INVARIANT SET THEORY

T.N.PALMER
DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

Abstract. Invariant Set Theory (IST) is a realistic, locally causal theory of fundamental
arXiv:1605.01051v1 [quant-ph] 30 Apr 2016

physics which assumes a much stronger synergy between cosmology and quantum physics
than exists in contemporary theory. In IST the (quasi-cyclic) universe U is treated as
a deterministic dynamical system evolving precisely on a measure-zero fractal invariant
subset IU of its state space. In this approach, the geometry of IU , and not a set of
differential evolution equations in space-time MU , provides the most primitive description
of the laws of physics. As such, IST is non-classical. The geometry of IU is based on Cantor
sets of space-time trajectories in state space, homeomorphic to the algebraic set of p-adic
integers, for large but finite p. In IST, the non-commutativity of position and momentum
observables arises from number theory - in particular the non-commensurateness of φ and
cos φ. The complex Hilbert Space and the relativistic Dirac Equation respectively are
shown to describe IU , and evolution on IU , in the singular limit of IST at p = ∞; particle
properties such as de Broglie relationships arise from the helical geometry of trajectories
on IU in the neighbourhood of MU . With the p-adic metric as a fundamental measure
of distance on IU , certain key perturbations which seem conspiratorially small relative
to the more traditional Euclidean metric, take points away from IU and are therefore
unphysically large. This allows (the ψ-epistemic) IST to evade the Bell and Pusey et
al theorems without fine tuning or other objections. In IST, the problem of quantum
gravity becomes one of combining the pseudo-Riemannian metric of MU with the p-
adic metric of IU . A generalisation of the field equations of general relativity which can
achieve this is proposed, leading to new suggestions the nature of the dark universe, space-
time singularities, and the fate of information in black holes. Other potentially testable
consequences are discussed.

1. Introduction
Ongoing debates about the nature of both the dark and inflationary universe, and of the
fate of information in black holes, are testament to the fact that we have yet to synthesise
convincingly our theories of quantum and gravitational physics. Whilst such synthesis
will likely require some revision to both quantum theory and general relativity, the basic
premise of this paper is that by far the more radical revision will have to be to quantum
theory, its success on laboratory scales notwithstanding. Here we propose a realistic locally
causal theory of quantum physics, Invariant Set Theory, from which quantum theory arises
as a singular limit [2]; it is ψ-epistemic in the sense that the variable of the theory which
corresponds to the quantum wavefunction, merely describes some distribution over deeper
ontic states. The theory differs from all previous attempts in this direction in that it is
assumed that there is a much greater synergy between cosmology and quantum physics
1
2 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

than is generally assumed in traditional approaches to formulating laws of physics. That

is to say, it is assumed that the nature of the world around us is as much determined by
‘top-down’ constraints [9] from cosmology, as from reductionist ‘bottom-up’ constraints
from elementary-particle physics.
The background to Invariant Set Theory is the seminal work of Lorenz [16] who showed
that the t → ∞ asymptotic behaviour of a (classical) nonlinear dynamical system is char-
acterised by a measure-zero fractal geometry in the system’s state space. As it turned out,
such fractal attractors characterise a generic class of nonlinear dynamical systems [31].
With this in mind, we imagine the (mono-)universe U to be described by a quasi-cyclic
cosmology, evolving repeatedly over multiple aeons [30, 23], such that the space-time tra-
jectory of U (over all aeons) describes a measure-zero fractal subset IU in the state-space
of U . The conditions for such a concept to be dynamically consistent are discussed below
(see especially Section 4). This ‘Cosmological Invariant Set Postulate’ [17, 18] implies that
the geometry of IU should be considered a more primitive expression of the laws of physics
than some set of differential evolution equations for fields within space-time MU , the latter
characterising the traditional reductionist approach to formulating fundamental physical
theory. The relationship between these two perspectives is mathematically non-trivial: in-
deed in nonlinear dynamical systems theory, the relationship between differential evolution
equations and fractal invariant set geometry is non-computational [3, 6].
Such a volte face has some immediate non-classical consequences. For example, any
expression of the laws of physics as deterministic differential equations for fields in MU
must necessarily be incomplete: by hypothesis, the behaviour of these fields is influenced
by the geometry IU in the state-space neighbourhood of MU and not just by fields within
MU itself. In the context of quantum physics, it will be shown in Sections 2 and 3 that such
geometric influence is the basis for the Heisenberg Uncertainty Principle, the de-Broglie
relations and of particle-antiparticle pairing. In the context of gravitational physics, it is
postulated in Section 4 that such geometric influence may be the basis of the dark universe
(both dark matter and dark energy).
Another consequence is that if the ontic states of the universe are precisely those that
lie on IU , any state-space perturbation which takes such a state off IU is, by definition,
inconsistent with the laws of physics. As discussed below, such perturbations are required
to show that a putative realistic locally causal theory is constrained by some quantum
no-go theorem (e.g. [15, 1, 4, 26]). By contrast, conventional ‘hidden-variable’ approaches
(deterministic or stochastic), where the space of ontic states is a continuum set, will nec-
essarily be constrained by these theorems. The Bell [1] and Pusey et al [26] theorems are
the focus of attention below.
As with Riemannian geometry, a key quantity needed to describe the geometry of IU
is its metric. Traditionally, it has been assumed that the relevant state-space metric is
Euclidean just as in space-time. Here we show that, because the geometry of IU is fractal,
the appropriate metric between two space-times on IU is instead what is referred to as the
‘locally p-adic’ metric gp (MU , M0U ) [27], where p is a large (Fermat) prime. By Ostrowski’s
theorem the Euclidean and p-adic metrics are the only inequivalent metrics in number
theory. The reason for the relevance of the p-adic metric, as discussed in Section 2, is
 INVARIANT SET THEORY 3

the homeomorphism between the algebraic set of p-adic integers and the types of Cantor
set C(p) which underpin IU . In particular, for large p, two space-times which are close
in the Euclidean metric, but where one does not lie on IU , are far apart in the p-adic
metric. Importantly, it is shown below that Invariant Set Theory is not fine-tuned with
respect to small p-adic amplitude perturbations, even though it may (erroneously) seem
conspiratorially fine tuned with respect to the Euclidean metric. Because of this Invariant
Set Theory can negate the Bell and Pusey et al no-go theorems without violating local
causality or realism.
It is shown that evolution on IU can be readily described by the relativistic Dirac equa-
tion (in the singular limit p = ∞). The problem of unifying quantum and gravitational
physics then becomes one of synthesising the pseudo-Riemannian metric of MU and the
p-adic metric of IU into a single geometric theory. A possible route for such a synthesis,
based on a comparatively minor revision to general relativity theory is outlined in Section
4.
In Section 2, the elements of Invariant Set Theory are described and developed. In
Section 3, the theory is applied to some of the standard quantum phenomena, focussing
on quantum no-go theorems and quantum interference. In Section 4, some speculations
are made about the application of Invariant Set Theory to situations where gravity is
important. To some extent, these Section can be read independently of one another.
Hence the results discussed in Section 3 could provide motivation for a deeper study of
Section 2. Some conclusions are given in Section 5.

2. Elements of Invariant Set Theory

2.1. The Cantor Set C(p) and Basins of Attraction on IU . As mentioned in the
Introduction, a quasi-cyclic cosmology is assumed, where U evolves through multiple aeons
[30, 23]. If IU is literally fractal, then these aeons never repeat. On the other hand, there is
nothing which fundamentally prevents IU from being some ‘fat fractal’ where evolution does
repeat after a very large but finite number of aeons, and IU is in fact a limit cycle. However,
the mathematics is more straightforward if a precise fractal is assumed (in a very similar
way to where the mathematics underpinning classical dynamics is more straightforward if
it assumed that phase space is some Euclidean space R2n than some discretisation thereof).
The geometry of IU is defined locally in terms of Cantor sets of trajectories segments
C × R. By definition, each trajectory segment defines a cosmological space-time MU (‘a
world’). In this respect, a neighbouring trajectory on IU does not define ‘another world’,
but merely defines U (‘our world’) at some earlier or later cosmic aeon. Using Bohmian
language, one could say that such worlds are close relative to the implicate order defined by
the p-adic metric on IU (see Section 2.3 below), but distant relative to the explicate order
defined by the pseudo-Riemannian space-time metric. Importantly, individual trajectories
of U never ‘branch’ or ‘split’ (as in the Everett Interpretation of quantum theory).
A Cantor Set
\
C= Ck
k∈N
4 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

Figure 1. A model for C(5) - topologically equivalent to the 5-adic integers

- based on disks at several levels of iteration.

is defined as the intersection of its iterations Ck . The simplest Cantor set is the well-known
ternary set C(2) based on iterates of the unit interval on R. For example, Ck (2) is obtained
by dividing each connected interval of Ck−1 (2) into 3 equal subintervals, and removing the
middle open subinterval. More generally, let C(p) denote a Cantor set where Ck (p) is
obtained by dividing each connected interval of Ck−1 (p) into 2p − 1 equal subintervals, and
removing every second open subinterval and thus leaving p pieces. The fractal similarity
dimension of C(p) is equal to log p/ log(2p − 1) ∼ 1 for large p.
In Fig 2.3, we illustrate an equivalent model for C(5), whose elements are disks rather
than intervals (2-balls rather than 1-balls). The zeroth iteration is the unit disk. For the
first iteration, create 5 copies of the disk, each shrunk in area, and place inside the original
disk as shown in the Figure. The second iteration is made by taking each disk of the first
iteration, creating 5 shrunken copies of it and placing inside the disk as before. For the
more general p, the kth iterate of C(p) is created by taking p shrunken copies of each disk
of the k − 1th iterate. C(p) is a topological model of the p-adic integers Zp [27], discussed
in Section 2.3.
Fig 2 shows schematically a fractal set of trajectory segments based on R × C(p), at
three levels of iteration (k − 1, k and k + 1 for arbitrary k), and projected onto a two
 INVARIANT SET THEORY 5

