)).
Hence, for a given quantum system, the difference of quantum entropies between the
postmeasurement and the premeasurement states works as a signature of the disturb-
ance of the state of a system due to the measurement. In particular, consideration of
local disturbances gives place to a type of quantum correlations measures and a way to
characterize them (see, e.g., Luo 2008, Horodecki et al. 2005). As the main ingredient
needed to guarantee the validity of these measures is the property expressed in
Eq. (16.27), these measures can be defined in terms of generalized quantum entropies
(see, e.g., Rossignoli, Canosa, and Ciliberti 2010, Bosyk, Bellomo, Zozor, et al. 2016).
In other words, since those squared coefficients coincide with the reduced states’
eigenvalues
336 Guido Bellomo and Gustavo M. Bosyk
AB LOCC
AB
ψ !
ϕ , Tr A
ψ AB ψ AB
≺ Tr A
ϕAB ϕAB
: (16.31)
We stress that the majorization relationship constitutes the necessary and sufficient
condition under which this transformation is allowed, without any reference to the
corresponding Schmidt bases.
As already mentioned, majorization gives a partial order over Δd and, as such,
Nielsen’s result does not hold in general, in the sense that there exists a pair of
states that neither of them majorizes each other: jψ i ↮ j ϕi. For instance, it is easy
to check that j ψi ↮ j ϕi by LOCC when the squared Schmidt coefficients are
σ ðψ Þ ¼ ½0:60; 0:15; 0:15; 0:10 and σ ðϕÞ ¼ ½0:50; 0:25; 0:20; 0:05, because
σ ðψ Þ ⊀ σ ðϕÞ and σ ðϕÞ ⊀ σ ðψ Þ. With this in mind, the celebrated result discussed
earlier, due to Nielsen, has subsequently been extended to the case of nondetermi-
nistic LOCC transformations by Vidal (1999). In that case, one looks for the
maximal probability of success. On the other hand, it is also possible to provide
further insight if one considers deterministic transformations but with approximate
target states. Vidal, Jonathan, and Nielsen (2000) have solved this problem by
invoking a criterion of maximal fidelity. The same problem has been tackled from
a different perspective, exploiting the lattice structure of majorization and showing
that both proposals are linked via a majorization relation (Bosyk, Sergioli, Freytes,
et al. 2017). The latter seems to be the first attempt to exploit the lattice character of
the majorization partial order in a quantum information context, beyond its well-
known partial-order properties.
Another generalization, proposed by Jonathan and Plenio (1999), consists in the
extension of the set of initial and final states by appealing to deterministic
entangled-assisted LOCC, that is, considering a shared catalytic entangled state
between both parts. In this protocol, we have a new partial-order relation that it is
called “trumping majorization” and reads as follows: Given x, y, z 2 Δd , it is said
that x is trumping majorized by y (and denoted by x ≺ T y) if and only if there exists
a catalytic r such that x ⊗ z ≺ y ⊗ z (see Daftuar and Klimesh 2001 for some
mathematical properties of trumping, and Müller and Pastena 2016 for an exten-
sion of this concept related to Shannon entropy). Although it is an open question
whether trumping majorization can be endowed with a lattice structure in the
general case (Harremoës 2004), it has been recently shown that the structure holds
for the minimal nontrivial case, namely the case of four-dimensional vectors and
two-dimensional catalysts (Bosyk, Freytes, Bellomo, et al. 2018).
It is notable that all these questions can be enclosed under the problem
of convertibility of one kind of physical resource into another. Lately, this
resource theoretic approach has been extensively applied to attack a bunch of
quantum information-related topics such as, for instance, nonlocal correlations
(see, e.g., Barrett et al. 2005, de Vicente 2014), quantum coherence and asymmetry
Majorization, across the (Quantum) Universe 337
(see, e.g., Ahmadi, Jennings, and Rudolph 2013, Piani et al. 2016), quantum
thermodynamics (see, e.g., Brandao et al. 2013, Gour et al. 2015) and super-
selection rules (see, e.g., Gour and Spekkens 2008). Remarkably enough, this
formalism has recently been applied out of the quantum domain, for instance, to
the study of polarization-coherence properties of classical electromagnetic fields
(Bosyk, Bellomo, and Luis 2018a, 2018b) as well as to the study of measures of
statistical complexity (Rudnicki et al. 2016).
value vanishes. For example, this happens when the quantum state is an eigenstate
of one the observables. In other words, the r.h.s of Eq. (16.32) is state-dependent
(that is, it is not universal) and does not fully characterize the incompatibility of the
observables.
For those reasons, several alternative uncertainty relations have been proposed
in order to overcome these issues. Among them, geometric (Landau and Pollak
1961, Bosyk et al. 2014), entropic (Deutsch 1983, Maassen and Uffink 1988,
Zozor, Bosyk, and Portesi 2014) and majorization (Partovi 2011, Friedland,
Gheorghiu, and Gour 2013, Puchała, Rudnicki, and Życzkowski 2013) uncertainty
relations have appeared as the most prominent ones. In general, an uncertainty
relation is an inequality of the form
2
0 1 þ c0
and ω1 ðc Þ ¼ with
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
c0 ¼ max jhiX jiY ij2 þ
i0X ji0Y
, (16.37)
where the maximum is taken over the all indexes iX ¼ i0X and iY 6¼ i0Y , and over the
all indexes iX 6¼ i0X and iY ¼ i0Y .
Finally, let us observe that from a majorization uncertainty relation one can always
obtain the corresponding entropic version by using Schur-concave additive entropies.
For instance, the Rényi entropic uncertainty relation obtained from Eq. (16.34) is
1 α α
H α ðpðX;ρÞÞ þH α ðpðY;ρÞÞ log ω1 ðcÞα þ ðω2 ðc0 Þ ω1 ðcÞÞ þ ð1 ω2 ðc0 ÞÞ :
1 α
(16.38)
It can be shown that for Shannon entropy (α ¼ 1) this entropic uncertainty relation
is stronger than the one derived by Deutsch (1983)
1þc
H ðpðX; ρÞÞ þ H ðpðY; ρÞÞ 2 log : (16.39)
2
Therefore, majorization-based uncertainty relations not only give adequate formu-
lations of the uncertainty principle, but also allow stronger entropic-based expres-
sions to be obtained.
Acknowledgments
We are extremely grateful to the organizers of the workshop Identity, indistinguish-
ability and non-locality in quantum physics (Buenos Aires, June 2017). This work
was partially supported by CONICET and UNLP and Grant 57919 from the John
Templeton Foundation.
340 Guido Bellomo and Gustavo M. Bosyk
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and the quantum resource theory of asymmetry,” New Journal of Physics, 15:
013057.
Alicki, R. and Lendi, K. (1987). Quantum Dynamical Semigroups and Applications.
Berlin: Springer-Verlag.
Arnold, B. (2007). “Majorization: Here, there and everywhere,” Statistical Science, 22:
407–413.
Barrett, J., Linden, N., Massar, S., Pironio, S., Popescu, S., and Roberts, D. (2005).
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Quantum Entanglement. Cambridge: Cambridge University Press.
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Tucumán Revista, Serie A, 5: 147–151.
Bosyk, G. M., Bellomo, G., and Luis, A. (2018a). “A resource-theoretic approach to
vectorial coherence,” Optics Letters, 43: 1463–1466.
Bosyk, G. M., Bellomo, G., and Luis, A. (2018b). “Polarization monotones of two-
dimensional and three-dimensional random electromagnetic fields,” Physical Review
A, 97: 023804.
Bosyk, G. M., Bellomo, G., Zozor, S., Portesi, M., and Lamberti, P. W. (2016). “Unified
entropic measures of quantum correlations induced by local measurements,” Physica
A: Statistical Mechanics and its Applications, 462: 930–939.
Bosyk, G. M., Freytes, H., Bellomo, G., and Sergioli, G. (2018). “The lattice of
trumping majorization for 4D probability vectors and 2D catalysts.” Scientific
Reports, 8: 3671.
Bosyk, G. M., Sergioli, G., Freytes, H., Holik, F., and Bellomo, G. (2017). “Approximate
transformations of bipartite pure-state entanglement from the majorization lattice,”
Physica A: Statistical Mechanics and its Applications, 473: 403–411.
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formulation of the uncertainty principle,” Physical Review A, 89: 034101.
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Majorization, across the (Quantum) Universe 341
17.1 Introduction
Decoherence is a process that leads to spontaneous suppression of quantum
interference. The orthodox explanation of the phenomenon is given by the environ-
ment-induced-decoherence approach (see, e.g., Zurek 1982, 1993, 2003),
according to which decoherence is a process resulting from the interaction of an
open quantum system and its environment. By studying different physical models,
it was proved that, when the environment has a huge number of degrees of freedom
and for certain interactions, the reduced state of the open system rapidly diagona-
lizes in a well-defined preferred basis.
