April 2024
The In omplete Foundation of Quantum Me hani s
Antonio Leon
Retired Professor. Independent resear her in the foundations of s ien e.
Links to other author's works.
Abstra t
.-This paper dis usses the in ompatibility between the formal logi
the ordinary logi
that underlies the mathemati s
s enario in whi h this in ompatibility
Keywords
derived from quantum phenomenology and
onstituting the formal language of quantum me hani s. A new dis rete
ould be resolved is also proposed.
: EPR paradox, GHZ states, laws of ordinary logi , dis rete magnitudes, dis rete spa e, dis rete time,
ellular
automata.
1 From EPR to GHZ
will analyze the reasons why this might not be the ase.
As is well known, and a ording to the Copenhagen
interpretation (CI), quantum me hani s assumes the
oexisten e (superposition, merge) of mutually ex lusive
states in quantum systems until a measurement is
performed on them. It is also well known that in
1935 a famous arti le was published that pointed to
the non- ompleteness of this interpretation of quantum
me hani s [6, p. 780℄:
But rst it is worth re alling the haoti use of
(ordinary) language that hara terizes ontemporary
physi s, a problem already denoun ed in 2015 by T.
Maudlin [16, p. xiv℄:
Unfortunately, physi s has be ome infe ted
with very low standards of larity and pre ision
on foundational questions, and physi ists have
be ome a ustomed (and even en ouraged) to
just shut up and al ulate, to ons iously
refrain from asking for a lear understanding
of the ontologi al import of their theories.
Previously we proved that either (1) the
quantum-me hani al des ription of reality
given by the wave fun tion is not omplete
or (2) when the operators orresponding
to two physi al quantities do not ommute
the two quantities annot have simultaneous
reality. Starting then with the assumption
that the wave fun tion does give a omplete
des ription of the physi al reality, we arrived
at the on lusion that two physi al quantities,
with non- ommuting operators, an have
simultaneous reality. Thus the negation of
(1) leads to the negation of the only other
alternative (2). We are thus for ed to on lude
that the quantum-me hani al des ription of
physi al reality given by wave fun tions is not
omplete.
Indeed, it is normal in today's physi s to nd
ontradi tory uses of language. For example, the word
something meaning the same as the word nothing;
or the word empty the same as the word full. In the
EPR-GHZ ase we are dealing with, it is very ommon
to nd things su h as [1, p. 5℄:
For example, on a subatomi s ale, individual
parti les
ommuni ate with ea h other
instantaneously despite being far apart, whi h
de es ommon sense.
Whi h obviously does not o ur, sin e the subatomi
parti les are not informed systems [11℄, they do not
elaborate, nor transmit, nor re eive, nor interpret
information. What orresponds to say is that, a ording
to CI, there seem to be instantaneous in uen es between
the entangled parti les; or a ausal ee ts, not lo ally
deterministi , between those parti les.
Some years later, in 1964, another famous paper was
published [4℄ in whi h its author J. Bell ( loser to
Einstein than to Bohr) proved the famous theorem that
bears his name, and whi h makes it possible to test in
experimental terms whether the inequalities in luded in
the theorem are violated or not. The experiments took
some time to arrive, but they arrived in 1982 [3, 2℄,
1998 [20℄, 2001 [9℄ and in all ases the violation of Bell's
inequalities was on rmed, whi h gave the reason to
Bohr's CI to the detriment of the previous argument of
Einstein, Podolsky and Rosen (EPR).
It has also been said that ordinary logi is not the logi
of quantum me hani s (e.g. [7, p. 60℄). In this onne tion
it is worth re alling E. S hrodinger's (who was also not
a supporter of CI) and his famous at [18, 5℄. We an
indeed a rm [14℄:
P: There is a living being inside S hrodinger's box.
Q: No living being an be at the same time alive and not
In 1990 a remarkable simpli ation of Bell's argument was published that avoided its inequalities, a
simpli ation also testable in experimental terms but
avoiding Bell's statisti al umbersomeness [8℄: the GHZ
states (by Greinberger, Horne and Zeilinger). Also in
this ase the orresponding experiments ended up being
arried out [5℄, whi h, again, gave the reason to Bohr's
CI. Sin e these last experiments, most authors onsider
the histori al Bohr-Einstein dispute to be over. Here we
alive.
Consequently:
CI ∧ P
Q
=⇒ ¬
whi h implies the violation of the Se ond Law of ordinary
logi .