Figure 2. A schematic illustration of state-space trajectories for the k−1th

(t0 < t < t1 ), kth (t1 < t < t3 ) and k + 1th (t3 < t) iterate trajectories of
R × C(p), with time varying instability, corresponding to a quantum system
undergoing repeated measurement. The lower magnifying glass shows the
kth iterate trajectories associated with a single k − 1th iterate trajectory;
the upper magnifying glass shows the k + 1th iterate trajectories associated
with a single kth iterate trajectory.

dimensional subset of state space. An evolving nonlinear dynamical systems will have tem-
porally varying stability and predictability. This is manifest in Fig 2, showing regimes of
metastability (with no divergence between neighbouring trajectories) punctured by inter-
mittent periods of instability. Associated with each such instability, the trajectories are
assumed attracted to two distinct quasi-stable regimes labelled ‘a’ and ‘a’,

or ‘b’ and ‘
b’
and trajectories bound for a or b are coloured red, whilst trajectories bound for a
or
b
are labelled blue. The schematic is presumed to illustrate evolution on IU associated with
repeated measurement of a quantum system.
6 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

Between t0 and t1 , Fig 2 shows a single trajectory of the k − 1 iterate of R × C(p) where
p = 2N + 1. For simplicity, we refer to this as a ‘k − 1th trajectory’. Under magnification
(i.e. by zooming to the kth iterate), this trajectory is found to comprise a bundle of 2N
kth trajectories. These are coloured red or blue according to whether they are attracted
(between t1 and t2 ) to the state-space regime a, or the state-space regime a.
The fraction
of trajectories attracted to a can be written as nk /2N where nk ∈ {0, 1, ....2N } takes one
of p values. (The fraction of trajectories attracted to b is equal to 1 − nk /2N .) In this way,
nk can be considered a label for one of the p disks in the kth iterate of C(p).
Having evolved to the regime a, Fig 2 shows a kth trajectory evolving through a second
period of metastability between t2 and t3 . Under magnification (i.e. by zooming to the
k + 1th iterate), this trajectory is found to comprise a bundle of 2N k + 1the trajectories.
As before, a fraction nk+1 /2N of these are coloured red (and 1 − nk+1 /2N blue) according
to whether they are attracted, between t3 and t4 , to regime b, or regime
b.
In Invariant Set Theory, these quasi-stable regimes are presumed to correspond to mea-
surement outcomes and the corresponding instability to the phenomenon of decoherence.
Hence, as discussed further in Section 2.2, the regions a and a
are related to the eigen-
states |ai and |ai

(similarly for b) of quantum theory. The notion of eigenstates as symbolic
representors of measurement outcomes is emphasised in Schwinger’s approach to quantum
theory [29]. In Section 4, we consider the notion that the regimes a and a
(and b and
b)
are gravitationally distinct (or ‘clumped’) regions of state space (c.f. [5, 22]).
A key question is how to account for experimenter choice in making a measurement. Here
one can note that the experimenter (and his or her neurons) are certainly part of IU and
therefore both constrain and are constrained by dynamics on IU . A crucial question is what
the structure of IU would have been had the experimenter chosen differently (in quantum
mechanical language, what the outcome of measurements in a different basis to |bi and |
bi
would have been)? Resolution of the ontological status of these types of counterfactual
experiments are absolutely central to Invariant Set Theory, and are discussed in detail in
Section 3 in various settings.
A key characteristic of fractal invariant sets is their noncomputability [6, 3]. One can
interpret this as implying that there will be no computational rule for determining the out-
come of a particular experiment, even though the underlying description of IU is completely
deterministic. (In the case where IU is some quasi-fractal limit cycle, the corresponding
result is that experimental outcomes cannot be estimated with a computational device
smaller than U itself). For this reason, the theory developed hereon is based on what
in nonlinear dynamical systems theory is referred to as ‘symbolic dynamics’ [35], implying
that the quantitative aspect of the analysis below focusses on the topological characteristics
of IU .
2.2. Complex Hilbert Vectors. Complex numbers play an essential role in quantum
theory. In Invariant Set Theory, the structure of the sample space of 2N trajectories (asso-
ciated with a single k − 1th trajectory) is guided by a permutation/negation representation
of the complex roots of unity. Consider the bit string
(1) Sa = {a1 , a2 , a3 , a4 , . . . a2N }
 INVARIANT SET THEORY 7

where ai ∈ {a, a}. N

That is to say, each of the 2 kth iterate trajectory is labelled by the
region a or a
to which it is eventually attracted. We now define two operators acting on
Sa . The first is the cyclic operator
(2) ζSa = {a3 , a4 , a5 , a6 , . . . a2N , a1 , a2 }
N −1
which permutes the elements around the bit string in pairs. That is, ζ 2 (Sa ) = Sa . The
second is the permutation/negation operator
(3) i(Sa ) = {a
2 , a1 , a
4 , a3 . . . a
2N , a2N −1 }

which operates on pairs of elements and satisfies i(i(Sa ) ≡ i2 (Sa ) = {a
1 , a
2 , a
3 , a
4 . . . a
2N } ≡

−Sa and so i2 (i2 (Sa ) ≡ i4 (Sa ) = Sa .
Now suppose Sa has the form
(4) Sa = Sa∗ ki(Sa∗ )ki2 (Sa∗ )ki3 (Sa∗ )
where
(5) Sa∗ = {a, a, a, . . . , a}
with 2N −2 elements and k is the concatenation operator. Then it is straightforwardly
checked that
N −3
(6) ζ2 (Sa ) = i(Sa )
Taking roots of (6) we have
N
(7) ζ(Sa ) = i8/2 (Sa )
From the elementary theory of complex numbers

(8) eiφ = i2φ/π

Hence, combining (7) and (8), we can write

(9) ζ n (Sa ) = eiφ (Sa ) ≡ Sa (φ)

providing 2φ/π = 8n/2N , or
φ n
(10) = N −1
2π 2
That is to say, it is necessary, in order to link these permutation/negation operators to
complex numbers eiφ and hence to angles φ on the circle, that φ, as a fraction of 2π, can
be described by N − 1 bits (and hence can be described by N bits). For other values
of φ, there is no correspondence between complex units and these permutation/negation
operators. As discussed in Section (2.5), the algebraic complex roots of unity arise from ζ
only as a singular limit [2] at N = ∞.
8 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

For example, with N = 4

Sa = {a, a, a, a, a,

a, a,

a, a,

a,

a,

a,

a, a,

a, a}


eiπ/4 (Sa ) ≡ ζ(Sa ) = {a, a, a,

a, a,

a, a,

a,

a,

a,

a, a,

a, a,
1/2

a, a} = i (Sa )
eiπ/2 (Sa ) ≡ ζ 2 (Sa ) = {a,

a, a,

a, a,

a,

a,

a,

a, a,

a, a,

a, a, a, a} = i(Sa )
eiπ (Sa ) ≡ ζ 4 (Sa ) = {a,

a,

a,

a,

a, a,

a, a,

a, a, a, a, a,

a, a,
2

a} = i (Sa )

Notice that Sa is partially correlated with eiπ/4 (Sa ), uncorrelated with eiπ/2 (Sa ) and anti-
correlated with eiπ (Sa ).
The strings Sa (φ) will be used to define a sample space for complex Hilbert vectors.
Recall that unit vectors in a real Hilbert space are a natural way to represent uncertainty
about an object of study, even in classical physics. That is to say, if i, j are orthogonal
vectors, then, by Pythagoras’s theorem,
p q
v(Pa ) = Pa i + Pa j


also has unit norm, for any probability assignments Pa , Pa of events a and a
respectively,


where Pa + Pa = 1. On this basis, the Hilbert vector


r r
nk nk
(11) |ai + 1 − N |ai
2N 2

will be used to represent a state of uncertainty as to whether a trajectory, randomly chosen
from a set of 2N trajectories belonging to some k − 1th trajectory of R × C(p), is attracted
to the quasi-stable a regime, or to the quasi-stable a
regime. Equivalently, if we write (11)
in the standard qubit form
θ θ
(12) cos |ai + sin |ai
2 2

then cos2 θ/2 must be expressible as a rational number of the form n/2N , where 0 ≤ n ≤ 2N ,
i.e. cos θ must be expressible in the form n/2N −1 − 1.
Using the permutation/negation operator ζ 2 and its relationship to complex roots of
unity, we extend this aspect of R × C(p) to complex Hilbert vectors. To do this let
(13) Sa (θ, φ) = Sa (φ)
but where the first 2N −1 cos θ occurrences of a
in Sa (φ) are set equal to a if 0 ≤ θ ≤ π/2,
or where the first −2N −1 cos θ occurrences of a in Sa (φ) are set equal to a
if π/2 ≤ θ ≤ π.
This implies for example
Sa (0, φ) = {a, a, a, . . . a}
Sa (π/2, φ) = Sa (φ)
(14) Sa (π, φ) = {a,

a,

a,

. . . a}


 INVARIANT SET THEORY 9

and the probability that an element, randomly chosen from Sa (θ, φ), is an a, is equal to
cos2 θ/2. In this way, when φ/2π and cos2 θ/2 are describable by N bits, we let
θ θ
(15) |ψa (θ, φ)i ≡ cos |ai + eiφ sin |ai
2 2

represent a sample space Sa (θ, φ) of trajectories corresponding to an iterate of R × C(p).
Now just as the quantum state is invariant under a global phase transformation, so Sa (θ, φ)
should be considered a sample space of bits, without order. That is to say, the set of global
phase transformations on |ψa i should be considered equivalent to the set of permutation
of elements of the bit string Sa (θ, φ).
As discussed in Section 3.2 below, the conditions that φ/2π and cos2 φ/2 are describ-
able by N bits, is in general, mutually incompatible. This non-commensurateness pro-
vides a number theoretic basis for a key property of quantum observables: operator non-
commutativity.
No matter how large is N , the mapping between bit strings and complex Hilbert vectors
is an injection. Complex Hilbert vectors where φ/2π and cos2 θ/2 are irrational never
correspond to bit strings. This has important physical implications. As discussed below,
certain counterfactual states allowed in quantum theory, are forbidden in Invariant Set
Theory. As such, the algebraically closed complex Hilbert Space of quantum theory (closed
under both multiplication and addition) arises as the singular (and not the smooth) limit
of Invariant Set Theory when N (and hence p) is set equal to ∞. As discussed by Berry
[2], old theories are typically the singular limit of theories which replace them.
Because of the correspondence between complex Hilbert vectors and sample spaces of
trajectories on IU , it can be seen that Invariant Set Theory is fundamentally a ψ-epistemic
theory of fundamental physics.

