#  Equivalence Relations in Quantum Theory:
An Objective Account of Bases and Factorizations111This paper is dedicated to
the memory of Newton da Costa (1929-2024)..

Christian de Ronde1,2,3, Raimundo Fernández Mouján2,4, César Massri5,6

###### Abstract

In orthodox Standard Quantum Mechanics (SQM) bases and factorizations are
considered to define quantum states and entanglement in relativistic terms.
While the choice of a basis (interpreted as a measurement context) defines a
state incompatible to that same state in a different basis, the choice of a
factorization (interpreted as the separability of systems into sub-systems)
determines wether the same state is entangled or non-entangled. Of course,
this perspectival relativism with respect to reference frames and
factorizations precludes not only the widespread reference to quantum
particles but more generally the possibility of any rational objective account
of a state of affairs in general. In turn, this impossibility ends up
justifying the instrumentalist (anti-realist) approach that contemporary
quantum physics has followed since the establishment of SQM during the 1930s.
In contraposition, in this work, taking as a standpoint the logos categorical
approach to QM —basically, Heisenberg’s matrix formulation without Dirac’s
projection postulate— we provide an invariant account of bases and
factorizations which allows us to to build a conceptual-operational bridge
between the mathematical formalism and quantum phenomena. In this context we
are able to address the set of equivalence relations which allows us to
determine what is actually the same in different bases and factorizations.

1\. Philosophy Institute Dr. A. Korn, University of Buenos Aires - CONICET
2\. Center Leo Apostel for Interdisciplinary Studies
Foundations of the Exact Sciences - Vrije Universiteit Brussel
3\. Institute of Engineering - National University Arturo Jauretche
4\. Philosophy Institute, Diego Portales University, Santiago de Chile
5\. Institute of Mathematical Investigations Luis A. Santaló, University of
Buenos Aires - CONICET
6\. CAECE University

Keywords: bases, factorizations, invariance, quantum mechanics, realism.

## Introduction

Since its formulation in the 1930s physicists have become accustomed to the
idea that the reference to a state in Standard Quantum Mechanics (SQM)
requires, to be meaningful, the specification of a particular experimental
context or “preferred” basis (given by a complete set of commuting
observables). This widely accepted perspectival relativization of the notion
of state of a physical system has immense consequences, for it precludes the
possibility in quantum theory to refer to a state of affairs as independent of
the particular choice of a reference frame (a condition that one would believe
is necessary for any objective physical representation). In contradistinction,
factorizations in SQM have been misleadingly conceived in atomistic
substantivalist terms, as informing the separation of quantum systems into
subsystems —disregarding the fact that this entails an operation provenly
inconsistent with the mathematical formalism itself. Let us add that this
understanding of factorizations linked to the notion of entanglement has
resulted also in a problematic relativistic view according to which the choice
of a factorization determines wether a state is entangled or not (see for a
detailed discussion [22]).

But are these sacrifices necessary? Is that renunciation to the conditions of
objectivity in physics in fact unavoidable? We would like to show that both
the account of bases and of factorizations mentioned above have been generated
due to the projection of presupposed classical concepts that have wrongly
determined, right from the start, the understanding of the formalism and of
quantum phenomena. However, by recognizing and escaping these inadequate
projections, it is possible to restore an account of QM that is operationally
invariant, that allows to account for a state of affairs independent of
reference frames, and that opens the door for the development of concepts
specifically adequate to this mathematical formalism [18, 19]. This conceptual
account must be grounded on the _intensive_ nature of the elements pointed
(already in Heisenberg’s matrix mechanics) as invariant by the formalism
itself. In this work we attempt, from a formal and conceptual reframing of the
theory, to provide the operational conditions that, bridging the gap between
the mathematical formalism and the conceptual representation of experience,
will allow us to understand what is actually observed according to the theory
of quanta. In short, the main attempt of this article is to provide a
consistent and coherent formal, conceptual and operational redefinition of the
notions of basis and factorization in invariant and objective terms.

##  1 The Relativist Account of Bases

A certain relativism seems to be implied in almost all accounts of QM. A
relativism we believe was introduced in QM by no other than Niels Bohr. This
was done through the redefinition of the notion of measurement within the
theory of quanta in terms of the interaction between quantum and classical
systems. A new interaction —that implied the experimental choice of the
observer— which was made a fundamental aspect of the theory itself. Of course,
as it is well known, measurement had never been part of any physical theory
before Bohr. Measurement in physics had always been understood as part of an
epistemic praxis which presupposed a human subject capable to produce a link
between the knowledge of an already “closed” theory and the highly complex
technical experience produced in the lab. As Einstein would tell Heisenberg
[29]: “it is only the theory which decides what can be observed.” The moment
of unity of a theory, that which the theory talks about, is —for the realist—
something to be defined through invariance and objectivity within the theory
itself. It is in this way that classical mechanics generates the formal-
conceptual notion of ‘particle’ and electromagnetism is able to come up with
the notion of ‘electromagnetic wave’. In short, we do not observe a unity
given to us in experience —we do not observe an electromagnetic wave— but a
multiplicity of phenomena —a lightning or two rocks attracting each other—
that need to be united through mathematical formalisms and concepts. As
Heisenberg would explain:

> “The history of physics is not only a sequence of experimental discoveries
> and observations, followed by their mathematical description; it is also a
> history of concepts. For an understanding of the phenomena the first
> condition is the introduction of adequate concepts. Only with the help of
> correct concepts can we really know what has been observed.” [30, p. 264]