The environment-induced approach has been extensively applied to many areas
of physics, such as atomic physics, quantum optics, and condensed matter, and has
acquired a great importance in quantum computation, where the loss of coherence
represents a major difficulty for the implementation of the information processing
hardware that takes advantage of superpositions. In the field of the foundations of
physics, this approach has been conceived as the key ingredient to explain
the emergence of classicality from the quantum world, because the preferred
basis identifies the candidates for classical states (see, e.g., Elby 1994, Healey
1995, Paz and Zurek 2002). It has been also considered a relevant element in
different interpretations or approaches to quantum mechanics (for a survey, see
Bacciagaluppi 2016).
The wide success of the environment-induced approach to decoherence over-
shadowed any conceptual difficulty: Only a few works were devoted to analyze the
assumptions and limitations of the orthodox approach. In resonance with this fact,
the different approaches to decoherence that have arisen to face those difficulties
were not taken into account with the care that they deserve. In this chapter we will
show that there is a different perspective to understand decoherence – a closed-
system approach – which not only solves or dissolves the problems of the orthodox
345
346 Sebastian Fortin and Olimpia Lombardi
which there is, in nature, a fixed set of mutually disjoint atomic quantum systems
that constitute the building blocks of all the other quantum systems (Bacciagaluppi
and Dickson 1999). Good candidates for elemental systems are those represented
by the irreducible representations of the symmetry group of the theory.
From this viewpoint, when quantum systems interact, their states may become
entangled: “By the interaction the two representatives [the quantum states] have
become entangled” (Schrödinger 1935: 555, when he coined the term ‘entangled’).
In this case, it is said that the composite system is an entangled state, because it
cannot be obtained as the tensor product of the components’ states. Entanglement
is, therefore, responsible for the correlations between the values of the observables
of the two subsystems.
This bottom-up ontological view leads us to first consider two particles, say, a
proton p and an electron e, represented by the Hilbert spaces H p and H e and in
states ψp 2 H p and ψe 2 H e , respectively. Then, the state ψ 2 H p ⊗H e of the
hydrogen atom as a composite system is said to be entangled when ψ 6¼ ψp ⊗ψe for
any pair of states ψp and ψe . This suggests that “entangled” is a property that
applies or not to the state of a composite system. However, the hydrogen atom can
also be represented as constituted by two different subsystems, the center-of-mass
system ψc 2 H c and the relative system ψr 2 H r , such that the state of the
hydrogen atom ψ 2 H c ⊗H r can be obtained as ψ ¼ ψc ⊗ψr : Now the state of
the composite system is not entangled. Although conceiving the hydrogen atom as
being composed of a proton and an electron seems more natural, there are group
reasons that may lead to considering that the decomposition in a center-of-mass
system and a relative system is more fundamental (see Ardenghi, Castagnino, and
Lombardi 2009). This means that it cannot be said that a state of a composite
system is entangled or not without first deciding which decomposition of the
system will be considered.
John Earman stresses this fact by saying:
[A] state may be entangled with respect to one decomposition but not another; hence,
unless there is some principled way to choose a decomposition, entanglement is a radically
ambiguous notion.
(Earman 2015: 303)
(2015: 324, 325, 327). A notion is ambiguous if it has more than one meaning; so,
in science and in philosophy ambiguity must be avoided. Therefore, if the notion of
entanglement is ambiguous, the need for a clear-cut decision about how to split the
composite system into subsystems seems completely reasonable. Nevertheless, a
different view is possible: The notion of entanglement is not ambiguous; it is
relative to the decomposition. The difference between ambiguity and relativity is
not irrelevant at all. Whereas the first is a conceptual problem to be solved, the
second is a common feature of physical concepts. In fact, the concept of velocity is
not ambiguous because it is relative to a reference frame. In the same sense,
entanglement is a notion that acquires a precise meaning when relativized to a
certain partition of the composite system and, as a consequence, no absolute
criterion to select the right decomposition is needed.
The relative conception of entanglement invites us to reverse the general
approach to quantum mechanics – from the traditional, classically inspired
bottom-up view, to a top-down view that endows the composite system with
ontological priority. From this perspective, even if two systems exist independ-
ently before interaction, after the interaction their existence is only derivative, they
become components of the composite system on a par with other subsystems
resulting from any different decomposition. This view finds a significant affinity
with the so called quantum structure studies, which deal with the different ways in
which a quantum system can be decomposed into subsystems according to differ-
ent tensor product structures (Harshman and Wickramasekara 2007a,b, Jeknić-
Dugić, Arsenijević, and Dugić 2013, Arsenijević, Jeknić-Dugić, and Dugić 2016,
Harshman 2016).
But the top-down view can be generalized a step further. Up to this point, the
relation between “top” and “down” was described in terms of decomposing the
composite system into its subsystems: The result of decomposition are subsystems,
represented by Hilbert spaces; the tensor product of the Hilbert spaces of the
subsystems is the Hilbert space of the composite system. But the top-down
relationship can also be conceptualized in terms of algebras of observables, in
resonance with the algebraic approach to quantum mechanics (Haag 1992). The
whole system, represented by its algebra of observables, can be partitioned into
different parts, identified by the subalgebras, even when these subalgebras do not
correspond to subsystems represented by Hilbert spaces. This perspective, released
from the subsystem-dependent view anchored in tensor product structures, was
proposed by Howard Barnum and colleagues (2003) as the basis for a
generalization of the notion of entanglement to partitions of algebras. This gener-
alized notion becomes the usual notion of entanglement when the partition of the
algebra of the whole system defines a decomposition of the system into subsystems
(Barnum et al. 2004, Viola et al. 2005; Viola and Barnum 2010). A further
A Closed-System Approach to Decoherence 349
crucial role in all the discussions of the emergent classicality. This issue was raised earlier,
but the progress to date has been slow at best. Moreover, replacing “systems” with, say,
“coarse grainings” does not seem to help at all, we have at least tangible evidence of the
objectivity of the existence of systems, while coarse-grainings are completely “in the eye
of the observer.”
(Zurek 2000: 338; see also Zurek 1998).
It is quite clear that the problem can be removed from a top-down closed-system
perspective as that delineated in the previous section.
In order to explain decoherence from a closed-system perspective, let us begin
by recalling the definition of the concept of reduced state, because the environ-
ment-induced-decoherence program decides to study the time behavior of the
reduced state of the system of interest. The reduced state ρr1 of a system S1 ,
subsystem of a system S, is defined as the density operator by means of which
the expectation values of all the observables of S belonging exclusively to S1 can
be computed. As Maximilian Schlosshauer emphasizes, strictly speaking, a
reduced density operator is only a “calculational tool” for computing expectation
values (Schlosshauer 2007: 48). This means that the description of decoherence in
terms of the reduced state of the open system is conceptually equivalent to the
description in terms of the expectation values of the observables of the open
system but viewed from the perspective of the whole closed system. This is the
path we will follow here.
where ρdU ðtÞ is not completely diagonal, but is diagonal in the preferred basis
of OS .
where the ρii and the Oii are the diagonal components, and the ρij and the Oij are the
nondiagonal components of ρ and O, respectively, in a certain basis. The second
sum of Eq. (17.5) represents the specifically quantum interference terms of the
expectation value. If those terms vanished, the expectation value would adopt the
structure of a classical expectation value, where the Oii might be interpreted as
possible values, and the ρii might play the role of probabilities, since they are
positive numbers that are less than or equal to one and sum to one.
In the light of this idea, the process of decoherence described by the evolution of
Eq. (17.2) leads to a classical-like expectation value, since ρdS ðtÞ is diagonal in the
preferred basis of OS :
tt D X
hOS iρS ðtÞ ! hOS iρd ðtÞ ¼ OSii ρdSii ðtÞ (17.6)
S
i
352 Sebastian Fortin and Olimpia Lombardi
where the ρSii and the OSii are the diagonal components of ρS and OS , respectively,
in the preferred basis.
However, the same move cannot be applied to the evolution as expressed in
Eq. (17.4), because ρdU ðt Þ is not completely diagonal: It is diagonal only in the
components corresponding to the preferred basis of OS . Nevertheless, decoherence
can be described from the closed-system perspective analogously to Eq. (17.6) if a
coarse-grained state ρG ðt Þ of the closed system U is defined as the operator such
that:
8ðOUS ¼ OS ⊗I E Þ 2 OU hOUS iρU ðtÞ ¼ hOUS iρG ðtÞ (17.7)
where ρdG ðtÞ remains completely diagonal for all times t t D . Now it can be said
that the expectation value also acquires a classical form from the closed-system
perspective since:
tt D X
hOUS iρU ðtÞ ! hOUS iρd ðtÞ ¼ OUSii ρdGii ðt Þ (17.10)
G
i
where the ρdGii and the OUSii are the diagonal components of ρdG and OUS ,
respectively, in the basis of decoherence. It is quite clear that ρG , although
operating onto OU , is not the quantum state of U: It is a coarse-grained state
of the closed system that disregards certain information of its quantum state.
However, ρG supplies the same information about the open system S as the
reduced state ρS , but now from the viewpoint of the composite system S. In fact,
if the degrees of freedom of the environment are traced off, the reduced state ρS is
obtained:
Tr E ρG ¼ ρS (17.11)
Therefore, the reduced density operator ρS can also be conceived of as a kind of
coarse-grained state of U, which disregards certain degrees of freedom considered
as irrelevant.