2.3. P-adic Integers and Cantor Sets.

‘We [number theorists] tend to work as much p-adically as with the reals
and complexes nowadays, and in fact it it best to consider all at once.’
(Andew Wiles, personal communication 2015.)
The discussion above, where an essential difference was made between numbers describ-
able or not describable by N bits, may seem like ‘fine tuning’ (especially when we are
considering very large values of N ). However, as will become clear, the distinction between
an experiment where a crucial parameter can be described by N bits, and one when it
cannot, is a distinction between two experiments in state space, not in space time.
To be more explicit, the natural metric of space-time is the pseudo-Riemannian metric
and we expect our physical theories to be continuous in this metric. Hence if two exper-
iments are performed in the real world, one with parameter settings φ1 , the other with
parameter settings φ2 , then we expect these experiments to give, statistically at least, the
same results as φ2 → φ1 . However, the nature of quantum paradoxes is that they involve
counterfactual experiments: experiments that might have been but weren’t. These are
separated from actual experiments in state space, not physical space-time. As before, we
should require that as the state-space distance between these experiments tends to zero,
10 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

the experiments should be statistically indistinguishable. However, what is an appropriate

metric on state space? Ostrowsky’s theorem [14] states that every nontrivial norm on Q is
either equivalent to the Euclidean norm or the p-adic norm. Here we argue that the metric
induced by the p-adic norm, rather than the Euclidean norm, is the appropriate measure
of distance in state space, transverse to state-space trajectories. This leads to a radically
different perspective on the fine-tuning issue above.
By way of introduction to the p-adic numbers, consider the sequence

{1, 1.4, 1.41., 1.414, 1.4142, 1.41421 . . .}

√
where each number is an increasingly accurate rational approximation to 2. As is well
known, this is a Cauchy sequence relative to the Euclidean metric d(a, b) = |a−b|, a, b ∈ Q.
Surprisingly perhaps, the sequence

(16) {1, 1 + 2, 1 + 2 + 22 , 1 + 2 + 22 + 23 , 1 + 2 + 22 + 23 + 24 , . . .}

is also a Cauchy sequence, but with respect to the (p = 2) p-adic metric dp (a, b) = |a − b|p
where
 −ord x
p p if x 6= 0
(17) |x|p =
0 if x = 0

and

the highest power of p which divides x, if x ∈ Z
(18) ordp x =
ordp a − ordp b, if x = a/b, a, b ∈ Z, b 6= 0

Hence, for example

(19) d2 (1 + 2 + 22 , 1 + 2) = 1/4, d2 (1 + 2 + 22 + 23 , 1 + 2 + 22 ) = 1/8

Just at R represents the completion of Q with respect to the Euclidean metric, so the
p-adic numbers Qp represent the completion of Q with respect to the p-adic metric. A
general p-adic number can be written in the form
∞
X
(20) ak pk
k=−m

where a−m 6= 0 and ak ∈ {0, 1, 2, . . . , p − 1}. The so-called p-adic integers Zp are those
p-adic numbers where m = 0.
It is hard to sense any physical significance to Zp and the p-adic metric from the defi-
nition above. However, they acquire relevance in Invariant Set Theory by virtue of their
association with fractal geometry. In particular, the map F2 : Z2 → C(2)
∞ ∞
X X 2ak
(21) F2 : ak 2k 7→ where ak ∈ {0, 1}
3k+1
k=0 k=0
 INVARIANT SET THEORY 11

is a homeomorphism [27], implying that every point of the Cantor ternary set can be
represented by a 2-adic integer. More generally,

∞ ∞
X
k
X 2ak
(22) Fp : ak p 7→ where ak ∈ {0, 1, . . . p − 1}
(2p − 1)k+1
k=0 k=0

is a homeomorphism between Zp and C(p) To understand the significance of the p-adic

metric, consider two points a, b ∈ C(p). Because Fp is a homeomorphism, then as d(a, b) →
0, so too does dp (ā, b̄) where F (ā) = a, F (b̄) = b. On the other hand, suppose a ∈ C(p),
b ∈/ C(p). By definition, if b ∈ / C(p), then b̄ ∈ / Zp . Let us assume that b ∈ Q. Then
b̄ ∈ Qp . This implies that dp (ā, b̄) ≥ p. Hence, d(a, b)  1 =⇒
6 dp (ā, b̄)  1. In particular,
it is possible that dp (ā, b̄)  0, even if d(a, b)  0. From a physical point of view, a
perturbation which seems insignificantly small with respect to the (intuitively appealing)
Euclidean metric, may be unrealistically large with respect to the p-adic metric, if the
perturbation takes a point on C(p) and perturbs it off C(p). The p-adic metric somehow
encompasses the primal ontological property of lying on the invariant set. The Euclidean
metric, by contrast, does not.
Let g(x, x0 ) denote the pseudo-Riemannian metric on space-time, where x, x0 ∈ MU . By
contrast let gp (MU , M0U ) denote a corresponding metric in U ’s state space, transverse to
the state-space trajectories. As above, we suppose that if MU ∈ IU , then gp (MU , M0U ) → 0
only in the p-adic sense, ie. only if M0U ∈ IU . Here, a space-time M is presumed to belong
to IU only when the relevant parameters cos2 θ/2 and φ/2π are of the form n/2N . When
the parameters are not of this form then the corresponding space-time M0 will not lie on
IU . Typically the latter occurs when considering the types of counterfactual experiment
needed to prove quantum no-go theorems. If these M0 do not lie on IU and are p-adically
distant from the corresponding M, then they are not admissible as feasible practically
realisable experiments and the sorts of conspiracy arguments used to argue against so-
called superdeterminist approaches to describe quantum physics fail. In this sense, use of
the p-adic metric in state space essentially nullifies such no-go theorems.
It can be noted in passing that many of the tools of analysis: algebra, calculus, Fourier
transforms, and indeed Lie group theory, can be applied to the set of p-adic numbers [27].
In addition, there is a natural measure on C(p), the (self-similar) Haar measure. This
provides a simple way to define the notion of probability and relate it to the concept of
frequency of occurrence (an issue which is problematic in quantum theory - see [34]).

2.4. Multiple Bit Strings. In Section 2.2, a correspondence was developed between a bit
string {a1 , a2 , . . . a2N } (modulo order) where ai ∈ {a, a},
and the complex Hilbert vector
|ψa i. It is natural to consider an extension of this to a correspondence between a pair
of bit strings {a1 , a2 , . . . a2N }, {b1 , b2 , . . . b2N }, where bi ∈ {b,
b}, and the tensor product
|ψa i ⊗ |ψb i = |ψab i.
12 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

To start, consider the three bit strings

Sa (θ1 , φ1 ) = {a1 , a2 , a3 . . . a2N } 7→ |ψa (θ1 , φ1 )
Sb (θ2 , φ2 ) = {b01 , b02 , b03 . . . b02N } 7→ |ψb (θ2 , φ2 )i
(23) Sb (θ3 , φ3 ) = {b001 , b002 , b003 . . . b002N } 7→ |ψb (θ3 , φ3 )i
where b0i , b00i ∈ {b,
b} and
θ1 θ1
|ψa (θ1 , φ1 ) = cos |ai + eiφ1 sin |ai
2 2