1
2
The In omplete Foundation of Quantum Me hani s
It might then be true that ordinary logi is not
the logi of quantum phenomena. But then a new
problem arises be ause the two fundamental laws of
ordinary logi (the Prin iple of Identity and the Law
of Non-Contradi tion) are in fa t fundamental laws
of the mathemati s with whi h quantum me hani s is
onstru ted. Consequently, CI would imply the use of a
logi in its mathemati al language that is not the logi
of the phenomena it des ribes. One ould then speak of
an in omplete foundation of quantum me hani s.
Before proposing the new s enario in whi h this
in omplete foundation of quantum me hani s ould be
resolved, it will be ne essary to deal, albeit very brie y,
with some formal issues related, above all, to the nature
of mathemati al in nity and to the ontologi al nature of
physi al spa e.
2 An in nity never questioned
Throughout the entennial history of quantum
me hani s, there have been, and still are, intense debates
about various aspe ts of the theory, but as far as I know,
there has never been a dis ussion about what should
have been the rst point to be dis ussed: the hypothesis
of the a tual in nity subsumed in the Axiom of In nity
that underlies the mathemati s of quantum me hani s
itself. Having been one of the most dis ussed hypotheses
in the history of human thought, it is very strange that
it has not been dis ussed for more than a entury, sin e
it was axiomati ally assumed by modern set theories.
These set theories have nally provided instruments
that allow us to analyze the in onsisten y of the a tual
in nite, as is the ase of the ω order of the natural
numbers, or the dense order of the rationals and the
reals. The spa etime ontinuum is one of those in nite
sets whose elements are densely ordered and whi h allow
us to onstru t proofs of their own in onsisten y, su h as
the one in luded in this se tion. An in onsisten y that
physi ists should be interested in, given the omnipresen e
of the spa etime ontinuum in a good part of their
theories, in luding quantum me hani s. But this is not
the ase, nor an I think of what to do to make it so. At
least I invite the reader to onsider the following proof,
whi h will take only a ouple of minutes:
(The next theorem is a very abbreviated version of the
argument [12, p. 59-63 Link℄, where the reader an nd
another 40 dierent proofs.)
Theorem 1 The Axiom of In nity is in onsistent.
Proof.-The interval of rational numbers Q01 = (0, 1) is
denumerable and densely ordered. So, it an be put in
one-to-one orresponden e f with the set N of natural
numbers in their natural order of pre eden e; and Q01 an
be rewritten as the set {f (1), f (2), f (3), . . . }. Let now x
be a rational variable initially de ned as f (1); and let
(the urrent value of) x be ompared with the su essive
elements f (1), f (2), f (3). . . so that x is rede ned as f (i)
if, and only if, f (i) is less than the urrent value of
x. Sin e, a ording to the Axiom of In nity, all elements
f (1), f (2), f (3), . . . of Q01 are rational numbers whi h
exist as a omplete totality, x an be su essively
1 Though
ompared with all of them:
∀n ∈ N : x is ompared with f (n), and
rede ned as f (n) i f (n) < x
(1)
On e ompared with all elements1 of Q01 , the urrent
value of x is the smallest rational of that set. Indeed, if
on e ompared with all elements of Q01 , the urrent value
of x were not the least rational of Q01 , there would exist
at least one element f (n) in Q01 su h that f (n) < x.
But this is impossible a ording to (1). Therefore, it
was ompared with f (n) and rede ned as f (n). So, it
is impossible that f (n) < x. But it is also immediate
to prove that: On e ompared with all elements of Q01 ,
the urrent value of x is not the smallest rational of that
set. In ee t, on e ompared with all elements of Q01 ,
and whatsoever be the urrent value of x, ea h element
of the in nite set {x/2, x/3, x/4 . . . } is an element of
Q01 less than x. This ontradi tion proves the Axiom of
In nity legitimizing the existen e of Q01 as an a tual (not
potential) in nite totality is in onsistent. Or in other
words: a omplete and ordered list, su h as the rational
interval (0, 1), without a rst element that starts the list
is in onsistent.
3 The reality of physi al spa e
In 2007, and therefore 8 years before the rst empiri al
dete tion of gravitational waves, R. Laughlin wrote [10,
p. 158℄:
Subsequent studies, arried out with large
parti le a elerators, helped to understand that
spa e is more like the glass of a window than
Newton's ideal va uum: it is full of "stu"that
is normally transparent but an be made
visible if stru k with su ient for e to dislodge
a part of it. In ontemporary physi s, the
va uum of spa e is understood as a relative
ether, a on eption that is on rmed every day
experimentally but whi h does not re eive that
name be ause it is taboo.
From 2015 to date, there have been about 100 episodes
of empiri al dete tion of gravitational waves in the global
network of interferometers (LIGO, Virgo, GEO 600,
TAMA 300, KAGRA). Of ourse, about these dete tions
it is possible to say [15, p. 2℄:
Of ourse, gravitational waves are real
vibrations that produ e real ee ts in real
instruments made of real ordinary matter.