A Closed-System Approach to Decoherence 353
where ρdG ðtÞ remains diagonal in the preferred basis for all times t t D . This
means that, although the off-diagonal terms of ρU ðt Þ never vanish through its
unitary evolution, it might be said that the system decoheres relatively to the
A Closed-System Approach to Decoherence 355
Acknowledgments
We are grateful to the participants of the workshop Identity, indistinguishability
and non-locality in quantum physics (Buenos Aires, June 2017) for their
A Closed-System Approach to Decoherence 357
interesting comments. This work was made possible through the support of Grant
57919 from the John Templeton Foundation and Grant PICT-2014–2812 from the
National Agency of Scientific and Technological Promotion of Argentina.
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A Closed-System Approach to Decoherence 359
18.1 Introduction
The description of the classical limit of a quantum system is one of the most
important issues in the foundations of quantum mechanics (see Cohen 1989). This
problem has been formulated in different ways and explained by appealing to
different interpretations (see Schlosshauer 2007). The attempts to explain the
classical limit go back to the correspondence principle, proposed by Niels Bohr.
This principle establishes a connection between quantum observables and their
classical counterparts when Planck’s constant is small enough in comparison with
relevant quantities of the quantum system. In particular, this happens in the limit of
large quantum numbers.
Nowadays, the most important approach to describe the classical limit is based
on the decoherence process (see Schlosshauer 2007). The general idea of this
approach is to explain the disappearance of the interference terms of quantum
states by appealing to the decoherence process induced by the environment. In this
way, the coherence needed for most typical quantum phenomena is lost, and the
classical features appear instead.
As is well known, the set of observables associated with a quantum system
forms a noncommutative algebra. This differs from the classical description of
physical systems, in which observables are represented by functions over a phase
space, which form a commutative algebra. This difference between quantum and
classical systems has a correlate in terms of the elementary properties of physical
systems. The elementary properties of quantum systems (also known as “yes-no
tests” or elementary experiments) are represented by orthogonal projectors acting
on a Hilbert space. These projectors form a non-Boolean lattice (more specifically,
a complete, atomic, atomistic, orthomodular lattice, satisfying the covering law,
see Kalmbach 1983). Instead, the elementary properties of a classical system are
the measurable subsets of the phase space, which form a Boolean lattice.
360
A Logical Approach to the Quantum-to-Classical Transition 361
or a large number of particles. A result that goes in line with the correspondence
principle is the Ehrenfest theorem. This approach is still important today, in particu-
lar for studying quantum phenomena in the semiclassical level.
Paul Dirac proposed another explanation of the classical limit, appealing to the
destructive interference among all the possible paths of the physical system (Dirac
1933). In this way, he showed that the classical action path has the dominant
contribution. This idea was subsequently elaborated by Richard Feynman (1942)
in his thesis, opening the door to the celebrated path-integral formulation of
quantum mechanics.
All these approaches presented problems, which where extensively discussed in
the literature. In particular, it is important to remark that Bohr himself did not
considered the classical limit as an explanation of the emergence of classical
reality. Quite on the contrary, Bohr believed that the classical realm exists inde-
pendently of quantum theory and cannot be derived from it. As is well known, the
discussion about the classical limit is subtle and problematic, and there is no real
agreement on a solution.
Nowadays, the most important approach for describing the classical limit is
based on the environment-induced decoherence. In this approach, it is considered
that the quantum-to-classical transition is the result of the loss of coherence of the
system due to the interaction with its environment (Schlosshauer 2007). Many
physicists considered this proposal as the correct explanation of the classical limit
(and also of the measurement process); however, some objections were raised,
because the decoherence process would not explain how the logical structure of the
elementary properties becomes a classical logic.
Another important approach to the study of the classical limit is based on
algebras’ deformation (see Landsman 1993). In this formalism, quantum commu-
tators (or equivalently, Moyal brackets) reduce to Poisson brackets, deforming the
algebra involved.
In what follows, we present an alternative approach to describe the classical
limit. This is a logical approach, based on the evolution of the quantum observ-
ables, and it allows describing the quantum-to-classical transition of the logical
structure of the quantum systems. In the next section, we review some basic
features about the lattice of the elementary properties of classical and quantum
systems, which are relevant to our logical approach of the classical limit.
In classical mechanics, physical systems are represented by the phase space, and
their properties are represented by subsets of the phase space (see Kalmbach 1983).
In quantum mechanics, the physical systems are represented by Hilbert spaces and
the properties are represented by closed vector subspaces or by their corresponding
orthogonal projectors (von Neumann 1932; for a recent discussion about the logic
approach to quantum mechanics, see Domenech, Holik, and Massri 2010, Holik,
Massri, and Ciancaglini 2012, Holik, Massri, Plastino, and Zuberman 2013, Holik,
Plastino, and Sáenz 2014, Holik and Plastino 2015; for applications to quantum
histories see Omnès 1994, Losada, Vanni, and Laura 2013, 2016, Griffiths 2014,
Losada and Laura 2014a, b). In both cases, the set of all the properties of a system
has an orthocomplemented lattice structure. This implies that there is an order
relation () such that for each pair of properties there are an infimum (∧) and a
supremum (∨), and each property p has a complement p⊥ with adequate proper-
ties. All orthocomplemented lattices satisfy certain inequalities, called distributive
inequalities (see Kalmbach 1983):
a∧ðb∨cÞ ða∧bÞ∨ða∧cÞ
(18.1)
a∨ðb∧cÞ ða∨bÞ∧ða∨cÞ
When the equalities hold, the lattice is distributive. An orthocomplemented and
distributive lattice is called a Boolean lattice. The distributive property is an
essential feature that differentiates classical and quantum lattices of properties.
In the classical case, the properties of the system are represented by the subsets
of its phase space. The partial-order relation is given by the inclusion () of sets.
The infimum and the supremum are the intersection (\) and the union ([) of sets,
respectively, and the complement of a property p is the complement of sets pc. The
set of classical properties is not only an orthocomplemented lattice, but also a
distributive one, i.e., classical properties satisfy the distributive equalities. There-
fore, the logical structure of a classical system is Boolean. This structure is usually
called classical logic.
The quantum case is very different. The properties are represented by closed-
vector subspaces (or by their corresponding orthogonal projectors; von Neumann
1932). Thus, the logical structure of quantum systems is given by the algebraic
structure of closed subspaces. The set of all quantum properties is also an
orthocomplemented lattice, and, as in the classical case, the partial order relation
is given by the inclusion of subspaces, and the infimum is given by the intersection
of subspaces. However, the supremum and the complement of properties are
different from the classical ones. The supremum is given by the sum of subspaces
and the complement of a property is its orthogonal subspace. The resulting lattice
is nondistributive (see Kalmbach 1983), and therefore, it is not Boolean. This
structure is called quantum logic (Birkhoff and von Neumann 1936).
364 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada
The distributive inequalities are the main difference between classical and
quantum logic. In the classical lattice, all properties satisfy the distributive equal-
ities, but in the quantum lattice, only distributive inequalities hold, in general.
However, for some subsets of quantum properties the equalities hold. When a
subset of properties satisfies the distributive equalities, they are called compatible
properties. It can be proved that a sufficient and necessary condition for a set of
properties to be compatible is that the projectors associated with the properties
commute. Moreover, it can be shown that properties associated with different
observables are compatible if the observables commute. If, on the contrary, two
observables do not commute, some of the properties associated with them are not
compatible. Therefore, by extension, commuting observables are called compatible
observables.
The differences between classical and quantum logic are of fundamental import-
ance for the classical limit problem. If a quantum system undergoes a physical
process such that its behavior becomes classical, then its logical structure of
properties should undergo a transition from quantum logic to classical logic, i.e.,
its lattice structure should become distributive. However, the description of the
classical limit of a quantum system usually focuses on the state of the system.
The mathematical description of this process does not explain how the logical
structure changes on time. Therefore, in these approaches it is not possible to
describe how the structure of quantum properties becomes classical. In order to
give an adequate approach to the classical limit, we need a description in which
observables and physical properties evolve over time, changing the logical struc-
ture of the system.
After time tD , the algebra of observables V ðtÞ becomes commutative, and the
corresponding orthomodular lattice L V ðtÞ becomes nondistributive. The logical
classical limit is expressed by the fact that, while L V ð0Þ is a nondistributive lattice,
L V ðtD Þ is a Boolean one. In this way, we obtain an adequate description of the
logical evolution of a quantum system.
In what follows, we discuss the dynamics of the quantum algebra of observables
and the logic structure of properties in some physical models.
We illustrate the logical approach with a simple example: the amplitude damping
channel.
The amplitude damping channel is useful for describing the energy dissipation
due to the environment effects. It is relevant for quantum information processing,
because it is an adequate model for quantum noise. In particular, this model can be
applied to the decay of an excited state of a two-level atom due to spontaneous
emission of photons. If the atom is in the ground state, no photon is emitted, and
the atom continues in the same state. But, if the atom is in the excited state, after an
interval of time τ, there is a probability p that the state has decayed to the ground
state and a photon has been emitted (see Nielsen and Chuang 2000).