But, instead of introducing the concepts specifically adequate to the quantum
formalism —and capable of producing an understanding of the observations
coherent with that formalism— Bohr would impose not only a reference to
‘waves’ and ‘particles’ that defined a necessary restriction to classical
concepts [32], but also —since quantum phenomena escaped a consistent
representation in terms of particles or waves— a contextual dependency of this
reference to the specific experimental setup. According to Bohr, that
reference —as repeatedly addressed within the double slit experiment— was
explicitly dependent on the choice of the specific measurement situation (see
for a detailed analysis [14]). Contrary to all pre-Bohrian physics, the nature
of the state of affairs would then come to depend on the choice of the
experimental context, and consequently, each different measurement would be
understood as referring to a different (context-dependent) state of affairs.
The complementary nature of quantum entities would then preclude a common
consistent reference to the same state of affairs. In short, Bohr’s new
understanding of measurement would define for each measurement context a
different “classical” state of affairs, establishing a relativist view of
quantum phenomena. And it is exactly this same relativist operation that would
be reproduced by Dirac in his redefinition of the notion of (quantum) state in
terms of a preferred reference frame (or basis). In his famous book [24], he
would extend complementarity from Bohr’s conceptual dualistic reference to
‘waves’ and ‘particles’ to the formal level of vectors and bases.222It is
important to notice that the notion of ‘observable’ was introduced by Dirac to
refer to linear operators (in the first edition from 1930) and to real
functions of dynamical variables (in the second edition from 1935). In this
way, the English engineer and mathematician would drastically subvert the
meaning of state of a system by imposing a direct link between vectors in
specific bases —namely, those in which a vector is written as a single ket,
$|x\rangle$— and actual observations of single ‘clicks’ in detectors. It is
this re-interpretation of the notion of state in terms of a measurement
outcome, mathematically represented by a ket vector, which would impose the
—today widely accepted— idea that the same system could be now represented in
terms of different states depending on the specific basis (i.e., the reference
frame).

> “[E]ach state of a dynamical system at a particular time corresponds to a
> ket vector, the correspondence being such that if a state results from the
> superposition of certain other states, its corresponding ket vector is
> expressible linearity in terms of the corresponding ket vectors of the other
> states, and conversely. Thus the state $R$ results from a superposition of
> the states $A$ and $B$ when the corresponding ket vectors are connected by
> $|R\rangle=c_{1}|A\rangle+c_{2}|B\rangle$.” [24, p. 16]

This is an essentially inconsistent account of the notion of state which not
only precludes its basic meaning as referring to something independent of any
basis, but also mixes the basis dependent definition of state with the
abstract reference to vectors. The state $|R\rangle$ is different to the state
$|A\rangle$ and different to the state $|B\rangle$, which means that when
considering different bases or experimental situations we obtain different
states, different outcomes. However, the state $|R\rangle$ is also considered
as being the same as the sum of the states $|A\rangle$ and $|B\rangle$. Thus,
through the identification of states with singular outcomes (taken to be
expression of specific particles), each of the states is completely
‘different’ from the others, and yet they are also ‘the same’. How is this
paradox created? As discussed in detail in [20], the contradiction is built
trough the implicit reference to two different (inconsistent) definitions of
the notion of state. The first basis-dependent definition is in terms of a
single term ket which, in turn, is also linked to the observation of a single
outcome —i.e., each state $|k\rangle$ generates a specific measurement outcome
‘k’.

######  Definition 1.1 (Operational Purity).

Given a quantum system in the state $|\psi\rangle$, there exists an
experimental situation linked to that basis (in which the vector is written as
a single term) in which the test of it will yield with certainty (probability
= 1) its related outcome.

In this case each different ket (e.g., $|R\rangle$, $|A\rangle$ or
$|B\rangle$) is interpreted as a different state. However, there is also an
implicit reference to a completely different notion of state in terms of the
‘abstract vector’ which grounds these different basis dependent
representations. But notice that this later definition is independent of
bases. In this case, $|R\rangle$ and $c_{1}|A\rangle+c_{2}|B\rangle$ are just
particular basis-dependent representations of the same abstract vector $\Psi$.

######  Definition 1.2 (Abstract Purity).

An abstract unit vector (with no reference to any basis) in Hilbert space,
$\Psi$, is a pure state. In terms of density operators $\rho$ is a pure state
if it is a projector, namely, if Tr$(\rho^{2})=1$ or $\rho=\rho^{2}$.

As demonstrated explicitly in [20], while the first definition is operational
but non-invariant, the latter mathematically abstract definition is invariant
but has no operational content. Furthermore, they are not equivalent, while
the first definition implies the second the inverse does not hold (for a more
detailed analysis see [21]).

It is interesting to notice that it is this self-contradictory relativist
account of state proposed by Dirac which would be applied by Bohr a few years
later in his famous reply to EPR. As Bohr [4, p. 696] would argue, the
condition to consider a subset of observables as definite valued required the
specification of the reference frame (i.e., the measurement context or basis):
“it is a well-known feature of the present formalism of quantum mechanics that
it is never possible, in the description of the state of a mechanical system,
to attach definite values to both of two canonically conjugate variables”. In
this way, complementarity would become extended in terms of the contextual
reference to states and bases.

> “the renunciation in each experimental arrangement of the one or the other
> of two aspects of the description of physical phenomena, —the combination of
> which characterizes the method of classical physics (…)— depends essentially
> on the impossibility, in the field of quantum theory, of accurately
> controlling the reaction of the object on the measuring instruments, i.e.,
> the transfer of momentum in case of position measurements, and the
> displacement in case of momentum measurements.” [4, p. 699]

Ever since, it has been dogmatically accepted that the reference to a state in
QM requires the specification of a basis. Different bases can be compared but
only in a complementary (i.e., inconsistent) fashion. Thus, quantum properties
(i.e., projection operators) cannot be assigned a global valuation and the
description of a state of affairs becomes intrinsically relative to the single
(preferred) context (or basis) in which the measurement is actually performed.
This has led many to argue that quantum states must be understood as
essentially relative [38] or perspectival [23] notions (see for a detailed
discussion and analysis [15]). As a consequence, it is commonly argued that
“the properties of a system are different whether you look at them or not”
[8].

##  2 The Substantialist Account of Factorizations

The orthodox account of QM established during the 1930s is grounded on Bohr’s
atomistic interpretation of the theory according to which there is a
microscopic realm composed of elementary particles which, even though are
actually responsible for the very existence of ‘tables’ and ‘chairs’ cannot be
represented because of the “uncontrollable disturbance” between quantum and
classical systems —due to the quantum of action— within the measurement
interaction. Dirac’s work would reinforce this substantialist reference to
particles within his intrinsically instrumentalist development of the theory.
In fact, the unjustified reference to particles would be used by Dirac in
order to justify the shift from the observation of intensive patterns —which
Heisenberg’s had considered within his matrix QM— to the unilateral attention
to single ‘clicks’ in detectors regarded —following Bohr’s “commonsensical”
reasoning— as the obvious consequence of (presupposed) “microscopic
particles”. This presupposition also resulted in Dirac’s ad hoc introduction
of the “collapse” process, imposed in order to bridge the gap between the
intensive values present within quantum superpositions and the observation of
the single measurement outcome —that had to be taken as the main experience to
explain, since it expressed the presence of that already presupposed
‘particle’.