θ2 θ2
|ψb (θ2 , φ2 ) = cos |bi + eiφ2 sin |
bi
2 2
θ3 iφ3 θ3
(24) |ψa (θ3 , φ3 ) = cos |bi + e sin |
bi.
2 2
Here we assume that the bit string {a1 a2 . . . a2N } is independent of {b01 , b02 , b03 . . . b02N } and
{b001 , b002 , b003 . . . b002N }. The three bits strings are reduced to two by setting
bi = b0i if ai = a
(25) bi = b00i if ai = a

whence 
{a1 , a2 , a3 . . . a2N }
7→ |ψab i
{b1 , b2 , b3 . . . b2N }
where we write |ψab i as
θ1 θ1
(26) |ψab i = cos |ai|ψb (θ2 , φ2 )i + eiφ1 sin |ai|ψb (θ3 , φ3 )i
2 2

equivalent to the general form

(27) |ψab i = γ02 |ai|bi + γ12 eiχ1 |ai|
bi + γ22 eiχ2 |ai|bi

+ γ13 eiχ3 |ai|


bi
of a 2-qubit state, where
θ1 θ2 θ1 θ2 θ1 θ3 θ1 θ3
γ0 = cos cos γ1 = cos sin γ2 = sin cos γ3 = sin sin
2 2 2 2 2 2 2 2
(28) χ1 = φ2 χ2 = φ1 χ3 = φ 1 + φ 3
It can be noted that there are six degrees of freedom associated with the two bit strings,
{a1 , a2 , b3 , . . . a2N }, {b1 , b2 , b3 . . . b2N } consistent with quantum theory. Note also that these
bit strings cannot in general both be written in the form Sa (θ, φ), Sb (θ, φ), consistent with
the fact that the state space S6 of the general 2-qubit cannot be expressed as the Cartesian
product S2 ×S2 of Bloch Spheres. Based on (25), it is easy to show that statistical relations
between the bit strings {ai } and {bi } are consistent with those from quantum theory. For
example, the probability that ai = a is equal to cos2 θ1 /2 from (23). Now if ai = a then
by definition bi = b0i from (25). The probability that b0i = b is equal to cos2 θ2 /2 from (23).
Hence the probability that ai = a and bi = b is equal to cos2 θ1 /2 cos2 θ2 /2 = γ02 from (28).
 INVARIANT SET THEORY 13

Consider two special cases. Firstly, if b0i = b00i for all i, then the probability that ai = a
is independent of the probability that bi = b. In other words, we can write

Sa (θ1 , φ1 )
(29) 7→ |ψab i
Sb (θ2 , φ2 )
consistent with the factorisation |ψab i = |ψa i|ψb i in quantum theory. In the second special
case, let b0i =
b00i for all i (so that cos2 θ3 /2 = sin2 θ2 /2). Then the probability that
ai = a and bi = b is equal to cos2 θ1 /2 cos2 θ2 /2, and the probability that ai = a
and
b00i =
b is equal to sin2 θ1 /2 sin2 θ3 /2 = sin2 θ1 /2 cos2 θ2 /2. Hence, the probability that
either ai = a and bi = b, or ai = a
and bi =
b (i.e. the labels ‘agree’) is equal to
cos2 θ1 /2 cos2 θ2 /2 + sin2 θ1 /2 cos2 θ2 /2 = cos2 θ2 /2. With cos2 θ1 /2 = 1/2, the correlation
between {a1 , a2 , . . . a2N }, {b1 , b2 , . . . b2N } are consistent with quantum theoretic correlations
of measurement outcomes on the Bell state
|ai|bi + |ai|
bi
(30) |ψab i = √

2
where θ2 denotes the relative orientation of EPR-Bell measurement apparatuses. The
constraint that cos2 θ/2 is describable by N bits takes centre stage in Invariant Set Theory’s
account of the Bell Theorem, as discussed below.
This entanglement construction is generalised to m qubits in Appendix A.

2.5. Evolution on IU and the Dirac Equation. In this Section we discuss how unitary
dynamical evolution is described in Invariant Set Theory. We start by describing evolution
of a uniformly moving particle of mass m in a space-time MU . It will be convenient to
discuss special relativistic theory from the outset.
With suggestive notation, let ψ(0) denote a set of four 2N element bit strings witten in
the form
 
S1
S2 
(31) ψ(0) = 
S3 


S4
To start, consider a frame where the particle is at rest. The time evolution of ψ(0) is
defined as
 n  
ζ S1
 ζ n  S2 
(32) ψ(n∆t) = 
ζ −n
 
  S3 
ζ −n S4

where ∆t = 2π/2N −1 ω, ~ω = mc2 and blank matrix elements denote zeros. From (9) we
can write
N −1
(33) ζ n ≡ e2πin/2 = eiωt
14 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

where t = n∆t. Hence the set {eiωt } of time evolution operators is isomorphic to the
multiplicative group of complex phases φ where φ/2π is describable by N − 1 bits. Writing
the evolution equation (32) as
 iωt 
e
 eiωt 
ψ(t) =  −iωt
 ψ(0)
 e 
e−iωt
the larger is N the larger the number of phases φ where eiωt is defined. However, for
any finite N , {eiωt } is not closed under addition. As discussed above, this is considered a
desirable property of Invariant Set Theory, making it counterfactually incomplete.
In the singular limit N = ∞ (or 1/N = 0), eiωt can be identified with the familiar
complex exponential function, in which case, eiω∆t ≈ 1 + iω∆t for small ∆t and
i∂t eiωt + ωeiωt = 0
Because the limit is singular, the derivative is undefined for any finite N , no matter how
big. In this sense, the Dirac equation for a particle at rest,
(34) i~γ0 ∂t ψ + mcψ = 0
can be treated as the singular limit of (32) at N = ∞. Of course, in (34), ψ is to be
considered some more abstract ‘wavefunction’, lying in complex Hilbert Space. However,
crucially, it should be noted that 32 is valid and well defined for finite N . That is to say,
from the perspective of Invariant Set Theory, (32) should be considered more fundamental
than (34).
We now equate the energy E of the particle in MU with a key property of the geometry
of the surrounding kth iterate trajectories on IU : its periodicity. That is to say, writing E =
~ω, then, because, ~ω = mc2 , (32) is consistent with the special relativistic formula E =
mc2 for a particle at rest. What is the physics behind this construction? As discussed in
the Introduction, the key idea in Invariant Set Theory is that the most primitive expression
of the laws of physics is a description of the geometry of the invariant set IU in state space.
That is to say, properties of our space-time M are determined by the geometry of IU in
state space, in the neighbourhood of M. Let us express the equation E = ~ω as an example
of the following more general form
Expression for a physical prop-
Expression based on the lo-
(35) erty in the locally Euclidean =
cally p-adic space IU
space MU
This is analogous to the geodesic equation in general relativity written as
duρ
(36) = −Γρµν uµ uν
dλ
where the left hand side is a physical property of a test particle (its acceleration) on a
geodesic and the right hand side involves derivatives transverse to the geodesic in space
time. A key difference between (35) and (36) is that whilst calculus in MU is based
on the pseudo Riemannian metric, the calculus on IU must be based on the locally p-adic
 INVARIANT SET THEORY 15

metric. Expression (35) is used to propose in Section 4 a generalisation of general relativity,

consistent with Invariant Set Theory.
It can be noted that (32) describes two pairs of helical trajectories, rotating with opposite
helicity. These can be associated with the two Weyl spinors of Dirac theory, and correspond
to ‘particles’ and ‘antiparticles’ in MU . This linkage is again a manifestation of the basic
notion expressed in (35).
In a non-rest frame, we generalise (32) to
 
S1
S2 
(37) ψ(t) = E0 E1 E2 E3 
S3 


S4
where
eiωt eik1 x
   
 eiωt   eik1 x 
E0 =  E1 = 
e−iωt

−ik

   e 1 x 
e−iωt −ik
e 1 x

ie−ik2 y eik3 z
   
 ieik2 y   e−ik3 z 
E2 =   E3 =  −ik

 ieik2 y  e 3 z 
ie−ik2 y e ik 3 z

where x = n∆x, ∆x = 2π/2N −1 k1 (etc for y and z) and ~2 ω 2 = ~2 |k|2 + m2 (leaving three
degrees of freedom to describe the particle’s momentum in MU ). Note that the Dirac
matrices

   
1 1
 1   1 
γ0 =   γ1 =  
 −1   −1 
−1 −1

   
−i 1
 i   −1
γ2 =   γ3 = 
−1

 i  
−i 1
are clearly evident in the form of the evolution operators E0 , E1 , E2 and E3 . With particle
momentum p in MU inherited from the periodicity of the surrounding geometry of IU , so
that p = ~k c.f. (35), then (37) implies that
E 2 = p2 + m2
as required by special relativity theory.
16 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

As before, in the singular limit N = ∞ (and only in this limit) we can write i∂x eik1 x +
k1 eik1 x = 0 and so on. Hence, (37) can be seen to be equivalent to evolution as described
by the full Dirac equation
(38) i~γi ∂i ψ + mcψ = 0
in the (singular) limit N = ∞. As an evolution equation for finite N , (37) is to be
considered (in Invariant Set Theory) as more fundamental than (38). Like the Dirac
equation, (37) is relativistically invariant. Interestingly, the evolution (37) in space and
time is fundamentally granular. The implications of this will be explored elsewhere.
Equation (37) describes the periods of metastable evolution on IU as shown schematically
in Fig 2 (between t0 < t < t1 and between t2 < t < t3 ), i.e. corresponding to unitary
evolution between preparation and measurement in quantum theory. In Section 4, the
periods of instability and attraction to regions a and a
(between t1 < t < t2 ) and b and
b
(between t3 < t < t4 ) are linked to explicitly gravitational processes.