The quantum operation of the amplitude damping channel can be expressed as
follows:
^ 0 ^ρ 0 E
ℰτ ð^ρ 0 Þ ¼ E ^ 1 ^ρ 0 E
^ †0 þ E ^ †1 , (18.7)
where the Kraus operators are
! pffiffiffi !
1 0 0 p
^0 ¼
E ^1 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , E : (18.8)
0 ð1 pÞ 0 0
If n ⟶ ∞, all the observables become proportional to the identity. This implies that
the algebra of observables becomes trivially commutative, and its corresponding
lattice of properties becomes a classical logic.
Gadella 2006) that dequantization may require two steps, one is a type of deco-
herence and the other is the notion of macroscopicity, which is implemented by the
limit ℏ ! 0. In the present section, we intend to give a brief account of another
notion of dequantization in the presence of unstable quantum systems or
resonances.
In a previous paper (Fortin et al. 2016), we argued that an essential characteristic
of the quantum-to-classical transition should be the transition from a noncommu-
tative algebra of observables to a commutative one, when t ⟶ þ ∞. This can be
rigorously formulated for unstable quantum systems, provided we considered the
linear space spanned by the resonance state vectors, also called Gamow vectors.
Resonances are usually characterized as poles of some analytic continuations of
a reduced resolvent or a scattering matrix. Both formulations are not always
equivalent – one may construct models for which the poles in one of these two
formulations are not poles in the other. In the energy representation, these poles
appear in complex conjugate pairs and have the form E R iΓ=2, where E R is the
resonance energy and Γ is related to the inverse of the mean lifetime (see Bohm
1993). Notice that Γ must be always positive.
From the observational point of view, single resonances show an exponential
decay, provided that the time intervals are not too short and not very large either
(Fonda, Ghirardi, and Rimini 1978). However, these deviations are very difficult to
observe. Therefore, most experiments with resonances show exponential decays
for practically all values of time (Fischer, Gutierrez-Medina, and Raizen 2001,
Rothe, Hintschich, and Monkman 2006).
Now, pure stable states have a mathematical representation in terms of vector
states. The difference between a stable state and a resonance state is just that
the value of the parameter Γ is equal to zero for stable states. Then, one is
tempted to introduce a definition of resonance states in such a way that, if the
resonance poles are ER iΓ=2, we have either H jψ D i ¼ ðE R iΓ=2Þjψ D i or
H jψ G i ¼ ðE R þ iΓ=2Þjψ G i (Nakanishi 1958). Here H ¼ H 0 þ V is the total
Hamiltonian which produces the resonance phenomenon. Note that, in the first
case, formal time evolution gives eitH jψ D i ¼ eiER t etΓ=2 jψ D i, which is an expo-
nential decay for t ⟶ þ ∞. On the other hand, a similar formal time evolution
gives eitH jψ G i ¼ eiER t etΓ=2 jψ G i, which decays exponentially as t ⟶ ∞. Vector
states jψ D i and jψ G i are known as the decaying and growing Gamow vectors,
respectively.
As a matter of fact, both vectors jψ D i and jψ G i (where the D stands for decay
and the G stands for growing) are equally suitable for a vector state for the
considered resonance. Nevertheless, the choice jψ D i seems more natural as the
time flows in the positive direction. The point is that both are time reversal of each
other and represent the same physical phenomenon. Note that both vector states
368 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada
describe the part of the resonance that behaves exponentially with time. Deviations
add a background term (Bohm and Gadella 1989), but here we can consider it as
negligible.
The previous considerations have an important mathematical flaw, however.
Gamow vectors are eigenvectors of the total Hamiltonian H with complex eigen-
values. This is not compatible with the assumption that H is self-adjoint. However,
this property is essential if we want the Gamow vectors to have an exponential
behavior with time. There are two possible remedies for this problem:
E
E
ψ j 2 Φ þ and
ψ j
G
2 Φ , where the index j stands for the number of reson-
ances in the system with resonance complex energies ERj iΓj =2. Decaying and
growing Gamow vectors have the desired time behavior, a fact that can be
rigorously proven (Bohm and Gadella 1989, Civitarese and Gadella 2004).
A nonrelativistic quantum system may have infinitely many resonances. This
means that only a finite number of resonances may be considered. We recall that
A Logical Approach to the Quantum-to-Classical Transition 369
resonances are determined by the poles of a complex analytic function which are
always isolated points in the complex plane. For large values of ER , the energies go
to the relativistic regime, so that we have to discard this possibility. But then,
resonances with large imaginary part are not observable, because their mean
lifetimes are extremely small. This means that only a finite number of resonances
may be considered within the nonrelativistic regime for a given unstable quantum
system.
In addition, if we only focus our attention on the resonance behavior, we may
consider the space spanned by the Gamow vectors. For decaying (growing)
Gamow vectors, this is a finite dimensional subspace of Φ þ Φ
. Let us assume
∗
that our system has N resonances with zj ≔E Rj þ iΓj =2 and zj being its complex
conjugate. We may consider the 2N dimensional space H G spanned by all Gamow
vectors n
o
ψ D ;
ψ G ;
ψ D ;
ψ G ; : . . . ;
ψ D ;
ψ G :::: (18.11)
1 1 2 2 N N
where δij is the Kronecker delta. We extend this pseudometric to the whole of H G
by linearity.
We may write the restriction of the total Hamiltonian H to H G as (Losada,
Fortin, Gadella, and Holik 2018)
XN
XN
∗
G
H¼ z
ψ
j¼1 j i
D
ψj
þ
G
z
ψ i
j¼1 j
ψDj
: (18.13)
Note that H in Eq. (18.13) is formally Hermitian. Using the pseudometric Eq.
(18.12), we find that
XN
XN n
n
D
G
Hn ¼ j¼1
z j
ψ i j
þ
ψG j¼1
z∗
j
ψ i ψDj
: (18.14)
U ðt Þ≔eitH ¼ j¼1
e
ψ i ψ j
þ e j
ψ G
i ψD
j
: (18.15)
o
I≔ j¼1
ψ D
j ψGj
þ
ψ j
G
ψD
j
: (18.16)
D
G
G
Using the pseudometric, we obtain that I
ψ D
j Þ ¼ ψ j Þ and I ψ j Þ ¼ ψ j Þ,
j ¼ 1, . . . , N, so that this is indeed the identity. With this identity, one possible
choice of the inverse of U ðtÞ is
370 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada
XN n
o
G
itz∗
U ðt Þ ≔ j¼1
e
ψ i ψ j
þ e
ψ G
j
i ψD
j
, (18.19)
which is Hermitian. In this case, we have that U ðtÞU † ðtÞ ¼ etΓ I. Nevertheless, we
should keep the definition of O at time t as Oðt Þ≔U † ðt ÞOU ðtÞ. In this case, we
have the following relationship for the commutator of two observables at time t:
XN n
o
2tΓj
D G
G
½O1 ðt Þ; O2 ðt Þ ¼ j¼1
e αj ðt Þ
ψ j ψ j
þ β j ð t Þ
ψ j ψD
j
, (18.20)
In the second period, the decoherence in open systems was studied. The main
characters of this period were Zeh (1970, 1973) and Zurek (1982, 1991). The
decoherence process is described as an interaction process between an open
quantum system and its environment. This process, called environment-induced
decoherence (EID), determines a privileged basis (usually called pointer basis or
moving decoherence basis), which defines the observables that acquire classical
features. Nowadays, this is the orthodox position on the subject (Bub 1997). The
decoherence times in this period were much smaller, solving the problem of the
first period.
In the third period, the arrival to equilibrium of closed systems was studied (Casati
and Chirikov 1995a, b, Ford and O’Connel 2001, Frasca 2003, Casati and Prosen
2005, Gambini, Porto, and Pulin 2007, Gambini and Pulin 2007, 2010). Within this
period, a new approach to the decoherence was presented by Castagnino et al.
According to this approach, the decoherence process can occur in closed systems,
and it depends on the choice of some observables with some particular physical
relevance (for example, the van Hove observables). This process, called self-induced
decoherence (SID), also determines which is the privileged basis, called the final
decoherence basis, that defines which observables acquire classical features.
In some works (Castagnino and Lombardi 2004, Castagnino et al. 2008, Cas-
tagnino and Fortin 2013), the common characteristics of the different approaches
to decoherence were summarized, and a general framework for decoherence was
proposed. According to the general framework, decoherence is just a particular
case of the general problem of irreversibility in quantum mechanics. Since the
quantum state follows a unitary evolution, it cannot reach a final equilibrium state
when time goes to infinity. Therefore, another element must be considered in such
a way that a nonunitary evolution is obtained. The way to introduce this nonunitary
evolution has to include the splitting of the whole space of observables O into a
relevant subspace OR O and an irrelevant subspace. Once the essential role
played by the selection of the relevant observables is clearly understood, the
phenomenon of decoherence can be explained in four general steps (reproduced
from Castagnino and Fortin 2013):
• First step:
The space of relevant observables OR is defined. For example, in the EID
approach the relevant observables are OR ¼ OS ⊗I E , where OS is an arbitrary
observable of the system S, and I E is the unit operator of the environment E.