Consequently, the vibrating medium, spa e,
must also be real. What does not exist an
neither vibrate nor physi ally intera t with
ordinary matter. In my opinion, this reality
of physi al spa e is the most important
onsequen e of the empiri al dete tion of
gravitational waves, although, as far as I know,
it has not even been onsidered. [...℄ And like
any real physi al entity, spa e will have its own
substantiality, whi h, for simpli ity, I will all
spa e matter.
it is not ne essary, this is formally proved by indu tion in [12℄, and
an also be proved by Modus Tollens and by supertask
theory.
T G
he
eneral
S
ien e
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ournal
April 2024
3
The In omplete Foundation of Quantum Me hani s
It is surprising then that in spite of the overwhelming
eviden e of its physi al reality, the majority of physi ists
still think that physi al spa e is not real, that it is
only a tion useful to express ertain relations between
material physi al obje ts. A tion that, on the other
hand, must be elasti , more rigid than metals, extensible,
deformable, vibrating and propagating its own vibrations
and the orresponding energies.
The s enario in whi h the in omplete foundation of
quantum me hani s introdu ed above ould be solved,
in ludes the reality of physi al spa e, the existen e of
a sort of spa e matter. A spa e matter that, a ording
to everything we know about the physi al world, must
be dierent from ordinary matter (baryoni old matter)
and transparent to this ordinary matter (Prin iple of
Inertia), whi h implies that both matters an oexist and
ease to oexist in the same pla es.
4 Dis rete magnitudes
Energy (of any kind), mass, ele tri harge and olor
harge are dis rete magnitudes with indivisible minimum
values (quanta). Consequently, any mathemati al
fun tion whose output is one of those dis rete magnitudes
must give a result ompatible with the dis reteness
of the orresponding dis rete magnitude: the result
(the output) of that fun tion an only be an integer
multiple of the quantum (indivisible minimum) of that
dis rete magnitude. As far as I know, physi ists have not
paid attention to this requirement of dis reteness, and
invariably they use ontinuous fun tions of ontinuous
output when that output should be dis rete, be ause it
must be the value of a dis rete magnitude.
Naturally, ontinuous variables annot enter a
dis rete output fun tion (the output would always be
ontinuous). However, it happens that in many fun tions
whose output is the value of a dis rete magnitude, for
example energy, ontinuous variables su h as spa e, or
time, or both (velo ity) intervene. In these ases it is not
possible to obtain the dis rete outputs that should be
obtained. It is only possible to obtain dis rete outputs
of a mathemati al fun tion if all the variables of that
fun tion are dis rete variables (see the last paragraph of
this se tion). Therefore, any onsistent physi al theory
must take this restri tion into a ount.
An immediate onsequen e of this restri tion is that
both spa e and time must be dis rete magnitudes, with
indivisible minima (quanta), whi h ould be alled qseats
for spa e and qbeats for time. In this model, whi h
is dis rete and onsistent with the dis rete magnitudes
(energy, mass, harge, ele tri harge, et .), the following
requirements (among others) must be veri ed [13, p. 2℄:
1. Nothing an be smaller than a quantum of spa e
(qseat).
2. Nothing an last less than a quantum of time (qbeat).
3. There is a maximum speed of a qseat per qbeat.
4. Any physi al obje t maintains its state for at least
one qbeat.
5. All physi al pro esses, in luding motions, must be
dis rete, not ontinuous, whi h is how we per eive
The General S ien e Journal
them.
The strange thing about quantum me hani s is that it
is a physi al theory about an essentially dis ontinuous
physi al world, but developed with a ontinuous, nondis rete, mathemati al language. Whi h, for the reasons
just given, is not formally a eptable: the dis rete
outputs of mathemati al fun tions annot be ontinuous.
Consequently, those dis rete output fun tions annot
be fed with ontinuous variables, whi h is the ase
in ontemporary quantum me hani s. An elementary
example of this ( ontinuous-dis rete) in ompatibility
never questioned by modern physi s is the elasti
potential energy E (for example of a spring), given by:
E=
1
k x2
2
(2)
where k is a onstant (a real number) and x a real
variable representing a spatial distan e. It is lear that
the output of 1/2kx2 an only be dis rete if the values
of the variable x are also dis rete, with a minimum
indivisible unit (quantum), somehow related to the
quantum of E so that the expression (2) is fun tional.
It seems that a new dis rete mathemati al physi s will
have to be onstru ted.
As far as I know, this in ompatibility between
ontinuous mathemati al fun tions and the ne essary
dis reteness of their outputs has never been onsidered.