3. Applications of Invariant Set Theory

3.1. The Bell Theorem. Invariant Set Theory is both realistic, and, as discussed below,
locally causal. Despite this, it robustly violates the Bell inequalities without fine-tuned
conspiracy, retrocausality or denial of experimenter free will. The CHSH [4] version of the
Bell Inequality can be written in the form
(39) |Corr(A1 , B1 ) − Corr(A1 , B2 )| + |Corr(A2 , B1 ) + Corr(A2 , B2 )| ≤ 2
Here Alice and Bob each choose one of two buttons labelled ‘1’ and ‘2’ to perform spin mea-
surements on entangled particle pairs prepared in the Bell state (30). Here ‘Corr’ denotes
the correlation between spin measurements over some ensemble of entangled particle pairs.
According to both Quantum Theory and Invariant Set Theory, Corr(A1 , B1 ) = cos θA1 B1 ,
where θA1 B1 denotes the relative orientation of the measuring apparatuses when Alice
presses button 1, Bob presses button 2 etc. With A1 , A2 and B1 collinear (not a vital
assumption; see [19, 20]) and θA1 B1 = θA1 A2 + θA2 B1 , then we can write
(40) cos θA1 B1 = cos θA1 A2 cos θA2 B1 − sin θA1 A2 sin θA2 B1 .
Now let us assume that cos θA2 B1 can be written in the form n1 /2N . This will be the case
in Invariant Set Theory over the subsample of measurements where Alice choses button 2
and Bob button 1. We can also assume cos θA1 A2 is in the form n2 /2N since Alice can,
having measured her particle relative to A2 , remeasure the same particle relative to A1 -
such a re-measurement experiment must also be described by a state on the Invariant Set.
However, then sin θA2 B1 and sin θA1 A2 cannot be written in the form n3 /2N and n4 /2N -
if they could then there would be integer solutions to n21 + n23 = n22 + n24 = 2N (we assume
none of n1 , n2 , n3 or n4 is identically zero). As first shown by Euclid, there are no non-zero
integer solutions to the Pythagorean equations a2 +b2 = c2 with c a power of 2. With θA1 A2
and θA2 B1 being independent angles, i.e. with no functional relationship, (40) implies that
cos θA1 B1 is not describable by N bits (no matter how large is N ).
 INVARIANT SET THEORY 17

However, if cos θA1 B1 is not describable by N bits, then the hypothetical experiment
where Alice presses A1 and Bob B1 on the particle pair where Alice actually presses A2
and Bob B1 , does not lie on IU . That is to say, if Alice chooses button 2 and Bob button
1, then the counterfactual world where, for the same entangled state, Alice chooses button
1 and Bob button 1, does not correspond to a state of the universe on IU . Similarly,
the counterfactual world where Alice chooses button 2 and Bob button 2 does not lie on
IU . Conversely, if Alice chooses button 1 and Bob button 1 (in MU ∈ IU ), then the
counterfactual worlds where, for the same entangled state, Alice chooses button 1 and Bob
button 2, or Alice button 2 and Bob button 1, do not correspond to states of the universe
on IU . Hence it is impossible to derive the CHSH inequality on any single sample of particle
pairs in Invariant Set Theory.
Does the Invariant Set constraint - that the cosine of relative orientation between Alice
and Bob’s apparatuses are describable by N bits - violate local causality? No. Let EA
and EB denote the setting of Alice and Bob’s measuring apparatus, assumed space-like
separated. The necessarily finite precision of these apparatuses mean that if N is large
enough, then whilst Alice and Bob together have complete control over the leading bits
of the cosine of their relative orientation, they have no control over the trailing bits, and
in particular have no control over whether or not the cosine of relative orientation is
describable by N bits. Since the choice of leading bits is irrelevant to the issue of whether
or not the relative orientation is describable by N bits, Alice’s choice of setting for the
orientation of her measuring apparatus does not in any way constrain Bob’s choice in
setting his apparatus: EA does not affect EB or vice versa. Rather, the uncontrollable
constraint that the cosine of the relative orientation must be describable by N bits is a
holistic property of the Invariant Set. Just as the causal ordering of events in Special
Relativity is not destroyed by the presence of space-time curvature (and hence distant
masses) in General Relativity, so neither does the geometry of IU in state space destroy
the causal structure of General Relativistic space-time: the holistic nature of the Invariant
Set postulate is not non-causal. Although, the Invariant Set postulate technically implies a
violation of the Measurement Independence condition [11, 12], at a deeper level it indicates
the limitation of conventional reductionist approaches to the formulation of physical theory:
in Invariant Set Theory the physics of the small is determined by the large-scale state-space
structure of the universe.
So what is actually measured when the CHSH inequality is shown to be violated ex-
perimentally? It is important to note that whilst the derivation of the CHSH inequality
assumes a single sample of particles over which different spin measurements are performed,
in a real-world experiment each of the four correlations in the CHSH inequality are esti-
mated from four sub-experiments using separate sub-ensembles of entangled particle pairs.
In Invariant Set Theory, each sub-experiment takes places in M on IU . This means that
experimentally the four relative angles for each of the four sub-experiments cannot be pre-
cisely the four angles θA1 B1 , θA1 B2 , θA2 B1 and θA2 B2 above, but must rather be, say, θA0 ,
1 B1
0
θA1 B2 , θA2 B1 and θA2 B2 where the cosine of each angle is describable by N bits. Here we
recognise the finite precision of real-world experiments. In particular, both |θA 0 − θA1 B1 |
1 B1
18 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

and |θA0 − θA2 B2 | can be assumed smaller than the finite precision within which these
2 B2
experiments can be performed.
Even though |θA 0 −θA1 B1 |  1, the argument above does not require the experimenters
1 B1
to be in any way precise in setting up quantum no-go experiments, i.e. in ensuring that the
relative orientation is θA 0 rather than θA1 B1 . It is worth discussing this in the context
1 B1
of Bell’s remark [1]:
Remember, however, that the disagreement between locality and quantum
mechanics is large . . . . So some hand trembling [by the experimenters] can
be tolerated without much change in the conclusion. Quantification of this
would require careful epsilonics.
Since by construction θA1 B1 defines a state which does not lie on IU , and θA 0 defines a
1 B1
state that does lie on IU , the p-adic distance between these states is large, even though
the Euclidean distance between the angles is small. In particular, with respect to the p-
adic metric, which we claim to be the physically relevant metric transverse to trajectories
in state space, there is no small perturbation (e.g. associated with experimenter hand
trembling) which can take a state of the universe from one where the relative orientation of
the measuring apparatuses is θA 0 to one where the relative orientation is θA1 B1 . Hence
1 B1
we conclude that application of ‘careful epsilonics’ in fact allows us to conclude that CHSH
violation is robustly consistent with local realism, even though the argument does appear
fine-tuned and hence conspiratorial with respect to the physically less relevant Euclidean
metric. This discussion is central to all so-called conspiracy arguments [1] concerning the
Bell inequalities: essentially there are no conspiracies if one measures distances in state
space p-adically.
One can phrase the issue in terms of robustness to noise. One can perturb a state on
IU with small amplitude noise. However, for the noise to be physical and hence respect
the Invariant Set postulate, then the noise must be added p-adically to an unperturbed
state (consistent with the property that the set of p-adic integers is closed under addition).
That is to say, one can p-adically perturb a state where θA 0 is describable by N bits with
1 B1
00
small amplitude noise and the relative orientation θA1 B1 of the perturbed state will still be
describable by N bits. Conversely, no amount of such noise will take the state where θA 0
1 B1
can be described by N bits, to one where θA1 B1 is not described by N bits - the states are
too far apart for this to be possible.
To conclude this Section, it is worth noting that the argument given here is not incom-
patible with a recent discussion by ’t Hooft [32] (based on a model [33] which, like Invariant
Set Theory, is deterministic, nonlinear, and eschews superpositions at a fundamental level):
. . . even in a superdeterministic world, contradictions with Bell’s theorem
would ensue if it would be legal to consider a change of one or a few bits
in the beables describing Alice’s world, without making any modifications
in Bob’s world. . . . [However,] it is easy to observe that, certainly in the
distant past, the effects of such a modification would be enormous and it
may never be compatible with a simple low-entropy Big Bang . . . Thus,
we can demand in our theory that a modification of just a few beables in
 INVARIANT SET THEORY 19

Alice’s world without any changes in Bob’s world is fundamentally illegal.

This is how an ontological deterministic model can ‘conspire’ to violate
Bell’s theorem.
Somewhat similar to Invariant Set Theory, this quote proposes a constraint on the physics
of the small from the physics of the large. On the other hand, the supposed conflict with
the low entropy state of the initial universe is speculative, and is not (as far as the author
can see) a direct consequence of the cellular automaton dynamics [33] proposed by ’t Hooft.
By contrast, in Invariant Set Theory, the illegality of modifying Alice’s world is a direct and
immediate consequence of the assumed Invariant Set postulate. There are other differences
with ’t Hooft’s theory. For example, although ’t Hooft’s model is not founded on differential
evolution equations, it is nevertheless based on a finite-difference cellular automaton rule
for dynamical evolution. That is to say, dynamical evolution falls into the generic class of
‘hidden-variable’ theories. In particular, its underpinning metaphysics differs conceptually
from the non-reductionist Invariant Set Theory philosophy.

3.2. Quantum Interferometry. It is now possible to show how the non-commutativity

of position and momentum observables in quantum theory is explainable in Invariant Set
Theory through the number theoretic incommensurateness between φ and cos φ. Consider
the experimental set up as in Fig 3a. According to quantum theory, the input state vector
|ai is transformed by the unitary operators
   
1 1 1 0
U= ; V =
1 −1 eiφ
according to
VU 1
(41) |ai 7→ √ (|bi + eiφ |
bi)
2
According to (15), the corresponding transformation between preparation and measure-
ment state on IU is
(42) Sa (0, 0) 7→ Sb (π/2, φ)
Consider, instead, the experimental set up as in Fig 3b (that is to say, a Mach-Zehnder in-
terferometer). Now, according to quantum theory, the input state vector |ai is transformed
according to
UV U φ φ
(43) |ai 7→ cos |ci + sin |
ci
2 2
According to (15), the corresponding transformation on IU is
(44) Sa (0, 0) 7→ Sc (φ, 0)
We now claim that transformations (42) and (44) are incompatible, in the sense that if
transformation (42) corresponds to a trajectory on the invariant set IU , then transformation
(44) does not, and vice versa. To see this we need a result from elementary number theory:
Theorem[13]. Let φ/π ∈ Q. Then cos φ ∈ / Q except when cos φ = 0, ±1/2, ±1.
20 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

Figure 3. a) A which-way experiment. b) A interferometric experiment

Proof . Assume that 2 cos φ = a/b where a, b ∈ Z, b 6= 0 have no common factors. Since
2 cos 2φ = (2 cos φ)2 − 2 then
a2 − 2b2
(45) 2 cos 2φ =
b2
Now a2 − 2b2 and b2 have no common factors, since if p were a prime number dividing
both, then p|b2 =⇒ p|b and p|(a2 − 2b2 ) =⇒ p|a, a contradiction. Hence if b 6= ±1, then
the denominators in 2 cos φ, 2 cos 2φ, 2 cos 4φ, 2 cos 8φ . . . get bigger without limit. On the
other hand, if φ/π = m/n where m, n ∈ Z have no common factors, then the sequence
(2 cos 2k φ)k∈N admits at most n values. Hence we have a contradiction. Hence b = ±1 and
cos φ = 0, ±1/2, ±1 QED.