SID-relevant observables were defined in Castagnino and Fortin (2013).
• Second step:
The expectation value hOR iρðtÞ , for any OR 2 OR , is obtained. This step can be
formulated in two different but equivalent ways:
372 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada
• Third step:
It is proved that hOR iρðtÞ ¼ hOR iρR ðtÞ reaches a final equilibrium value hOR iρ∗ :
lim t!∞ hOR iρðtÞ ¼ lim t!∞ hOR iρR ðtÞ ¼ hOR iρ∗ , 8OR 2 OR : (18.22)
This also means that the coarse-grained state ρR ðt Þ evolves, with a nonunitary
evolution, toward a final equilibrium state:
• Fourth step: n o
The moving preferred basis j jð~t Þi is defined. This basis is the eigenbasis of a
state ρP ðtÞ such that
lim t!∞ hOR iðρR ðtÞρP ðtÞÞ ¼ 0, 8OR 2 OR : (18.24)
where γ1
i are the characteristic times of the system, which are associated with the
complex eigenvalues of the effective Hamiltonian. Then, it is easy to see that the
commutator between two relevant observables is (Fortin and Vanni 2014)
OR ; O0R ρðtÞ ⟶0: (18.26)
This means that, when t ⟶ þ ∞, the expectation value of the commutator between
OR and O0R becomes zero. Therefore, the Heisenberg uncertainty relation becomes
undetectable from the experimental viewpoint.
A Logical Approach to the Quantum-to-Classical Transition 373
in which the “+” sign corresponds to Fermi-Dirac statistics and the “” to Bose-
Einstein. The parameter α is related to the particle number according to the
condition
X X 1
s
ns ¼ s exp ðα þ βϵ Þ 1
¼N: (18.28)
s
When the concentration of the gas is made sufficiently low, quantum effects
should be important. This limit corresponds to small N. Equivalently, we should
have ns 1 (or exp ðα þ βϵ s Þ 1).
If we now assume that the particle number is fixed, and we increase the
temperature (this is equivalent to β⟶0), we obtain that the most important terms
are those satisfying βϵ s α. Under these conditions, we obtain that
exp ðα þ βϵ s Þ 1. Or equivalently, that ns 1. This is the condition for the
classical limit. In other words, the condition under which quantum effects are
negligible. In this limit, and for both cases, Fermi-Dirac and Bose-Einstein, we
obtain
ns ¼ exp ðα βϵ s Þ: (18.30)
This constraint reduces to
X
s
exp ðα βϵ s Þ ¼ N; (18.31)
only parameter that allows us to observe a logic transition. The algebraic aspects of
this transition will be discussed in a future work, but we can advance some
points here.
First, some interpretations of quantum mechanics suggest that, when the clas-
sical limit is obtained, an irreversible process should be observed. Under this
perspective, this can be related to the mathematical formalism of Gamow vectors.
Second, the approach of dynamical logics can be useful to interpret the quant-
ization deformation formalism under a new light. Indeed, some authors (see, e.g.,
Landsman 1993) have proposed to study the classical limit and the quantization of
a given theory by appealing to the formalism of deformation quantization. In this
approach, one starts with a classical (commutative) algebra of observables A0 ,
endowed with a pointwise product ∙ and a Poisson bracket f;g. Then, a family of
algebras Ah is introduced, indexed with a parameter h 0. The parameter h is
intended to represent a dimensionless combination of some characteristic param-
eters associated with the system and Planck’s constant. An associative product ?h
is introduced in the indexed algebras, and it is required that (see Landsman 1993
for details)
i
lim h!0 ½ f ; g h ¼ f f ; gg (18.33)
h
and
1
lim h!0 ½ f ; g hþ ¼ f
g: (18.34)
2
The examples shown in this section suggest that the parameters involved in the
classical limit process could be time, temperature, particle number, or others. Thus,
our dynamical logics approach could be connected in a natural way with the
formalism of deformation of algebras. We will discuss this possibility elsewhere.
18.5 Conclusions
In this chapter, we have presented a logical approach for the description of the
quantum-to-classical transition of physical systems. This approach consists in
describing the system as a collection of observables that evolve over time,
according to the Heisenberg picture, but with a nonunitary evolution.
In turn, the algebra of observables determines a lattice of elementary physical
properties with a logical structure. In the classical case, the properties have a
classical logic structure, and in the quantum case, they have a quantum logic
structure. The time evolution of the algebra induces a time evolution of the lattice
of properties. Therefore, in this approach, the classical limit is attained when the
A Logical Approach to the Quantum-to-Classical Transition 375
Acknowledgments
We are grateful to the participants of the workshop Identity, indistinguishability
and non-locality in quantum physics (Buenos Aires, June 2017) for their useful
comments. This work was made possible through the support of Grant 57919 from
the John Templeton Foundation and Grant PICT-2014–2812 from the National
Agency of Scientific and Technological Promotion of Argentina.
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378 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada
19.1 Introduction
Since its birth, quantum mechanics has enjoyed high prestige thanks to its success
in the explanation and prediction of phenomena at the atomic and molecular scales.
Indeed, this theory began by explaining the emission lines observed in the hydro-
gen atom and, after a few years after its first formulation, it could explain the
energy spectrum of simple molecules. This type of success quickly leads scientists
to suppose that all chemistry can be explained by physics. The famous claim by
Paul Dirac is an example of such an assumption:
The underlying physical laws necessary for the mathematical theory of a large part of
physics and the whole of chemistry are thus completely known, and the difficulty is only
that the exact application of these equations leads to equations much too complicated to be
soluble. It therefore becomes desirable that approximate practical methods of applying
quantum mechanics should be developed, which can lead to an explanation of the main
features of complex atomic systems without too much computation.”
(Dirac 1929: 714)
However, as time went by, it turned out to be clear that the attempt to explain
chemistry from physics leads to complications that allow us to question Dirac’s
claim. One of these problems is to explain molecular structure from quantum
mechanics. There are several ways to approach this problem, but in this work we
will do it by means of a particular case: optical isomers and the Hund paradox.
When young Kant meditated upon the distinction between his right hand and
his left hand, he could not foresee that the problem of incongruent counterparts
would be reborn in the twentieth century under a new form. The so-called Hund
paradox points to the difficulty of giving a quantum explanation to chirality, that
is, to the difference between the members of a pair of optical isomers or
enantiomers. The question about whether the quantum formalism can account
for chirality concerns philosophy of science for (at least) three reasons. First, it
379
380 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin
introduces an interesting case for the debate about the relation between physics
(quantum mechanics) and chemistry (molecular chemistry), which has been the
focus of many philosophical works in recent years. Second, and related to the
previous point, the analysis of the paradox can enrich the discussion about
whether quantum mechanics can provide an explanation of molecular structure.
Third, since some approaches attribute the origins of the paradox to a focus on
isolated molecules, the solution is believed to be found in considering molecules
in interaction; these views pose a relevant question to the ontology of chemistry:
Is chirality an intrinsic property of a molecule? These three problematic points
make the resolution of the Hund paradox an issue of the utmost importance for
the philosophy of science.
On this basis, in this chapter we will analyze the problem of optical isomerism
by proceeding in the following steps. In Section 19.2, the Hund paradox will be
presented in formal terms. Section 19.3 will be devoted to showing the relevance
of the paradox to the relation between physics and chemistry, to the explanation of
molecular structure, and to the ontology of chemistry. In Section 19.4 the paradox
will be conceptualized as a case of quantum measurement, stressing that decoher-
ence does not offer a way out for this problem. Finally, in Section 19.5 we will
argue for the need of adopting a clear interpretation of quantum mechanics; in
particular, we will claim that the modal-Hamiltonian interpretation, which con-
ceives measurement as a breaking symmetry process, supplies the tools required to
solve the Hund paradox.
depends only on the internuclear distances, the Hamiltonian is exactly the same for
the two members of the pair. Consequently, quantum mechanics provides the same
description for two chemical species that can effectively be differentiated in
practice by their optical activity (Harris and Stodolsky 1981, Wolley 1982, Berlin,
Burin, and Goldanskii 1996, Quack and Stohner 2005, Schlosshauer 2007).
In the quantum domain, the parity operator P ^ is associated with spatial reflec-
tion: if jDi and jLi are the states of isomers D and L, respectively, P ^ transforms jDi
into jLi and vice versa: P ^ jLi ¼ jDi, P^ jDi ¼ jLi. Let us suppose that the molecule
consists of A atomic nuclei and N electrons. Then, the Coulombic Hamiltonian of
the complete molecule reads
!