> “When we make the photon meet a tourmaline crystal, we are subjecting it to
> an observation. We are observing whether it is polarized parallel or
> perpendicular to the optic axis. The effect of making this observation is to
> force the photon entirely into the state of parallel or entirely into the
> state of perpendicular polarization. It has to make a sudden jump from being
> partly in each of these two states to being entirely in one or other of
> them.” [24, p. 7]

It is at this point, that we need to recognize that the unjustified reference
to particles was never “just a way of talking” [16] but —on the very contrary—
implied from the beginning a series of essential methodological operations
that would even determine the formalism of the theory itself. As Faraday
explained long ago: “the word _atom_ , which can never be used without
involving much that is purely hypothetical, is often intended to be used to
express a simple fact; but good as the intention is, I have not yet found a
mind that did habitually separate it from its accompanying temptations” [35,
p. 220]. Schrödinger rephrases this idea for the quantum case: “We have taken
over from previous theory the idea of a particle and all the technical
language concerning it. This idea is inadequate. It constantly drives our mind
to ask information which has obviously no significance” [39, p. 188]. It is
not difficult to understand that if one dogmatically applies a series of
categorical principles —such as those of particle metaphysics, e.g.,
separability, individuality, locality, etc.— to a mathematical formalism that
was never meant to be understood under the constraints of such representation,
the result of this methodology can only lead to paradoxes and dead ends. An
excellent example of how pseudo-problems can be built by following this
methodology —where dogma is imposed over reason and experience— is the
substantialist account of factorizations, that in SQM are interpreted as the
“separability of systems into sub-systems”, an interpretation put forward by
the Russian mathematician and physicist Lev Landau during the late 1920s.
Landau’s account of factorizations imposed the dogmatic application of the
principle of separability derived from the metaphysics of particles inherent
to classical physics. In his work from 1927 the projection of subspaces was
incorrectly interpreted in terms of the “separation of a system into sub-
systems”. Importing the classical way a reasoning about classical systems to
QM, Landau would write in [34] about “coupled systems” and the necessity of a
“probability ensemble”. According to Landau [34, p. 8] : “A system cannot be
uniquely defined in wave mechanics; we always have a probability ensemble
(statistical treatment). If the system is coupled with another, there is a
double uncertainty in its behaviour.” He pointed out that if a system is
coupled with another, then the uncertainty grows. If the first system can be
described by quantities $\\{a_{i}\\}$,

$|x\rangle=\sum_{i}a_{i}|x_{i}\rangle$

and the second system by $\\{b_{j}\\}$,

$|x^{\prime}\rangle=\sum_{j}b_{j}|x^{\prime}_{j}\rangle,$

then the two systems together are described by $\\{c_{ij}=a_{i}b_{j}\\}$,

$|xx^{\prime}\rangle=\sum_{ij}a_{i}b_{j}|x_{i}x_{j}^{\prime}\rangle.$

If the two systems are coupled, the coefficients $c_{ij}$ depend on time and
can no longer be resolved as a product. Now, let us see why this
interpretation is completely meaningless within the formalism of QM.

The notion of separability is grounded on the modern metaphysical
representation provided by classical physics according to which physical
reality is composed of independent separated individual entities which exist
within space and time. According to this supposedly “commonsensical” picture,
a system can be understood in terms of its parts and the knowledge of these
parts implies the knowledge of the whole individual. This is of course a
direct consequence of the underlying Boolean logic that is a prerequisite to
define the notion of actual entity in terms of the principles of existence,
non-contradiction and identity —also grounding classical mechanics. Indeed, as
a consequence, the propositions derived from classical mechanics can be
arranged in a Boolean lattice (see for a detailed discussion [17]). According
to classical logic, and following set theory, the sum or union of the elements
of a system imply its complete characterization as a whole. It is easy to
picture all of this in terms of sets and the logical relations that we learned
in primary school:

Figure 1: Union, intersection and complement in Boolean logic.

However, as it is also well known since the famous paper by Birkhoof and von
Neumann [3], the underlying logic of QM is not Boolean, it is not
distributive. Thus, the basic classical way of reasoning about systems becomes
precluded right from the start. This is an obvious consequence of the fact
that vectorial spaces do not relate between each other following the same
rules as the elements of a set through union and conjunction. In the quantum
case the equivalent to the union of two vectors is not the sum of the
individual vectors considered as lines, but instead what they are capable to
generate in terms of subspaces. Thus, in the particular case when we consider
the sum of two vectors what we obtain is the whole plane.

|  |
---|---|---
$\langle v_{1}\rangle$ | $\langle v_{2}\rangle$ | $\langle v_{1}\rangle\oplus\langle v_{2}\rangle$

As it could have been easily foreseen, the artificial ad hoc introduction of a
set of logical relations completely alien and even incompatible with the
mathematical formalism of the theory could only lead to confusions,
contradictions and pseudo-problems. Sadly enough, this is exactly what
happened with the introduction of the notion of separability in the context of
QM. As we just explained, the _union_ of two vectors was inadequately
understood as a _sum_ when, in fact, it is a _generation_. Analogously, the
_projection_ of a subspace was incorrectly interpreted as a separation of the
whole set and the choice of a subset of elements, when, as a matter of fact,
its correct interpretation is that of shadow [22]. In figure 2 we can see that
while the shadow of $|\Psi\rangle$ in the $x$-axis is the vector $|x\rangle$,
the shadow of $|\Psi\rangle$ in the $y$-axis is the vector $|y\rangle$

Figure 2: The shadow of $|\Psi\rangle$ in the $x$-axis, $|x\rangle$, and in
the $y$-axis, $|y\rangle$.