Suppose an experimenter performs a ‘which way’ experiment as in Fig 3a. As before, the
experimenter is assumed to have control only on the leading bits of φ and has no control
on the trailing bits, or on the Invariant Set condition that φ/2π is describable by N bits.
It is exponentially unlikely (with N ) that cos φ would correspond precisely to one of the
exceptional values in the theorem. Hence, in general, one can state that in Invariant Set
Theory the number-theoretic incommensurateness between φ and cos φ implies that if a
 INVARIANT SET THEORY 21

position measurement is made, then a momentum measurement cannot be made, and vice
versa.
Feynman famously claimed [10] that quantum interference demonstrates that the Lapla-
cian laws for combining probabilities necessarily fail in quantum physics. We can use the
discussion above to argue otherwise. Let us assume φ ≈ 0, and let Pc ≈ 0 denote the
probability of detection by Dc in Fig 3b, Pb the probability of detection by Db and Pb the

probability of detection by Db in Fig 3a. In general, Pc 6= Pb + Pb = 1. However, as dis-
 
cussed above, the notion of probability is defined from the natural Haar measure on IU . As
discussed above, in situations where momentum measurements are made (cos φ describable
by N bits), counterfactual position measurements (φ/π describable by N bits) do not lie
on IU . That is to say, the sample space from which the probability Pc is estimated is com-
pletely distinct from the sample space where position measurements are made. As such,
one can assert in Invariant Set Theory that particles do travel either through the upper
of lower branch. However, if a momentum measurement is being made, then the positions
of such particles are fundamentally unobservable - in Bell’s language, their positions are
merely ‘beables’.
If particles do travel through either the upper or lower arm in situations were an in-
terference experiment is being conducted, then a key ‘conceptual’ question is: How does
a particle ‘know’ whether to behave like a wave rather than a classical particle? This
question becomes potentially problematic when one considers a delayed-choice experiment,
where the experimenter only decides whether to perform an interferometric or which-way
experiment after the particle has entered the apparatus - and perhaps passed through the
phase shifter. However, in Invariant Set Theory, this isn’t problematic at all: the structure
of IU at some time t0 is determined by events to the future of t0 . This isn’t to be confused
by the notion of retrocausality, but is again indicative of the holistic causal nature of IU .
The situation is fundamentally no different to asking whether a point p ∈ MU lies on a
black hole event horizon [19], defined as the boundary of null rays which escape to (future
null) infinity. Like IU , the event horizon is an entirely causal and realistic entity. However,
because of the global definition of the event horizon, it is impossible to determine its posi-
tion from a knowledge of the Riemann tensor in the neighbourhood of the event horizon;
its position at some time t0 in space-time can be influenced by an event potentially in the
future of t0 , e.g. whether or not a massive object falls into the black hole at t1 > t0 . As
such, the position of the event horizon at t0 can also depend on the state of some exper-
imenter’s neurons at some future time, if these neurons are responsible for determining
whether or not the massive object falls into the black hole.
At the practical level, the finite-precision with which an experimenter can set φ means
that it is impossible to ascertain by experiment whether cos φ is describable by N bits, or
φ/π is describable by N bits. Moreover, at the computational level there is no algorithm
for determining the precise value of φ. More specifically, since the geometric properties IU
are non-computable [3, 6], there is no algorithm for determining what type of experiment
(which way or interferometric) will be performed ahead of time. Because of this, Invariant
Set Theory explains naturally why, in quantum physics, statements about the future have
22 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

to be cast in overtly probabilistic language, whilst statements about the past can be cast
in quite definite non-probabilistic terms [7].
3.3. PBR. The recent PBR theorem [26] is a no-go theorem casting doubt on ψ-epistemic
theories (where the quantum state is presumed to represent information about some un-
derlying physical state of the system). Unlike CHSH where Alice and Bob each choose
measurement orientations A or B, here Alice and Bob, by each choosing 0 or 1, prepare a
quantum system in one of four input states to some quantum circuit: |ψ0 i|ψ0 i, |ψ0 i|ψ1 i,
|ψ1 i|ψ0 i or |ψ1 i|ψ1 i, where
θ θ
|ψ0 i = cos |0i + sin |1i
2 2
θ θ
|ψ1 i = cos |0i − sin |1i
2 2
In addition to the parameter θ, the circuit contains two phase angles α and β; as dis-
cussed below, the phase angle α most closely plays the role of the phase angle φ in the
Mach-Zehnder interferometer in Section 3.2. The output states of the quantum circuit
are characterised as ‘Not 00’, ‘Not 01’, ‘Not 10’ and ‘Not 00’. The α and β are chosen
to ensure that (according to quantum theory), if Alice and Bob’s input choices are {IJ}
where I, J ∈ {0, 1}, then the probability of ‘Not IJ’ is equal to zero. However, if physics
is governed by some underpinning ψ-epistemic theory, then, so the argument goes, at least
occasionally the measuring device will be uncertain as to whether, for example, the input
state was prepared using 00 and 01 and on these occasions it is possible that an outcome
‘Not 01’ is observed when the state was prepared as 01, contrary to quantum theory (and
experiment). How does Invariant Set Theory, which is indeed a ψ-epistemic theory, avoid
this problem?
Working through the algebra, it is found that the probabilities of various outcomes are
trigonometric functions of α − β, α − 2β, β and θ. For example, if Alice and Bob chose 00,
then, according to quantum theory (and therefore experiment), the probability of obtaining
the outcome ‘Not 01’ is equal to
θ θ θ θ
(46) X = cos4 + sin4 + 2 cos2 sin2 cos(α − 2β) 6= 0
2 2 2 2
On the other hand, if Alice and Bob chose 01, then the probability of obtaining the outcome
‘Not 01’ would be equal to
θ θ θ θ θ θ
(47) Z = X − 4 cos2 sin2 − 4 cos3 sin cos(α − β) − 4 cos sin3 cos β = 0
2 2 2 2 2 2
The key point is that X contains the trigonometric term cos(α − 2β), whilst Z contains
the terms cos(α − β) and cos β. Now one can clearly find values for α, β and θ such that
X is described by N bits. That is to say, for large enough N , Invariant Set Theory can
predict the quantum theoretic probability of outcome ‘Not 01’ when Alice and Bob chose
00. However, in general it is impossible to find values for these angles such that X and Z
are simultaneously describable by N bits. The number-theoretic argument is exactly that
used to negate the Bell Theorem. For example, if cos(α − 2β) and cos β are describable by
 INVARIANT SET THEORY 23

a finite number of bits, then cos(α − β) = cos(α − 2β) cos β + sin(α − 2β) sin β is not. This
means that if in reality Alice and Bob chose 00 in preparing a particular quantum system
and the outcome was ‘Not 01’ , then there is no counterfactual world on IU where Alice and
Bob chose 01 in preparing the same quantum system, and the outcome was again ‘Not 01’.
That is to say, it is not the case that Z = 0 for this counterfactual experiment - rather, Z
is undefined. Conversely, if in reality Alice and Bob chose 01, then there exist values for
α, β and θ such that Z is described by N bits and equal to zero (to within experimental
accuracy) and ‘Not 01’ is not observed.
Let {αX , βX , θX } denote a set of angles such that X is describable by N bits, and
{αZ , βZ , θZ } a set of angles such that Z is describable by N bits, i.e. these correspond to
experiments on IU . Now, as before, we can find values such that the differences αZ − αX ,
βZ − βX and θZ − θX are each smaller than the precision by which these angles can be set
experimentally. Hence, Invariant Set Theory can readily account for pairs of experiments
performed sequentially with seemingly identical parameters, the first where Alice and Bob
choose 00 and the outcome is sometimes ‘Not 01’, and the second where Alice and Bob
choose 01 and the outcome is never ‘Not 01’. That is to say, the Invariant Set Theoretic
interpretation of the PBR quantum circuit reveals no inconsistency with experiment. Even
though Invariant Set Theory is ψ-epistemic, the holistic structure of the invariant set IU
ensures that the measuring device will never be uncertain as to whether, for example, the
input state was prepared using 00 and 01.
It was shown above that Invariant Set Theory evades the Bell theorem by violating the
Measurement Independence assumption. Here it has been shown that Invariant Set Theory
evades the PBR theorem by violating an equivalent Preparation Independence assumption.
As before a more revealing reason for the failure of these theorems is that Invariant Set
Theory does not fit into the traditional reductionist approach to hidden-variable theory.
As before, this does not conflict at all with the experimenter’s sense of free will. Neither
does it imply fine-tuning with respect to the physically relevant p-adic metric on IU .