XA
P 2 X A
Z Z X i
P 2 XA
Z X N
1
^ ¼
H
g
þ e2
g h
þ
i
e2
g
þ e2
g
2mg g<h
2mg N
2me g
^r ig i<j
^r ij
(19.1)
where P^ g , Z g , and mg are the momentum operator, the atomic number, and the
mass of the nucleus g, respectively, with g = 1, 2, . . ., A; e and me are the charge
and the mass of the electrons, respectively; P ^ i is the momentum operator of the
electron i, with i = 1, 2, . . ., N; ^r ij is the operator “distance” between the electron i
and the electron j, and ^r ig is the operator “distance” between the electron i and the
nucleus g. Since the Coulombic interaction only depends on the distance between
the interacting particles, it is symmetric under spatial reflection; therefore, the
Hamiltonian commutes with the parity operator P: ^
^ H
P; ^ ¼0 (19.2)
This means that the eigenstates j ωn i of the Hamiltonian have definite parity. In
particular, the ground state j ω0 i is invariant under space reflection:
^ j ω0 i ¼ j ω0 i. As a consequence, j ω0 i cannot be a chiral state j Di or j Li, but
P
it is a superposition of them:
1
j ω0 i ¼ pffiffiffi ðjDi þ jLiÞ (19.3)
2
The question is, then, why chiral molecules are never found in this superposition
state. The states obtained in the laboratory are j Di and j Li, which are not
eigenstates of the Hamiltonian and do not correspond to the ground state. So,
why do certain chiral molecules display an optical activity that is stable over time,
associated with a well-defined chiral state? Why do chiral molecules have a
definite chirality? (Berlin et al. 1996). The Hund paradox points to the core of
certain traditional problems of the philosophy of chemistry. Let us consider them
briefly.
382 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin
If many measurements are performed on identical systems with the same initial
conditions, it is possible to define an ensemble, whose state is represented by a
density operator:
1
^ρ collapsed ¼ ðjDihDj þ jLihLjÞ (19.5)
2
This state is interpreted as stating that there is a probability 0.5 of finding the
system in the state j Di and a probability 0.5 of finding the system in the statej Li:
the state is a mixture of an equal number of definite chiral states. The collapse
hypothesis is very successful in reproducing the experimental results, but it has no
explanatory power, to the extent that it is an ad hoc hypothesis specifically
designed to account for the quantum measurement problem. Moreover, collapse
is a nonunitary process that breaks the Schrödinger evolution; however, the
hypothesis does not explain why or when the process happens. For this reason,
during the last decades, quantum measurements have been approached from
different perspectives; one of them is that given by the theory of decoherence.
According to the orthodox approach – the so-called environment-induced deco-
herence (Zurek 1981, 1993, 2003) – decoherence is a phenomenon resulting from
the interaction between an open quantum system and its environment. Let us
consider a closed system U with two subsystems: the open system S in the initial
state ^ρ S , and the environment E in the initial state ^ρ E . Then, the initial state of
the total system is ^ρ U ¼ ^ρ S ⊗^ρ E . This state evolves in a unitary way according to
the Schrödinger equation. But, the theory of decoherence studies the behavior
of the reduced state of the open system, ^ρ reduced ¼ Tr E ð^ρ U Þ, obtained by applying
the partial trace on the state of the whole closed system; the partial trace is an
operation that removes the degrees of freedom of the environment from ^ρ U . As
a consequence, the reduced state of the open system is no longer governed
by the Schrödinger equation, but is instead ruled by a master equation: ^ρ reduced
may evolve in a nonunitary way. Moreover, when the number of degrees
of freedom of the environment is very high, the reduced state may become
diagonal and mimic the ^ρ collapsed , obtained by means of the collapse hypothesis
(see Eq. (19.5)).
In his “Editorial 37” in Foundations of Chemistry, Eric Scerri (2011) explicitly
relates the problem of optical isomerism to the quantum measurement problem.
According to Scerri, the Hund paradox would be dissolved if the interaction of the
molecule with its environment were taken into account:
The study of decoherence has shown that it is not just observations that serve to collapse
the superpositions in the quantum mechanics. The collapse can also be brought about by
molecules interacting with their environment.
(Scerri 2011: 4; see Scerri 2013 for a similar claim)
Quantum Mechanics and Molecular Structure: The Case of Optical Isomers 387
The idea is that the enantiomer molecule is in interaction with the environment (air,
particles, other molecules, etc.). If the initial states of the molecule and the
pffiffiffi are j ω0 i and j ε0 i, respectively, the initial state of the whole system
environment
is 1= 2 ðjDi þ jLiÞ ⊗ j ε0 i. The interaction between the molecule and its envir-
onment define the evolution of the total system, which, in some cases, produces a
correlation between the possible states of the system and the environment:
1 1 1
pffiffiffi ðjDi þ jLiÞ ⊗ jε0 i ! pffiffiffi j Di ⊗ jεD i þ pffiffiffi j Li ⊗ jεL i (19.6)
2 2 2
Decoherence occurs when, as the result of the evolution, the states of the environ-
ment become rapidly orthogonal: hεL jεL i ! 0. As a consequence, after an
extremely short decoherence time, the reduced state of the molecule acquires the
same structure as that of the mixed state after collapse (see Eq. (19.5)):
1
^ρ decohered ¼ ðjDihDj þ jLihLjÞ (19.7)
2
As in the case of quantum measurement, this state is interpreted as stating that the
molecule is in one of the states j Li or j Di, and that probabilities measure our
ignorance about which state it is. In this way, the theory of decoherence would
solve the problem underlying the Hund paradox.
Although there was a time when, as stressed by Anthony Leggett (1987) and
Jeffrey Bub (1997), decoherence was considered the “new orthodoxy” in the
physics community to explain quantum measurements, at present it is quite clear
that decoherence does not solve the measurement problem. In fact, collapse is the
change of the state of the system, from a superposition to a definite state; on this
basis, ^ρ decohered can be interpreted as a legitimate mixture. On the contrary, in the
case of decoherence, the state of the whole system never collapses, but always
evolves according to the Schrödinger equation – the superposition never vanishes
through the unitary evolution. Therefore, it cannot be supposed that what is
observed at the end of the decoherence process is one of two definite events: either
that associated with j Li or that associated with j Di (see Adler 2003). Bub (1997)
even claims that the assumption of a definite event at the end of the process is not
only unjustified, but also contradicts the eigenstate-eigenvalue link. These conclu-
sions about decoherence can also be drawn from the traditional distinction between
a proper mixture – the mixed state of a closed system – and an improper mixture –
the reduced state of an open system. As Bernard d’Espagnat (1966, 1976) repeat-
edly stressed, improper mixtures cannot be interpreted in terms of ignorance (for
additional arguments, see Fortin and Lombardi 2014).
Summing up, at present some authors still consider that decoherence, by itself,
solves many conceptual problems in quantum physics (e.g., Crull 2015).
388 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin
now suppose that we want to measure one of those observables, say, the compon-
ent Py in direction y. For this purpose, we have to place a wall normal to the
direction y, in such a way that the new Hamiltonian is the original one plus a term
that represents the asymmetric potential barrier. It is precisely this term that breaks
the symmetry of the original Hamiltonian and renders the observable Py actual and
definite-valued and, as a consequence, accessible to measurement. But the point to
stress here is that now the system is no longer the free particle; it is a new system,
whose Hamiltonian is not symmetric with respect to displacements in direction y.
In light of these interpretive ideas, the Hund paradox can now be rephrased in
MHI’s language. As stressed in Section 19.2, the exact Hamiltonian H ^ of the
enantiomeric molecule (see Eq. (19.1)) is symmetric under spatial reflection – it
commutes with the parity operator P ^ (see Eq. (19.2)). Now, let us consider the
^ whose eigenstates are j Di and j Li. The eigenvalues d and l
observable chirality C,
^
of C represent the properties dextro-rotation or levo-rotation,
respectively. It is
^ ^ ^ ^
easy to see that C does not commute with P: P; C 6¼ 0. In this case, as in the
^ would determine
example of the free particle, the actualization of the observable C,
the chirality of the molecule in a completely arbitrary way: It would introduce in
the molecule an asymmetry not contained in its Hamiltonian. As a consequence,
from the MHI viewpoint, C ^ has no actual value: Chirality is indefinite in the
isolated molecule.
If the observable C^ is to be measured, the parity symmetry of the molecule has
to be broken. For this purpose, the molecule must interact with another system M,
which plays the role of the apparatus, in such a way that the Hamiltonian H ^ T of the
new composite system is no longer parity invariant. For instance, this is obtained
when
H ^ þH
^T ¼ H ^ M, (19.8)
where the Hamiltonian ^ M of the new system breaks the original parity
H
invariance: H ^ 6¼ 0 ) H
^ M; P ^ T; P
^ 6¼ 0. A good candidate for H ^ M is the Hamil-
tonian usually introduced in quantum chemistry to describe the interaction between
molecules and polarized light (see Shao and Hänggi 1997), which is a function of
the electric field E and the magnetic field B ^
of the polarized light. Additionally, C
must commute with the total Hamiltonian H ^ T in order to obtain a stable reading of
chirality. Under these conditions, according to the MHI the observable C ^ acquires
a definite actual value: We measure dextro-rotation or levo-rotation, but now the
system is no longer the isolated molecule, but the molecule in interaction with the
polarized light. In a certain sense, this answer to the Hund paradox agrees with
the view according to which the solution must be looked for in the interaction of
the molecule with its surroundings: Chirality is not an intrinsic property of the
molecule, but of the composite system molecule plus polarized light. However, our
390 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin
Acknowledgments
This work was made possible by the support of Grant 57919 from the John
Templeton Foundation and Grant PICT-2014–2812 from the National Agency of
Scientific and Technological Promotion of Argentina.