Neither has the problem of the arithmeti ompatibility
of the dierent quanta of the dierent dis rete
magnitudes, a ompatibility ne essary for all the outputs
of all the fun tions that relate them to be dis rete. A
new and ex iting quantum problem to be solved. A very
simple, and also very spe ulative, solution is that all
quanta of all physi al magnitudes are integer multiples
of a universal minimum magnitude.
5 A quantum lo al universe
It does not seem reasonable to assume one logi for
quantum phenomenology, and a dierent logi for the
mathemati al language with whi h this phenomenology
is expressed. But that is exa tly what happens with
some interpretations of quantum me hani s, as is the
ase with the most ommonly a epted of them all: the
Copenhagen interpretation (CI). If lassi al logi is not
valid for CI, it should not be valid for its mathemati al
language either. There should be the same fundamental
logi for both. Until this situation is resolved, we will have
to admit an in orre t foundation of quantum me hani s.
About mathemati s, it is worth remembering that it is
in onsistent due to the hypothesis of the a tual in nity
that axiomati ally founds it. Therefore, all bran hes of
mathemati s should be nitist. With respe t to the CItype interpretations of quantum me hani s, a question
seems inevitable: an a s ien e be in ompatible with
the Se ond Law of logi ? A ording to that law, an
atom annot be at the same time disintegrated and not
disintegrated, nor an it be su essively disintegrated and
not disintegrated, be ause on e it disintegrates it annot
disintegrate again. It is important to remember that the
violation of the Se ond Law of logi makes it possible
to prove any proposition, so that the orresponding
April 2024
Referen es
4
onsistent
emergent from quantum me hani al events. [17,
p. 97℄.
Perhaps the only legitimate onsideration is the
onsideration of the whole universe as a single obje t
evolving dire tionally in the sense of its isotropization,
i.e. in the sense of its entropy in reasing. A dire tional
evolution that has been produ ing the same obje ts
(galaxies, stars, et .) for billions of years, and has
been doing so without observers ollapsing the state
superpositions of the quantum obje ts involved. Perhaps
experimental quantum strangeness only o urs in the
realm of laboratory and thought experiments deta hed
from the universe as a whole. The superposition of
the living and non-living state of a living being is
in ompatible with known organi nature. And if that
superposition is linked to the disintegrated atom/nondisintegrated atom superposition, the latter would have
to ollapse with the presen e of the at and the rest of
the setup of S hr
odinger's famous thought experiment,
otherwise the Se ond Law of logi would be violated.
From the perspe tive of emergentism, physi al
laws are rules of olle tive behaviour that follow
from more primitive behaviour rules. [10, p. 111℄
violating theory would be anything but a
s ienti theory.
A new perspe tive for quantum phenomena ould be
oered by ellular automata (CAs), of whi h there is
already at least one model developed along these lines
[19℄, albeit with the same in onsistent mathemati s of
the a tual in nity. As in CAs, quantum obje ts ould
somehow be present, and in periodi ally varying ways,
in all qseats in the universe. Their presen e in qseats
ould add and subtra t giving rise to onstru tive and
destru tive interferen es. A new perspe tive to explain
wave- orpus le duality.
On the other hand, every observation is an intera tion
that has to be in luded in the explanation, and that an
ause hanges in the observed obje ts ( ollapse of the
wave fun tion). The wave presen e of quantum obje ts
would ollapse with observational intera tions. Moreover,
the entangled obje ts ould be onsidered as a single
obje t, so they would vary simultaneously. This ould be
a new perspe tive to explain the instantaneous in uen e
at a distan e and the quantum indeterminism linked to
entanglement.
The re ursive fun tioning of CAs and their evolution
in unison, as a whole, ould also be the ause of the
emergen e of nonlinear dynami s, of new obje ts, and of
new laws governing their evolution. Thus, the fun tioning
of a CA ould serve to explain ertain behaviors of the
quantum world and of the universe that so far seem
inexpli able to us. We would also have a CA version
of Bhor's Prin iple of Complementarity: the behavior of
individual qseats versus that of the obje ts of the CA
de ned by those qseats. And in addition to obje ts, the
ontents of the qseats would also de ne the dierent CA
for e elds.
Despite the CA proposal for quantum me hani s ited
above, there is still mu h work to be done in this regard.
One should even onsider the role of omputational
languages in CA models. Problems may arise when
extrapolating observations and on lusions about the
world of ma ros opi physi al obje ts to the world of
subatomi ontents of CA qseats. The former emerge
from the latter. We would have two levels of laws: those
of the CA and those of the emergent obje ts:
... and the stru ture of the everyday obje ts is
The General S ien e Journal
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