4. Quantum Gravity
‘Despite impressive progress . . . towards the intended goal of a satisfactory
quantum theory of gravity, there remain fundamental problems whose so-
lutions do not appear to be yet in sight. . . . [I]t has been argued that Ein-
steins equations should perhaps be replaced by something more compatible
with conventional quantum theory. There is also the alternative possibil-
ity, which has occasionally been aired, that some of the basic principles of
quantum mechanics may need to be called into question.’ (Roger Penrose,
1976 [21]
Using Invariant Set Theory as a guide, we speculate here on one such ‘alternative possibil-
ity’, and in so doing propose a generalisation of Einstein’s equations. This in turn suggests
some potentially observable consequences of Invariant Set Theory.
As discussed, quantum theory emerges from Invariant Set Theory as the singular limit
where 1/N = 0. As Michael Berry has noted, old theories are typically the singular limits
24 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

of new theories: classical physics and quantum theory, Newtonian gravity and general
relativity being two pairs of relevant examples. As in both of these pairs of examples, the
old theories often work well under limited parameter regimes, but can fail catastrophically.
We speculate here that quantum theory may itself fail catastrophically under situations
where gravity is not negligible. For example, it seems plausible to speculate that the relative
stability of the two identifiable regions a and a,
and the corresponding instability of the
basin boundary - both geometric properties of IU - is a manifestation of the phenomenon
we call ‘gravity’. This is consistent with the notion [5] [22] that measurement eigenstates
in quantum theory are gravitationally distinct. If this is correct it will mean that gravity
will itself be inherently ‘decoherent’ - quite different from the effects of the other gauge
fields. This idea should be testable experimentally in the coming years.
How are the gauge fields representable in Invariant Set Theory? Instead of thinking of
these as fields on a fixed background space-time, one should instead think them in terms of
the collective properties of helical geometry of space-time trajectories on IU . For example,
as discussed in Section 2.5, these helical structures can be linked with the spinors which
provide the most basic representations of the Maxwell equations [24]. (It can be noted that
the theory of p-adic manifolds as Lie Groups is well established [28].) Hence, for example,
the vacuum fluctuations of quantum field theory can be considered as describing variations
in the kth iterate symbolic labels associated with a particular k −1th iterate trajectory (c.f.
Fig 2). As discussed below, relative to the p-adic metric, the distance between individual
kth iterates is a factor p smaller than the distance between the ‘gravitationally clumped’
regions a and a
discussed above. This suggests that these vacuum fluctuations should be
considered as gravitationally indistinct. This has important implications, discussed below.
The fundamentally different representations of gauge fields and gravity on IU sug-
gests that it is misguided to imagine that the synthesis of quantum and gravitational
physics can be brought about by applying quantum field theoretic ansätze to a gravita-
tional Lagrangian. Indeed one can go further: Invariant Set Theory predicts the non-
existence of the graviton and hence one will not be discovered in a particle accelerator
or elsewhere (c.f. http://home.cern/about/physics/extra-dimensions-gravitons-and-tiny-
black-holes). The existence of the graviton has also been called into question by Dyson
[8].
The existence of the spin-2 graviton is sometimes used to support the concept of su-
persymmetry - the argument being that it is unlikely that there is a gap in the spin k/2
particles from k = 0 to k = 2. However, this argument fails if in fact there is no spin 2
particle, as predicted is the case above. Suppose indeed that the highest spin elementary
particle is spin 1 and supersymmetry is not part of the laws of physics. How then can
one account for the dark universe? Here one can draw on the basic concept of Invariant
Set Theory described in (35) (and used to motivate the de Broglie relationships): that
the physics of our space-time MU is determined by the geometry of state space in the
neighbourhood N of MU on IU . This suggests a generalisation of the dynamical equations
of general relativity from
8πG
(48) Gµν (MU ) = 4 Tµν (MU )
c
 INVARIANT SET THEORY 25

to
Z
8πG
(49) Gµν (MU ) = 4 Tµν (M0U )F (MU , M0U ) dµ
c N (MU )

where F (MU , M0U ) is some propagator (to be determined) and dµ a suitably normalised
Haar measure in some neighbourhood N (MU ) on IU . Just as the presence of distant matter
(as represented by General Relativity Theory) does not destroy the notion of causal order
in Special Relativity Theory, so the generalisation (49) does not affect the notion of causal
order in MU . That is to say, (49) is a locally causal extension of the field equations of
General Relativity.
The generalisation (49) predicts that the integrated (or ‘smeared out’) effect of energy-
momentum in space-times neighbouring MU on IU would influence the curvature of MU .
Such an influence could be interpreted (wrongly, according to Invariant Set Theory) as
implying the existence of ‘dark matter’ in MU ; as such, Invariant Set Theory predicts that
dark-matter particles will not be discovered experimentally. Extension (49) also provides a
potential explanation for dark energy. That is to say, a basic aspect of the geometry of IU , as
shown in Fig 2, is the exponential divergence of nearby space-times. Such divergence occurs
ubiquitously in state-space in the neighbourhood of MU . Through (49), this exponential
divergence should leave an imprint on our space-time as a positive cosmological constant.
This raises the question as to why, in Invariant Set Theory, vacuum fluctuations do not
contribute to a vastly larger value of dark energy than is observed (as they do when
computed using standard quantum theory). The answer has already been suggested above:
the p-adic metric gp (MU , M0U ) on IU does not vary continuously as MU → M0U , but
rather jumps by factors of p. This means that when M0U is sufficiently close to MU it
effectively has no distinct role in contributing to the integral on the right hand side of
(49). As mentioned, the space-times M0U associated with what in quantum field theory
would be described as vacuum fluctuations in MU , are examples where the p-adic distance
gp (MU , M0U ) is so small that it has no impact in (49). Whether these speculations stand
up to quantitative analysis remains to be seen - in any case, one should recall that these
speculations arose from an attempt to formulate a realistic theory of quantum physics, and
not to solve the dark universe problem per se.
Invariant Set Theory also provides a novel explanation of the fate of information in
black holes. For IU to be a compact set in U ’s state space, there must be corresponding
regions of state space where trajectories are converging. It seems plausible to suppose
that this occurs on Planck scales (an aspect of Invariant Set Theory not discussed in this
paper). Consistent with this, Penrose [23] argues that state-space trajectory convergence
is generic at final space-time singularities. However, such convergence need not imply
loss of information (as Penrose assumes to be the case). The concept of information is
closely linked to entropy and conventionally requires one to consider a coarse-graining of
state-space volumes. In this way, one can equate information loss to state-space volume
shrinkage associated with trajectory convergence. However, the volume of IU is strictly
zero, and hence volumes cannot shrink on IU . That is to say, the resolution of the black hole
suitably information paradox may actually be no different to the resolution of the other
26 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

quantum paradoxes discussed in this paper. For example, in Section 3.2, the counterfactual
Mach-Zehnder experiments that would, if physically real, contradict Laplacian addition of
probability, lie off IU . Hence Laplacian addition of probability is not really violated -
even though it may seem to be. Similarly, the counterfactual experiments necessary to
derive the Bell inequality also lie off IU and do not correspond to states of physical reality.
Hence, local causality is not really violated, even though it may seem to be. Finally,
the counterfactual experiments necessary to infer state space volume shrinkage and hence
black-hole information loss, also lie off IU and do not correspond to states of physical
reality. Hence, ‘information’ (suitable defined on IU ) is not really lost, even though it may
seem to be. Notice that we have managed to retain information without needing a firewall
or any such structure for the (locally non-computable) event horizon.
In a late universe with many black holes forming, this suggests would be strong state-
space convergence and hence a reversal of dark energy similar to quintessence fields [36].
This would lead to a strong reduction in what could be described as total entropy. The
convergence of space time trajectories associated with a universe collapsing towards a big
crunch would therefore be in a low-entropy state before it reemerged into the next epoch.
As such, Invariant Set Theory has no need for an inflationary epoch at the time of the Big
Bang.
One of the reasons for seeking a quantum theory of gravity is that it should eliminate
precise singularities in MU . Here we speculate that the ‘smearing effect’ of the generalisa-
tion (49) could do just this: space-times M0U in the neighbourhood MU where a massive
star would otherwise collapse towards a singularity, would be collapsing in subtly different
ways and the integrated impact of these alternatives on the space-time curvature of MU
could prevent the singularity in MU from forming. Such a speculation, like the others in
this Section, must of course be put on a proper quantitative footing.