The concept of “subsystem” and that of “product state” lead to the set-
theoretic idea that the “system” is the union of its parts —which, as it is
very well known, is not the case in the quantum formalism and vectorial spaces
where there are no ‘parts’ (i.e., elements of a set) but ‘shadows’ (i.e.,
projection of subspaces). To say that the projections of subspaces should be
understood —following the classical line of reasoning— as “subsystems” in SQM
leads then to essential contradictions. In the early 1980s Diederik Aerts
would derive a non-separability theorem in the context of quantum logic which
demonstrated that quantum systems are essentially non-separable [1, 2]. Later
on, in the mid-1990s Rob Clifton would show in [9] the inconsistencies of
imposing the notion of separability to the orthodox formulation. Taking into
consideration the following states:

$\displaystyle|\psi\rangle$ $\displaystyle=\sqrt{w_{1}}|\text{Decay after time
}t\rangle_{\text{Atom}}\otimes|\text{Dead}\rangle_{\text{Cat}}$
$\displaystyle+\sqrt{w_{2}}|\text{No decay after time
}t\rangle_{\text{Atom}}\otimes|\text{Alive}\rangle_{\text{Cat}}$

for $w_{1}+w_{2}=1$, $0<w_{1}<w_{2}<1$ he would show how it is easy to find a
contradiction when attempting to consider the definite values of states.
Clifton begins by making explicit two central claims to which any realist
interpretation should subscribe:

  * MI1:

Schrödinger’s cat is definitely alive or dead despite its entanglement in
$|\psi\rangle$.

  * MI2:

At any given time, only a proper subset of all possible properties of a given
system corresponds to the definite properties it actually possesses.

Let $\text{Def}_{|\phi\rangle}(S)$ be the set of particular intrinsic
properties a system $S$ possesses in the state $|\phi\rangle$, let
$\\{P(S)\\}$ be the set of all properties of $S$ and let $P_{|\phi\rangle}$ be
the projection operator with one dimensional range generated by the vector
$|\phi\rangle$. Then, we can translate the assertions mathematically as

  * MI1:

$\\{P_{|\text{Alive}\rangle_{\text{Cat}}},P_{|\text{Dead}\rangle_{\text{Cat}}}\\}\subseteq\text{Def}_{|\psi\rangle}(\text{Cat})$

  * MI2:

$\text{Def}_{|\psi\rangle}(\text{Atom + Cat})\subseteq\\{P(\text{Atom +
Cat})\\}$.

And using the standard formalism of QM we obtain the following,

  * CP1:

If $\text{Tr}(P|\psi\rangle)=1$ then $P\in\text{Def}_{|\psi\rangle}(\text{Atom
+ Cat})$.

  * CP2:

$\text{Def}_{|\psi\rangle}(\text{Atom + Cat})$ is an ortholattice.

  * CP3:

If $P\in\text{Def}_{|\psi\rangle}(\text{Cat})$ then $I\otimes
P\in\text{Def}_{|\psi\rangle}(\text{Atom + Cat})$.

Clifton proves that MI1 negates MI2, hence both claims cannot coexist. Notice
that CP3 formalizes the fact that the intrinsic properties of a composite
system should at least include the intrinsic properties of its parts. Then he
provides an example of the inconsistency leading to the lack of this necessary
condition to account for systems and their parts in any meaningful manner:

> “the intrinsic properties of a composite system should at least include the
> intrinsic properties of its parts. For example, whenever its true
> (respectively, false) that ‘the left-hand wing of the 747 has the property
> of being warped’, then it must surely (classically, anyways!) also be true
> (respectively, false) that ‘The 747 has the property that its left-hand wing
> is warped’. It may seem that the difference between these two propositions
> is inconsequential; but in fact that we take the former to entail the latter
> makes all the difference to whether we are confident flying the 747!” [9]

Maybe a simple image can expose why the attempt to understand shadows as
subsystems is obviously misleading.

“Dad, those are our subsystems, right?”

##  3 The Logos’ Realist Approach to QM: Operational-Invariance

The relativization of the notion of physical state in SQM has precluded the
understanding of physics as a discipline which is capable of producing an
invariant-objective representation of a state of affairs independent of
reference frames or empirical viewpoints, opening the door to the
establishment of the 20th century instrumentalist account of physics. This is
expressed in the widely accepted claim according to which, since we cannot
understand what the theory talks about nor produce a detached representation
of physical reality, we must content ourselves with a pragmatic approach to
the quantum formalism understood as a “recipe” to predict measurement
outcomes. As Bohr [5] would state explicitly: “Physics is to be regarded not
so much as the study of something a priori given, but rather as the
development of methods of ordering and surveying human experience.” Today,
this continues to be the orthodox contemporary understanding of the theory of
quanta. As recently described by Tim Maudlin:

> “What is presented in the average physics textbook, what students learn and
> researchers use, turns out not to be a precise physical theory at all. It is
> rather a very effective and accurate recipe for making certain sorts of
> predictions. What physics students learn is how to use the recipe. For all
> practical purposes, when designing microchips and predicting the outcomes of
> experiments, this ability suffices. But if a physics student happens to be
> unsatisfied with just learning these mathematical techniques for making
> predictions and asks instead what the theory claims about the physical
> world, she or he is likely to be met with a canonical response: Shut up and
> calculate!” [36, pp. 2-3]

However, most of these claims regarding the impossibility of theoretical
representation are just consequence of imposing a dogmatic atomist picture. In
fact, in QM it is —and always has been— possible to determine theoretically a
state of affairs independent of reference frames. Right from the beginning —in
Heisenberg’s matrix mechanics— QM was constructed in invariant terms allowing
to determine, within its formalism, physical quantities that would remain the
same for different reference frames. This invariant character of the formalism
has been however generally ignored or casted aside, due to the projection of
classical concepts, incompatible with the quantum formalism. Instead of
developing the new concepts adequate to the invariant content of the
formalism, the efforts were directed —first by Schödinger and later on by
Dirac— towards an (impossible) restoration of a “common sense” view of the
physical world, which was classical, and which was believed to be the only
reasonable one. This mandatory projection of an understanding incompatible
with QM has made us leave aside that invariant aspect of the formalism. But
let us explain this further.

In principle, at least in formal terms, it is operational-invariance which
defines what has to be considered as _the same_ in a physical theory. The
essential role played by invariance consists in that it allows to consider
experience from different perspectives in a consistent manner as referring to
the same state of affairs. In turn, as Einstein constantly remarked, this
makes possible to address the existence of a state of affairs as detached from
particular empirical observations and reference frames. Given a mathematical
formalism of a theory which has an invariant transformation, it does not
really matter which particular reference frame we might choose to describe the
state of affairs simply because there will be a consistent translation between
any of them. We can go from one representation to another without loosing
consistency and coherency regarding what is considered to be the same
state.333Of course, thsi does not mean that the properties considered from
different reference frames will possess the same values (e.g., position and
velocity in classical mechanics), it means there will exist a consistent
translation of these properties from one reference frame to the other. This is
surely realized in Newtonian mechanics, in Maxwell’s electromagnetism, and
also —via the Lorentz’ transformations— in relativity theory. But is it
possible to find invariance in QM? Contrary to what is usually believed, the
answer is yes, and —as we already mentioned above— right from the start, in
Heisenberg’s matrix mechanics.