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392 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin
393
394 Index
Classical (cont.) Earman, John, 271, 347
mechanics, 71, 76–77, 81–82, 89, 93, 108, 130, 191, Egg, Matthias, 229
226, 236–237, 239, 242, 275, 278, 280, 283, 346, Eigenstate-eigenvalue link, 9, 13–15, 23, 153, 280,
363 387
physics, 11, 67, 73, 76, 81, 94, 96, 137–139, 158, Einstein, Albert, 45, 77, 90, 121, 123, 150, 164, 168,
177, 205, 235–237 176–178, 219–220, 229, 245–247, 249, 263
statistical mechanics, 89, 236–237 Electron density, 384–385
statistics, 209, 214 Ensemble
universe, 237 canonical, 211, 214
variables, 76–77, 83 grand canonical, 212, 214
Clifton, Rob, 14 Entanglement
Collapse as a relationship between algebras of observables,
hypothesis, 385–386 349
of the wave function, 87–88, 97, 225–227, 233 generalization of, 348
relativistic, 9, 16, 20, 22, 24, 26 relativity of, 348
theories, 9–10, 13, 15–16, 20, 22, 25, 95, 100, 104, Entropy
125, 165–167, 173–174, 224, 228 Burg, 327
Collier, John, 179 quantum, 333
Contextuality, 32, 44, 75–76, 279, 388 Rényi, 327
Correlations, 36–37, 45, 53, 63, 94, 168, 175, Shannon, 327, 330, 334, 336, 339
237–239, 245, 247, 262, 315–316, 329, 334, 336, Tsallis, 327
339, 346–347 von Neumann, 330
Costa de Beauregard, Oliver, 286 Ergodicity, 245, 248–249, 255–256, 259–261
Cramer, John, 220 Esfeld, Michael, 3, 136, 168, 222
Cushing, James, 124, 126, 128 Euclid, 294
Everett, Hugh, 54, 126, 133, 135, 161, 239
d’Espagnat, Bernard, 387 Experiment
Dalton, John, 346 Aspect, 127
de Broglie, Louis, 80, 124, 126, 229, 231 double-slit, 114–115
Decision theory, 103 EPR, 125, 219
Decoherence loophole-free, 245, 248–249, 255, 257
environment-induced, 345–346 noninteracting, 44
relativity of, 353 Stern–Gerlach, 38, 84, 86, 125, 170, 222
self-induced, 375
top-down, closed-system approach to, 346 Faye, Jan, 154
Democritus, 261 Feynman, Richard, 72, 77, 180, 362
Deutsch, David, 103, 121, 129 Finkelstein, Jerry, 21
Dieks, Dennis, 2, 51 Fleming, Gordon, 23
Diosi, Lajos, 95 Folse, Henry, 121
Dirac, Paul, 124, 133, 135, 157, 362, 379, 382 Forman, Paul, 124, 126
Discernibility, 187, 192 Fortin, Sebastian, 4, 345, 360, 379
Distinguishability, 205, 209, 215, 219–220, 228, Frauchiger, Daniela, 58–60, 63, 65
296 French, Steven, 193–194, 196, 206, 208, 220
Distribution Frigg, Roman, 145
Bose-Einstein, 186, 209, 213, 373 Fuchs, Christopher, 136
equilibrium, 83, 85
Fermi-Dirac, 186, 209, 213, 217, 373 Gadella, Manuel, 4, 360
flash, 238 Galilean
Maxwell–Boltzmann, 208–210, 212, 216, 219, 373 covariance, 277, 279
Planck, 210 invariance, 291
Poisson, 226 transformations, 272, 278, 280, 282, 289, 300
Rayleigh-Jeans, 209 Galileo Galilei, 71, 245
Dürr, Detlef, 166 Galois, Évariste, 270
Dynamics Gamow vectors, 367–369, 374
collapse, 21, 167, 227, 230, 232 Gasiorowicz, Stephen, 285
deterministic, 242 Gell-Mann, Murray, 107
Markovian, 16, 18 Generalized contexts, 107, 112
nonlinear, 16 Ghirardi, GianCarlo, 9, 13, 15, 22–23, 25, 95, 226
Index 395
Girardeau, Marvin, 316 Clauser-Horne-Shimony-Holt (CHSH), 250
Gisin, Nicolas, 16, 18 Eberhardt’s, 250
Gleason, Andrew, 75 Mermin’s, 255
Goethe, Johann, 155 Instrumentalism, 122, 129, 233
Goldstein, Sheldon, 15, 136, 166, 247 Interpretation of quantum mechanics
Grassi, Renata, 9, 13, 15, 22–23, 25 atomic modal, 346
Griffiths, Robert, 55, 107 Bohr’s, 85
Group consistent-histories, 55
Galilean, 34, 39–42, 269, 272–274, 276–279, Copenhagen, 72, 77, 83, 121–122, 133, 245,
281–282, 284, 286, 288, 291, 295, 310, 247
312 de Broglie-Bohm pilot-wave, 122, 281
Lie, 272 Everett, 54, 122, 134, 173–174
maximal kinematic, 305, 314 many worlds (MWI), 59, 68, 98–99, 101–104, 126,
Poincaré, 42, 282, 291, 295 165–167, 224
Schrödinger, 35 modal, 32–33, 36, 54, 57, 234,
GRW 280–281
flash theory (GRWf ), 230, 233, 235, 238, 240–241 modal-Casimir, 41
matter density theory (GRWm), 227, 229, 232–233, modal-Hamiltonian (MHI), 33–34, 39–40, 46, 54,
235, 238, 240 281, 356, 388
theory, 9, 53, 161, 165, 173, 220, 226–230 perspectival, 59
Guiding equation, 78–80, 82–84, 86, 231–232, 234, QBist (quantum Bayesian), 141, 160
238, 241 relational, 55, 57
statistical, 247–248, 262
Haag, Rudolf, 297 transactional, 220
Hacking, Ian, 123 Invariance
Haecceity, 193–197, 200 dynamical, 304, 308, 313, 317
Halvorson, Hans, 3, 133, 145 Galilei, 41, 269
Hamilton, William, 247 Parity, 389
Hamiltonian Poincaré, 42
effective, 372 time-reversal, 278, 282, 286–288,
non-Hermitian, 368, 372 290–291
Harshman, Nathan, 4, 294 Irreducible representations (irreps), 40, 295,
Hartle, James, 107 347
Healey, Richard, 136, 140–141, 159–160 Ismael, Jenann, 175
Hegel, Georg W. Friedrich, 155
Heisenberg, Werner, 57, 133, 135, 157, 185, 337 Jaimes Arriaga, Jesús, 4, 379
Hendry, Robin, 383 Jaynes, Edwin, 245
Hettema, Hinne, 384
Hidden variables, 75, 78, 87, 123–124, 165–166, 171, Kant, Immanuel, 124, 155, 379
174, 195, 200, 251, 253, 255, 260 Kastner, Ruth, 3, 205
Hnilo, Alejandro, 4, 245 Kelvin, Lord, 93–94, 96
Holik, Federico, 4, 360 Kent, Adrian, 17
Howard, Don, 164, 168 Kepler, Johannes, 294
Huggett, Nick, 205, 207–208, 211, 214 Klein, Felix, 279
Humeanism, 169, 195, 233, 236 Kochen, Simon, 75, 234
Hund paradox, 379, 381, 383, 386–390 Krause, Décio, 3, 185, 193–194, 196
Hund, Friedrich, 380
Ladyman, James, 3, 121, 179
Improper mixture, 56, 387 Lakatos, Imre, 126, 130
Indiscernibility, 187–188, 197–198, 203 Landau, Lev, 76
Indistinguishability, 44–46, 198, 203, 214, 312, Lange, Marc, 173
315 Laplace, Pierre-Simon, 93, 135
Individuality Laplace’s demon, 237
non-, 186, 188, 191, 193–196 Lattice
transcendental, 193, 208 Boolean, 108, 360, 363
Inequality distributive, 108, 111–112, 363, 365
Clauser-Horne (CH), 251 orthocomplemented, 108, 111–113, 116,
Clauser-Horne inequality (CH), 250 363–364
396 Index
Laue, Hans, 288 Noncommutativity, 73
Laura, Roberto, 2, 107 Nonlocality, 45, 74, 80, 90, 222, 229, 232, 238–239
Leegwater, Gijs, 65 Nonseparability, 214
Leggett, Anthony, 387 North, Jill, 282, 287
Leibniz law (principle), 198, 200, 203, 205 Nozick, Robert, 279
Leibniz, Wilhelm, 179
Lewis, David, 100 Ohanian, Hans, 270
Lewis, Davis, 205 Olkin, Ingram, 323
Lewis, Peter, 167, 176 Omnès, Roland, 107
Lieb, Elliott, 316 Ontology
Lifshitz, Evgeny, 76 distributional, 9–10
Liniger, Werner, 316 of properties, 43, 46
Lo, Hoi-Kwong, 335 particle, 159, 222, 226, 231, 233, 240