5. Conclusions
A motivating belief underpinning this work is that quantum mechanics is a fundamen-
tally inaccurate theory with which to describe and hence incorporate the phenomenon
of gravity. An alternative geometric ψ-epistemic theory of quantum physics - Invariant
Set Theory - has been described which, it is claimed, can incorporate Einstein’s causal
geometric theory of gravity straightforwardly. As discussed, this has a number of po-
tentially observable consequences: gravity as an inherently decoherent phenomenon, the
non-existence of the graviton, no need for supersymmetry to explain the dark universe,
and no need for an inflationary phase of the universe.
These predictions run completely counter to those of contemporary physical theory. The
reason for this is Invariant Set Theory does not conform to the conventional reductionist
approach to theory. Instead, Invariant Set Theory is based on a ‘top-down’ postulate that
the universe U is a deterministic dynamical system evolving precisely on a measure-zero
fractal subset IU of U ’s state space. That is to say, it is proposed that the most primitive
expression of the laws of physics describe the global fractal geometry of IU , rather than
differential evolution equations in space-time.
 INVARIANT SET THEORY 27

Any ψ-epistemic theory of quantum physics, is necessarily constrained by no-go theorems

(e.g. [26]), which appear to make it inconsistent with experiment. However, making use of
the homeomorphism between IU and the space of p-adic integers, for large but finite p, it
is shown that these theorems are nullified by using the p-adic metric (rather than the more
intuitive Euclidean metric) in state space to distinguish between physically allowable and
physically inconsistent counterfactual perturbations. Importantly, neither local causality
nor realism are abandoned in Invariant Set Theory. Fractal geometry has many links
with number theory, and elementary number theory plays an important role in Invariant
Theory, in explaining the principal features of quantum theory, including, for example, the
non-commutativity of quantum observables, the violation of the Bell inequalities, and the
relationship between probability and frequency of occurrence in space-time. The problem
of synthesising quantum and gravitational physics becomes one of combining the locally
Euclidean structure space-time with the locally p-adic structure of state space. A way to
do this has been proposed in the paper.
The complex multi-qubit Hilbert Space and the Schrödinger equation (here discussed in
relativistic form) emerges from Invariant Set Theory in the singular limit [2] where (and
only where) the otherwise finite p is set equal to infinity. This notion of a singular limit
should not be considered pathological. Indeed, Berry [2] notes that singular limits are
commonplace in science, and provide insight into how a more general theory (e.g. the
Navier-Stokes theory of viscous fluids) can reduce to a less general theory (e.g. the Eu-
ler theory of inviscid fluids), and therefore how higher-level phenomena can emerge from
lower-level ones. In such situations, the approximate theory can describe reality very ac-
curately for many purposes, but fail catastrophically in other situations. Many aspects of
fluid turbulence in high Reynolds number flows can be described well by the Euler equa-
tions. However, Euler theory fails utterly in describing the phenomenon of heavier-than-air
flight! Here it is suggested that whilst quantum theory is an excellent predictor of labo-
ratory experiments, it may fail catastrophically in situations where the effects of gravity
are nontrivial. Whilst many of these failures may only be strongly apparent on the cosmo-
logical scale, the author looks forward to the execution of laboratory-scale experiments in
coming years which can test the potentially decoherent nature of gravity during quantum
measurement.

Acknowledgement
My thanks to Harvey Brown, Shane Mansfield, Simon Saunders, Kristian Strommen and
David Wallace and anonymous reviewers for helpful comments on an earlier version of this
paper.

References
[1] J.S. Bell. Speakable and unspeakable in quantum mechanics. Cambridge University Press, 1993.
[2] M Berry. Singular limits. Physics Today, 55:10–11, 2002.
[3] L. Blum, F.Cucker, M.Shub, and S.Smale. Complexity and Real Computation. Springer, 1997.
[4] J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt. Proposed experiment to test local hidden
variable theories. Phys. Rev. Lett., 23:880–884, 1969.
28 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

[5] L. Diósi. Models for universal reduction of macroscopic quantum fluctuations. Phys. Rev., A40:1165–74,
1989.
[6] S. Dube. Undecidable problems in fractal geometry. Complex Systems, 7:423–444, 1993.
[7] F.J. Dyson. Science and Ultimate Reality: Quantum Theory, Cosmology and Complexity., chapter
Thought experiments in honour of John Archibald Wheeler. Cambridge University Press, 2004.
[8] F.J. Dyson. Is a graviton detectable? Int. J. Modern Physics A, 28:1330041, 2013.
[9] G.F.R. Ellis. Top-down causation and emergence: some comments on mechanisms. Interface, 2:126–
140, 2011.
[10] R.P. Feynman and A.R.Hibbs. Quantum Mechanics and Path Integrals. McGraw-Hill, 1965.
[11] M.J.W. Hall. Local deterministic model of singlet state correlations based on relaxing measurement
independence. Phys. Rev. Lett., 105:250404, 2011.
[12] M.J.W. Hall. Relaxed bell inequalities and kochen-specker theorems. Phys. Rev. A, 84:022102, 2011.
[13] J. Jahnel. When does the (co)-sine of a rational angle give a rational number? arXiv:1006.2938, 2010.
[14] S. Katok. p-adic Analysis compared with Real. American Mathematical Society, 2007.
[15] S. Kochen and E. Specker. The problem of hidden variables in quantum mechanics. J. Mathematics
and Mechanics, 17:59–87, 1967.
[16] E.N. Lorenz. Deterministic nonperiodic flow. J.Atmos.Sci., 20:130–141, 1963.
[17] T.N. Palmer. The invariant set postulate: a new geometric framework for the foundations of quantum
theory and the role played by gravity. Proc. Roy. Soc., A465:3165–3185, 2009.
[18] T.N. Palmer. Lorenz, Gödel and Penrose: new perspectives on determinism and causality in funda-
mental physics. Contemporary Physics, 55:157–178, 2014.
[19] T.N. Palmer. Bell’s conspiracy, Schrödinger’s black cat and global invariant sets. Phil. Trans. A,
A373:20140246, 2015.
[20] T.N. Palmer. Invariant set theory: Violating measurement independence without fine tuning, conspir-
acy, constraints on free will or retrocausality. EPTCS, 195:285–294, 2015.
[21] R. Penrose. The nonlinear graviton. Gen. Rel. Grav., 7:171–176, 1976.
[22] R. Penrose. The Road to Reality: A Complete Guide to the Laws of the Universe. Jonathan Cape,
London, 2004.
[23] R. Penrose. Cycles of Time: An Extraordinary New View of the Universe. The Bodley Head, 2010.
[24] R. Penrose and W. Rindler. Spinors and space-time. Cambridge University Press, 1992.
[25] I.C. Percival. Quantum spacetime fluctuations and primary state diffusion. Proc. Roy. Soc., A451:503–
513, 1995.
[26] M.F. Pusey, J. Barrett, and T. Rudolph. On the reality of the quantum state. Nature Phys., 8:474–477,
2012.
[27] A. M. Robert. A Course in p-adic Analysis. Springer ISBN 0-387-98660-3, 2000.
[28] P. Schneider. p-Adic Lie Groups. Springer, 2011.
[29] J. Schwinger. Quantum Mechanics: Symbolism of Atomic Measurements. Springer, 2001.
[30] P.J. Steinhardt and N. Turok. Endless Universe: Beyond the Big Bang. Phoenix, 2007.
[31] S.H. Strogatz. Nonlinear dynamics and chaos. Westview Press, 2000.
[32] G. ’t Hooft. Models on the boundary between classical and quantum mechanics. Phil. Trans. A,
A373:20140236, 2015.
[33] G. ’tHooft. The cellular automaton interpretation of quantum mechanics. arXiv:1405.1548, 2015.
[34] D. Wallace. The Emergent Multiverse. Oxford, 2014.
[35] S.G. Williams. Symbolic Dynamics and its Applications. American Mathematical Society, 2004.
[36] I. Zlatev, L. M. Wang, and P.J. Steinhardt. Quintessence, cosmic coincidence and the cosmological
constant. Phys. Rev. Lett., 82:896, 1999.
 INVARIANT SET THEORY 29

Appendix A. Multiple Qubits

A general m qubit state can be built from a general m − 1 qubit state using the following
inductive formula:
|ψa, b . . . d (θ1 , . . . θ2m −1 ; φ1 , . . . φ2m −1 )i =
| {z }
m
θ1
cos 2 |ai × |ψb, c . . . d (θ2 , . . . θ2m−1 ; φ2 . . . φ2m−1 )
| {z }
m−1
θ1 iφ1
(50) + sin 2e |ai

× |ψb, c . . . d (θ2m−1 +1 . . . θ2m −1 ; φ2m−1 +1 . . . φ2m −1 )i
| {z }
m−1

The correspondence with bit strings is similarly defined inductively. Let

|ψb,c...d (θ2 , . . . θ2m−1 ; φ2 . . . φ2m−1 )i
(51)
Sb (θ2 , φ2 ) = {b01 , b02 , b03 , . . . b02N }




S (θ , φ ) = {c0 , c0 , c0 , . . . c0 N }

c 3 3 1 2 3 2
(52) ←[

. . .

S (d; θ m−1 , φ m−1 ) = {d0 , d0 , d0 , . . . d0 }

d 2 2 1 2 3 2N

and
|ψb,c...d (θ2m−1 +1 , . . . θ2m −1 ; φ2m−1 +1 . . . φ2m −1 )i
(53)
Sb (θ2m−1 +1 , φ2m−1 +1 ) = {b001 , b002 , b003 , . . . b02N }




S (θ m−1 , φ m−1 ) = {c00 , c00 , c00 , . . . c00N }

c 2 +2 2 +2 1 2 3 2
(54) ←[
. . .


S (θ m , φ m ) = {d00 , d00 , d00 , . . . d00 }

d 2 −1 2 −1 1 2 3 2N

define a pair of m − 1 qubit correspondences (where b0i , b00i ∈ {b,
b} etc), then with
θ1 θ1
(55) cos |ai + sin eiφ1 |ai

←[ {a1 , a2 , a3 . . . a2N },
2 2
an independent bit string with ai ∈ {a, a},
the m qubit correspondence can be written as


{a1 , a2 , a3 , . . . a2N }

{b1 , b2 , b3 , . . . b2N }



(56) |ψa,b...d (θ1 , . . . θ2m −1 ; φ1 . . . φ2m −1 )i ←[ {c1 , c2 , c3 , . . . c2N }




...

{d1 , d2 , d3 , . . . d2N }

30 T.N.PALMER DEPARTMENT OF PHYSICS, UNIVERSITY OF OXFORD, UK

where
bi = b0i ci = c0i . . . di = d0i if ai = a
(57) bi = b00i ci = c00i . . . di = d00i if ai = a

and bi ∈ {b,
b}, ci ∈ {c,
c} etc. This reduces to (25) when m = 2.
E-mail address: tim.palmer@physics.ox.ac.uk