For some years, Heisenberg had followed Bohr’s guide, focusing on the question
of describing the trajectories of electrons inside the atom. But the critical
reaction of Wolfgang Pauli and Arnold Sommerfeld led him to think that he
should take a different path [7]. So, instead of trying to describe
trajectories of unseen, presupposed, corpuscles, Heisenberg reframed the
problem in terms of observable quantities. As explained by Jaan Hilgevoord and
Joos Uffink [31]: “His leading idea was that only those quantities that are in
principle observable should play a role in the theory, and that all attempts
to form a picture of what goes on inside the atom should be avoided. In atomic
physics the observational data were obtained from spectroscopy and associated
with atomic transitions. Thus, Heisenberg was led to consider the ‘transition
quantities’ as the basic ingredients of the theory.” That same year, he would
present his groundbreaking results in the following manner [27]: “In this
paper an attempt will be made to obtain bases for a quantum-theoretical
mechanics based exclusively on relations between quantities observable in
principle.” Emancipating himself completely from the atomist picture,
Heisenberg was able to create a completely new mathematical formalism. As he
would recall in his autobiography:

> “In the summer term of 1925, when I resumed my research work at the
> University of Göttingen —since July 1924 I had been Privatdozent at that
> university— I made a first attempt to guess what formulae would enable one
> to express the line intensities of the hydrogen spectrum, using more or less
> the same methods that had proved so fruitful in my work with Kramers in
> Copenhagen. This attempt lead me to a dead end —I found myself in an
> impenetrable morass of complicated mathematical equations, with no way out.
> But the work helped to convince me of one thing: that one ought to ignore
> the problem of electron orbits inside the atom, and treat the frequencies
> and amplitudes associated with the line intensities as perfectly good
> substitutes. In any case, these magnitudes could be observed directly, and
> as my friend Otto had pointed out when expounding on Einstein’s theory
> during our bicycle tour round Lake Walchensee, physicists must consider none
> but observable magnitudes when trying to solve the atomic puzzle.” [29, p.
> 60]

Heisenberg was capable of developing matrix mechanics following two ideas:
first, to leave behind the classical notion of particle-trajectory, as it did
not seem required by QM —and it rather appeared as a classical habit that was
inadvertently coloring a priori the approach to the understanding of the new
theory—, and second, to take as a methodological standpoint Ernst Mach’s
positivist idea according to which a theory should only make reference to what
is actually observed in the lab. And what is actually observed is well known
to any experimentalist: a spectrum of line intensities. This is what was
described by the tables of data that Heisenberg attempted to mathematically
model and that finally led him —with the help of Max Born and Pascual Jordan—
to the development of the first mathematical formulation of the theory of
quanta. Let us stop to take note once again of some of the conditions that
were fundamental for the development of the quantum formalism. First,
Heisenberg’s abandonment of Bohr’s atomist narrative and research program
which focused in the description of unobservable trajectories of presupposed
yet irrepresentable quantum particles. Second, the consideration of Mach’s
observability principle as a methodological standpoint that —even if
Heisenberg didn’t fully embraced the positivist credo— allowed him to find a
starting point unburdened of those classical presuppositions. That
methodological standpoint led him finally to the replacement of Bohr’s
fictional trajectories of irrepresentable electrons by the consideration of
the intensive quantities appearing in line spectra that were actually observed
in the lab. And these quantities, once detached from a supposedly necessary
reduction to atomic elements, were what the formalism was indicating as
invariant. Radically new, and of fundamental importance to produce a
consistent and invariant quantum formalism, was this idea that we should
accept intensive values as basic, as perfectly good “substitutes”, that are in
no need whatsoever to be reconduced to binary values. Intensities appeared as
basic and sufficient. But Heisenberg’s intuition, according to which we should
take as mainly significant the intensive patterns, was mostly discarded. As we
explained earlier, the replacement of intensive patterns by single outcomes as
what was most meaningful took place especially in Dirac’s work, and as a
consequence of a dogmatic presupposition: we must refer to particles, and
those single outcomes must be the expression of specific particles.

Perhaps it may also be of interest to consider an unexpected conclusion
emerging from what we have just discussed. The orthodox understanding of
current physics is largely justified on an empiricist foundation, wherein the
theoretical work of physics must always start from observations, considered as
uncontaminated data, stripped of any theoretical presupposition. And it is on
this foundation, on the supposedly theoretically innocent nature of
observations, that the objectivity of physics is somehow justified. Beyond the
problems behind this idea of physics (especially in its conception of
observation), the truth is that what has been said so far seems to suggest
that this condition may not have been met, as commonly believed. One could
argue that the only one who remained firmly rooted in what is actually
observed, the only radical empiricist, the only one who managed to shed
inadequate assumptions and consider anew what was actually observed (and, from
that, develop the formalism of quantum mechanics), was Werner Heisenberg. He
managed to free himself from the atomistic narrative that imposed a series of
dogmatic presuppositions and could thus confront what was observed directly,
concluding in the invariance of intensities suggested by his matrix mechanics.
From all of this, it becomes apparent that what happened afterward was, in
large part, a gradual departure from the radical attention to what is actually
observed. By reinstating the atomistic dogma and reintroducing the
fundamentals of classical concepts, invariance was destroyed, and a series of
presuppositions were imposed on observations that were not only foreign to
them but, even worse, were incompatible with them. The history of quantum
mechanics seems to describe, after Heisenberg’s matrix mechanics, an
increasing departure from the focus on what is observed. Less empiricism, not
more —at least if we decide to take it seriously.