Loewer, Barry, 13, 37, 169 primitive, 15, 91, 166–167, 178, 233, 235–236
Lombardi, Olimpia, 2, 4, 32, 269, 345 Optical isomerism, 38, 380, 386
London, Fritz, 52–57
López, Cristian, 4, 269 Parmenides, 261
Lorenz, Max, 323 Particles
Losada, Marcelo, 2, 4, 107, 360 Bohmian, 97–98, 234
Bohmian Dirac sea of, 241
Mach-Zehnder interferometer, 116, 309 Dirac sea of, 241–242
Majorization, 323 identical, 44–45, 78, 304–305, 313, 316, 319
Marshall, Albert, 323 indistinguishable, 311, 314–317
Martínez González, Juan Camilo, 4, 379 individual, 167, 222, 226
Mass density, 13, 15, 22, 25 numerical difference of, 191–192
Matter density point, 186, 225–226, 231–232, 239, 241
field, 226–229, 232, 235, 238 Pauli, Wolfgang, 122
ontology, 25, 167, 229 Pearle, Philip, 9, 14, 22, 24, 28, 95
Maudlin, Tim, 21, 95, 121, 130, 220, 223–224, 230, Peirce, Charles Sanders, 124
234, 239 Penrose, Roger, 95, 133
Maxwell equations, 94, 128, 156, 158, 261, 277 Peres, Asher, 136
Measurement Perspectivalism, 51, 57–59, 61, 64, 67–68
as a symmetry breaking process, 33, 39, 390 Picture/representation
determinate outcomes of, 223 Heisenberg, 90, 108, 110–113, 127, 361, 365, 374
frequency, 36 interaction, 17
reliable, 37 Schrödinger, 111
single, 36–37, 246, 385 Tomonaga–Schwinger, 17, 20
von Neumann model of, 36–38, 55 Planck constant, 79, 360–361, 374
Mermin, David, 21, 180, 254 Plato, 294
Mill, John Stuart, 123 Popescu, Sandu, 335
Minkowski spacetime, 10, 16, 28, 64, 66 Popper, Karl, 126, 261
Molecular Positive-operator-valued measures, 86
chemistry, 38, 380, 382 Principle
structure, 379–380, 382–385, 390 of compositionality, 202
Møller, Christian, 155 of correspondence, 246
Monton, Bradley, 14 of identity of indiscernibles, 191
Motion reversal, 290–291 of impenetrability, 191
Mott, Nevill Francis, 124 of individuality, 45, 189, 191–192
Muller, Fred, 192 of local causality, 172
Musgrave, Alan, 126 of metaphysical continuity, 10–11, 14
Myrvold, Wayne, 2, 9, 65, 67, 168, 173, 176 of superposition, 246, 261, 263
of uncertainty, 78, 323–324, 337–339
Newton laws, 94 Pauli exclusion, 213
Newton, Isaac, 71, 178–179, 346 Projection postulate, 161, 219, 385
Ney, Alyssa, 3, 136, 164–165 Properties
Nguyen, James, 145 atomic, 108–109, 111–112
Nielsen, Michael, 323, 335 bundles of, 43–44, 46, 356
No-go theorems, 43, 65–66, 123 case-, 43, 45
Index 397
contrary, 107, 113, 116 Schrödinger, Erwin, 72, 80, 83, 186–191, 193–196,
definite, 65 201, 261, 331
elementary, 360–362 Schrödinger’s cat, 72, 88, 223, 232
natural physical, 148–149 Schur, Issai, 323, 326, 328
possible, 44, 46 Schur-concavity, 326, 333
relational, 57–58, 280 Sebens, Charles, 103
type-, 43, 45 Self-induced decoherence, 371
Semi-group, 19
Quantum Shimony, Abner, 13, 171
chemistry, 128, 380, 382, 384, 389 Signalling, 16, 18–19
computation, 72, 129, 345 Spacetime
information, 72, 86, 297, 324, 332, 336, 339, Galilean, 16, 127
366 globally hyperbolic, 16
noise, 366 relativistic, 10, 16–17, 20
potential, 81–82 Specker, Ernst, 75, 234
statistics, 45, 78, 205, 207, 209–211, 214, 373 Spontaneous localization, 9, 227, 229–230
Quantum field theory, 11, 21, 42, 47, 73, 78, 90, 128, Spurrett, Don, 179
225, 239, 242, 279, 282, 291, 295 Standard model, 11, 128, 201, 240–241, 297, 346
Quantum Theory of Atoms in Molecules (QTAiM), State
384–385 actual, 54, 137
Quasi-set theory, 200 Bell, 254, 334
Quine, Willard V. O., 157 entangled, 16, 18, 45, 51, 53–54, 62, 89, 99, 170, 187,
223, 251, 256–257, 261, 330, 335–336, 347, 349
Racah, Giulio, 286 extrinsic, 20–22
Randomness, 78, 83–84, 87, 100, 124, 249, 256, 259, GHZ, 253–255
318 perspectival, 56
Realism potential, 137
about the quantum state, 135, 159 reduced, 12, 20–22, 331, 334–335, 345, 350–352,
entity, 123, 129 386–387
local, 247, 251 relational, 56
metaphysical, 129 Suárez, Mauricio, 39, 279
ontic structural, 168 Sudbery, Anthony, 59
scientific, 72, 121–122, 124–125, 127–130, 143, Superselection rules, 304, 337
149 Symmetrization, 46, 214, 219, 317
spacetime state, 151, 168 Symmetry
Redhead, Michael, 206, 208, 220 abelian, 302
Reichenbach, Hans, 145 dynamic, 300, 305
Relativity Galilean, 79, 281, 299
general, 150, 185, 201, 228 gauge, 42
special, 64, 66, 96, 176 group of, 40, 295, 299
Renner, Renato, 58–60, 63, 65 kinematic, 300, 304–305, 313–315, 317
Rigged Hilbert space, 301, 366, 368 of particle permutations, 304
Rimini, Alberto, 226 of the Hamiltonian, 35, 38, 296, 388
Roberts, Bryan, 283 permutation, 103, 188, 192
Ross, Don, 179 Poincaré, 299
Rovelli, Carlo, 55, 57, 135, 141, 160 rotational, 35, 299, 304
Ruffini, Remo, 270 space-time, 299
SU(2), 295
Sachs, Robert, 285
Sakurai, Jun, 285, 287, 290 Tails problem, 13
Saunders, Simon, 126, 134, 136, 142, 192 Tappenden, Paul, 103
Scerri, Eric, 386 Tarski, Alfred, 44
Schaffer, Jonathan, 175 Teller, Paul, 168
Schlosshauer, Maximilian, 350 Tensor product structure, 34, 306–311, 313–316, 348,
Schmidt decomposition, 330–331 353
Schrödinger equation Theorem
covariance of the, 274–277 Birkhoff’s, 326
invariance of the, 41, 269–270, 275–276 Ehrenfest, 362
398 Index
Theorem (cont.) Wave function
equipartition, 215–217 as a field on configuration space, 136
Horn’s, 328 as a mere calculational device, 140
Kochen-Specker, 41, 45, 139, 279, 388 as a multi-field on physical space, 136
Schrödinger, 329, 333 as an abstract mathematical object, 143
Schur’s, 331 conditional, 85, 88
Uhlmann’s, 324 realism, 136, 151, 158
Zanardi’s, 309 representational status of the, 157
Timpson, Christopher, 142–143, 149, 151–153, 158, 168 Weber, Tullio, 226
Tumulka, Roderich, 25, 166, 230 Weihs, Gregor, 259
Typicality measure, 231, 236–237, 241–242 Weyl, Hermann, 187–188, 191–192, 200, 279
Wheeler, John, 121
Uncertainty relations, 52, 123, 130, 215–216, 226, Wigner, Eugene, 287–290, 295, 299
237, 240, 324, 337–339, 372 Wigner’s friend, 55, 57–58, 65, 67
Urelemente, 199 Wilson, Jessica, 175
Wiseman, Howard, 171–172
Vaidman, Lev, 2, 93, 135, 253
van Fraassen, Bas, 32, 129, 149, 151 Yang, Chen Ning, 316
Vanni, Leonardo, 2, 107
von Helmholtz, Hermann, 155 Zanardi, Paolo, 297, 308, 310, 315
von Neumann, John, 52–54, 56, 75, 95, 122 Zanghí, Nino, 2, 71, 136, 166
Zeeman effect, 38
Wallace, David, 103, 134, 136, 142–143, 148–149, Zeh, Heinz-Dieter, 371
151–153, 158, 161, 168 Zeilinger, Anton, 121
Watanabe, Satosi, 286 Zurek, Wojciech, 349, 353, 355, 371