In any case, a physical theory is not only an invariant mathematical
formalism. Equally fundamental is the development of a conceptual
representation that allows us to qualitatively understand what is physically
real according to the formalism, and to give meaning to the observations that
are predicted by the formalism. Together with formal invariance, conceptual
objectivity is a fundamental aspect of any physical theory. But this
conceptual representation cannot be an arbitrary addition. An objective
conceptual representation has to be developed in strict accordance with the
conditions established by the formalism. Specifically, it should be grounded
on the moment of unity produced through the invariant content of the
formalism. Constructing, in turn, an objective conceptual representation of
what appears as invariant, producing the physical concepts that correspond to
the invariant aspects of the formalism. If, as we said, what appears as
invariant are the intensive quantities, and if this invariance is lost when we
attempt to redirect intensities to binary values, we should start by producing
the concept of an originally intensive element of physical reality, which is
sufficient, which does not entail the redirection to other elements that could
be understood in a binary manner. In this respect, we propose the concept of
_power of action_ , or _intensive power_ , which represents those invariant
intensive quantities present in the quantum formalism (without the need of
adding any projection postulate). The intensive —or probabilistic— value of
each power is termed its _intensity_ or _potentia_. If we think about it, in
fact the reference to a physical reality of an intensive nature is already
present in the way Max Planck formulated his original discovery: the _quantum
of action_. Action surely is no particle; action represents a reality perhaps
more intuitively conceived in intensive terms. The reference is also found in
configuration space which, in fact, is nothing but a space which encapsulates
degrees of action. In this representation, QM talks about “action”, about
powers of action, quantitatively defined by their intensity or potentia. This
representation entails a rejection of the atomistic image of the world, that
forces us constantly to take the probabilistic values as a diminished,
insufficient representation, as evidently not the ‘real thing’, and it also
entails that it is not necessary to redirect those intensive quantities in
each case to binary values in order to determine a state of affairs. Let us
add that the notion of a physical element that is in itself intensive implies
a different understanding of observation. For instance, in the particular
experimental situations where we obtain a single outcome at a time, it is only
possible to obtain a measure of the physical element considered within the
theory through repetition. This means that, in those situations, a single
outcome is not what is mainly meaningful, but, on the contrary, an
insufficient information, a minimal and partial measure. This is an example of
how the understanding of observation is conditioned by theoretical
presuppositions —contrary to the contemporary widespread empiricist
understanding according to which observations, taken as pure, uninterpreted
data, define an uncontaminated starting point.

When we stick to the intensive values of action, we are able in fact to refer
to a state of affairs that is independent of the particular representation in
a reference frame (or basis), escaping thus the relativism with which most
accounts of QM have contented themselves. In contraposition to an Actual State
of Affairs (ASA), defined classically in terms of a set of true definite
valued binary properties, we propose to relate the reference of QM to an
Intensive State of Affairs (ISA) —also called elsewhere Potential State of
Affairs (PSA) [19]. What has been demonstrated is that by considering an
intensive, rather than binary, state of affairs, it is possible to restore a
consistent global valuation for all projection operators independently of the
basis. Let us recall some results from [19]. While a _Global Binary Valuation_
(GBV) is a function from a graph to the set $\\{0,1\\}$, a _Global Intensive
Valuation_ (GIV) is a function from a graph to the closed interval $[0,1]$. We
term projection operators as intensive powers.444For a detailed introduction,
analysis and discussion of the notion of ‘intensive power’ we refer the
interested reader to [13], and more specifically, [19, Sect. 8] and [18, Sect.
3]. Let $H$ be a Hilbert space and let $\mathcal{G}=\mathcal{G}(H)$ be the set
of observables. We give to $\mathcal{G}$ a graph structure by assigning an
edge between observables $P$ and $Q$ if and only if $[P,Q]=0$. We call this
graph, _the graph of powers_. Among all global intensive valuations we are
interested in the particular class of ISA.

######  Definition 3.1.

Let $H$ be a Hilbert space. An _Intensive State of Affairs_ is a global
intensive valuation $\Psi:\mathcal{G}(H)\to[0,1]$ from the graph of powers
$\mathcal{G}(H)$ such that $\Psi(I)=1$ and

$\Psi(\sum_{i=1}^{\infty}P_{i})=\sum_{i=1}^{\infty}\Psi(P_{i})$

for any piecewise orthogonal projections $\\{P_{i}\\}_{i=1}^{\infty}$. The
numbers $\Psi(P)\in[0,1]$, are called intensities or potentia and the nodes
$P$ are called _powers_. Hence, an ISA assigns a potentia to each power.

Intuitively, we can picture an ISA as a table,

$\Psi:\mathcal{G}(H)\rightarrow[0,1],\quad\Psi:\left\\{\begin{array}[]{rcl}P_{1}&\rightarrow&p_{1}\\\
P_{2}&\rightarrow&p_{2}\\\ P_{3}&\rightarrow&p_{3}\\\
&\vdots&\end{array}\right.$

######  Theorem 3.2.

Let $H$ be a separable Hilbert space, $\dim(H)>2$ and let $\mathcal{G}$ be the
graph of powers with the commuting relation given by QM.

  * •

Any positive semi-definite self-adjoint operator of the trace class $\rho$
determines in a bijective way an ISA $\Psi:\mathcal{G}\to[0,1]$.

  * •

Any GIV determines univocally a set of powers that are considered as truly
existent.

###### Proof.

  1. 1.

Using Born’s rule, we can assign to each observable $P\in\mathcal{G}$ the
value $\mbox{Tr}(\rho P)\in[0,1]$. Hence, we get an ISA
$\Psi:\mathcal{G}\to[0,1]$. Let us prove that this assignment is bijective.
Let $\Psi:\mathcal{G}\to[0,1]$ be an ISA. By Gleason’s theorem there exists a
unique positive semi-definite self-adjoint operator of the trace class $\rho$
such that $\Psi$ is given by the Born rule with respect to $\rho$.555As
remarked in [40]: “Prior to the Bell and Kochen-Specker theorems, Gleason’s
theorem demonstrated that, for any quantum system of dimension at least three,
the unique way to assign probabilities to the outcomes of projective
measurements is via the Born rule. In particular, Gleason’s theorem excludes
any deterministic probability rule given by a {0, 1}-valued assignment of
probabilities to all the self-adjoint projections on the system’s Hilbert
space.”

  2. 2.

Consider the function $\tau:[0,1]\to\\{0,1\\}$, where $\tau(t)=0$ if and only
if $t=0$. Now, given a GIV $\Psi:\mathcal{G}\to[0,1]$, the map
$\tau\Psi:\mathcal{G}\to\\{0,1\\}$ is a well-defined map.

∎

######  Definition 3.3.

Let $\mathcal{G}$ be a graph. We define a _context_ as a complete subgraph (or
aggregate) inside $\mathcal{G}$. For example, let $P_{1},P_{2}$ be two
elements of $\mathcal{G}$. Then, $\\{P_{1},P_{2}\\}$ is a contexts if $P_{1}$
is related to $P_{2}$, $P_{1}\sim P_{2}$. Saying it differently, if there
exists an edge between $P_{1}$ and $P_{2}$. In general, a collection of
elements $\\{P_{i}\\}_{i\in I}\subseteq\mathcal{G}$ determine a context if
$P_{i}\sim P_{j}$ for all $i,j\in I$. Equivalently, if the subgraph with nodes
$\\{P_{i}\\}_{i\in I}$ is complete. A _maximal_ context is a context not
contained properly in another context. If we do not indicate the opposite,
when we refer to contexts we will be implying maximal contexts.

For the graph of powers, the notion of context coincides with the usual one; a
complete set of commuting operators. However, all projection operators can be
assigned a consistent value bypassing in this way the famous Kochen-Specker
theorem,

######  Theorem 3.4.

(Intensive Non-Contextuality Theorem) Given any Hilbert space $H$, then an ISA
is possible over $H$.

###### Proof.

See [19]. ∎

This theorem restores the possibility of an invariant physical representation
of any quantum wave function $\Psi$. Thus, contrary to the orthodox
interpretation of QM in terms of systems with properties (which impose a
binary valuation), our conceptual representation of quantum physical reality
is not relative to any particular context, it is global and essentially
intensive. We refer the reader to [18, 19] for a detailed discussion and
analysis. Taking this approach as a standpoint, it was possible to derive in
[22], two important theorems: the Basis Invariance Theorem and the
Factorization Invariance Theorem to which we now turn our attention.

##  4 An Invariant Account of Bases and Factorizations

Let us begin by recalling the Basis Invariance Theorem.

######  Theorem 4.1.

(Basis Invariance Theorem) Let $H$ be a Hilbert space, $\mathcal{G}$ its graph
of powers and let $\Psi:\mathcal{G}\to[0,1]$ be an ISA. Let
$\mathcal{C}_{1},\mathcal{C}_{2}\subseteq\mathcal{G}$ be two contexts. The
intensities of $\Psi$ over $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ may be
different, but both determine in a unique way the whole ISA. In other words,
the GIV defined by $\Psi$ does not depend on the basis for $H$.

###### Proof.

Let $\rho$ be the density matrix associated to $\Psi$ in the basis
$\mathcal{C}_{1}$. Then, the intensity of a particular power $P$ can be
computed with the Born rule $Tr(\rho P)$. If $\rho^{\prime}$ is the density
matrix associated to $\Psi$ in the basis $\mathcal{C}_{2}$, then there exists
a unitary matrix $U$ such that $\rho^{\prime}=U\rho U^{\dagger}$. Hence, $P$
is transformed to $P^{\prime}=U^{\dagger}PU$ and the intensities of $P$ and
$P^{\prime}$ are the same,

$Tr(\rho
P)=Tr(U\rho^{\prime}U^{\dagger}P)=Tr(\rho^{\prime}U^{\dagger}PU)=Tr(\rho^{\prime}P^{\prime}).$

∎

As a consequence of the Basis Invariance Theorem, the choice of a basis or
context becomes in this scheme just the choice of a viewpoint of analysis
which is compatible with the choice of any other viewpoint. Thus, escaping
contextuality and relativism, all reference frames provide a consistent
account —of the different values of projection operators which are part— of
the same (intensively) defined state of affairs.

Turning now our attention to the relativism of factorizations exposed in [12]
it is possible to reach an equivalent result [22]. Let us denote by ${\cal
ISA}(H)$ the set of all possible ISAs on the Hilbert space $H$. We can
identify this set (after fixing a basis) with the space of density matrices
over $H$. In fact, given any completely positive trace-preserving map
$T:B(H_{1})\to B(H_{2})$, we obtain a map $T_{*}:{\cal ISA}(H_{1})\to{\cal
ISA}(H_{2})$. An example of a completely positive trace-preserving map is the
partial trace. Notice that for each factorization $H=H_{1}\otimes H_{2}$, we
have a different partial trace, $T:B(H)\to B(H_{1})$.

######  Definition 4.2.

Let $T:B(H)\to B(H^{\prime})$ be a completely positive trace-preserving map
between (trace-class) operators on Hilbert spaces $H,H^{\prime}$ and let
$\Psi:\mathcal{G}(H)\to[0,1]$ be an ISA over $H$. We say that
$\Psi^{\prime}:\mathcal{G}(H^{\prime})\to[0,1]$ is a _shadow_ 666It it argued
orthodoxly that if $\rho$ is a density matrix in $A\otimes B$, then the matrix
associated to the “sub-system” in $A$ is the first partial trace of $\rho$.
However, the resulting density matrix $Tr_{1}(\rho)$ is not a submatrix of
$\rho$ but a projection, an image. In order to make explicit our departure
from the orthodox interpretation in terms of systems and sub-systems which has
been already shown to be inconsistent (see [9, 10]), we choose the word
“shadow” —instead of “subsystem”— which seems a much more appropriate notion
to capture this mathematical aspect of the formalism. of $\Psi$ if
$\Psi^{\prime}=T_{*}(\Psi)$. Specifically, if $\Psi$ is given by a density
matrix $\rho$. Then, $\Psi^{\prime}$ is given by $\rho^{\prime}=T(\rho)$.

######  Theorem 4.3.

(Factorization Invariance Theorem) Let $H$ be a Hilbert space with
factorizations $H=H_{1}\otimes H_{2}=H_{1}^{\prime}\otimes H_{2}^{\prime}$.
Let $T:B(H)\to B(H_{1})$ and $T^{\prime}:B(H)\to B(H_{1}^{\prime})$ be the
partial traces of these factorizations. Let $\Psi$ be an ISA over $H$ and let
$\Psi_{1}=T_{*}(\Psi)$ and $\Psi_{1}^{\prime}=T_{*}^{\prime}(\Psi)$ the
corresponding ISAs over $H_{1}$ and $H_{1}^{\prime}$. Assume there exists a
completely positive trace-preserving map $U:B(H_{1})\to B(H_{1}^{\prime})$
such that $UT=T^{\prime}$. Then, $\Psi_{1}^{\prime}=U_{*}(\Psi_{1})$.

###### Proof.

Straightforward,
$\Psi_{1}^{\prime}=T^{\prime}_{*}(\Psi)=(UT)_{*}(\Psi)=U_{*}T_{*}(\Psi)=U_{*}(\Psi_{1})$.
(For an example see [22]) ∎

The previous theorem implies that all factorizations are consistent and that
all corresponding shadows of the ISA are all compatible between each other. In
fact, if we have a compatibility $U$ between the factors $H_{1}$ and
$H_{1}^{\prime}$, then this compatibility translates itself in a compatibility
between $\Psi_{1}$ and $\Psi_{1}^{\prime}$.

######  Corollary 4.4.

All factorizations are compatible with respect to the same ISA.

###### Proof.

Follows from the previous Theorem. ∎

Thus, the choice of the factorization has no incidence in the intensities or
potentia already contained in the considered ISA. All choices of different
factorizations remain consistent with the same global account of the same ISA.
Taking these mathematical facts as a standpoint, we are now in conditions to
provide a natural conceptual account of both ‘bases’ and ‘factorizations’
operationally linked to the intensive relations between the multiple degrees
of freedom that can be considered within a laboratory when analyzing quantum
phenomena.

##  5 A Conceptual-Operational Account of a Quantum Lab

Marking a radical departure with respect to the orthodox Bohrian-positivist
contemporary account of observations not only as unproblematic givens of
experience but also as the kernel standpoint from which scientific theories
can be developed, Einstein writes in the following terms:

> “From Hume Kant had learned that there are concepts (as, for example, that
> of causal connection), which play a dominating role in our thinking, and
> which, nevertheless, can not be deduced by means of a logical process from
> the empirically given (a fact which several empiricists recognize, it is
> true, but seem always again to forget). What justifies the use of such
> concepts? Suppose he had replied in this sense: Thinking is necessary in
> order to understand the empirically given, and concepts and ‘categories’ are
> necessary as indispensable elements of thinking.” [26, p. 678] (emphasis in
> the original)

As it should be recognized, in a truly realist framework, the moments of unity
are not presupposed as exterior (pre-theoretical) givens but —instead— are
always theoretically constructed in formal-conceptual terms, and that is the
reason why “it is only the theory which decides what can be observed.” It is
only with adequate concepts that we can establish what has been observed. A
physical theory requires an invariant formalism, as well as an objective
conceptual representation that is consistent with it. Furthermore, both
conceptual and formal representations must be operational. This is, they must
contain the conditions under which we can understand observations in
accordance to what is presented in formal-conceptual terms. We have to be able
to consistently connect the theory —namely, the mathematical-conceptual
system— with the phenomena in question. For as stated clearly by Einstein [25,
p. 26], “The concept does not exist for the physicist until he has the
possibility of discovering whether or not it is fulfilled in an actual case.”
If, as what just said, the understanding of observation requires concepts, at
the same time, the physical concepts require a link to experience; i.e., the
specific operational conditions that allow to observe what the concepts are
referring to. So let’s advance in this direction. In order to bridge the gap
between, on the one hand, the mathematical formalism and the conceptual
objectivity as we presented it —without the projection postulate and replacing
the reference to binary outcomes by intensive patterns and values— and, on the
other, a consistent and coherent understanding of experience, we must produce
a set of meaningful operational notions. Let us remark that, contrary to the
Bohrian doctrine of classical concepts, these notions should not be considered
as relying on a presupposed classical representation. Even though we might use
the same words, in the context of QM these physical notions need to be
understood in internal terms with respect to the theory —there is no necessary
link to the classical representation of physics. As remarked by Heisenberg in
an interview by Thomas Kuhn:

> “The decisive step is always a rather discontinuous step. You can never hope
> to go by small steps nearer and nearer to the real theory; at one point you
> are bound to jump, you must really leave the old concepts and try something
> new… in any case you can’t keep the old concepts.” [6, p. 98]

Of course, this does not mean you cannot keep the same words. For example, the
words ‘space’ and ‘time’ refer to specific concepts in classical mechanics
which differ significantly from the concepts, related to exactly the same
words, discussed in relativity theory. This is a consequence of the fact that
concepts acquire their meaning in each case from the place they occupy in the
specific conceptual system of the particular theory they are part of, this is,
their meaning is determined through the relation with the other concepts
inside the system. As Heisenberg explained:

> “New phenomena that had been observed could only be understood by new
> concepts which were adapted to the new phenomena. […] These new concepts
> again could be connected in a closed system. […] This problem arose at once
> when the theory of special relativity had been discovered. The concepts of
> space and time belonged to both Newtonian mechanics and to the theory of
> relativity. But space and time in Newtonian mechanics were independent; in
> the theory of relativity they were connected.” [28, pp. 97-98]

We are now ready to introduce the notions of screen, detector, experimental
arrangement and quantum laboratory that will help us to build a conceptual-
operational bridge between the mathematical formalism and quantum phenomena.
Let us start with the simplest case. A screen with $n$ detectors corresponds
to the vector space $\mathbb{C}^{n}$. Choosing a basis, say
$\\{|1\rangle,\dots,|n\rangle\\}$, is the same as choosing some specific $n$
detectors. For example, in the next two figures we picture $\mathbb{C}^{2}$,
and $\mathbb{C}^{2}$ with a chosen basis,

Figure 3: A screen with a place for two detectors.

Figure 4: A screen with two specific detectors.

A factorization $\mathbb{C}^{i_{1}}\otimes\dots\otimes\mathbb{C}^{i_{n}}$ is
the specific number $n$ of screens, where the screen number $k$ has $i_{k}$
places for detectors, $k=1,\dots,n$. Choosing a basis in each factor
corresponds to choosing the specific detectors; for instance
$|\uparrow\rangle,|\downarrow\rangle$. After choosing a basis in each factor,
we get a basis of the factorization
$\mathbb{C}^{i_{1}}\otimes\dots\otimes\mathbb{C}^{i_{n}}$ that we denote as

$\\{|k_{1}\dots k_{n}\rangle\\}_{1\leq k_{j}\leq i_{j}}.$

The basis element $|k_{1}\dots k_{n}\rangle$ represents the power of action
(or simply power) that produces an intensive global effect in the $k_{1}$
detector of the screen $1$, in the $k_{2}$ detector of the screen $2$ and so
on until the $k_{n}$ detector of the screen $n$. In general, any given power
will produce a unitary multi-screen non-local effect that has an intensive
content.777It might be stressed that, once we give up Bohr’s correspondence
principle or the so called “quantum to classical limit”, namely, the idea that
QM must be related to a spatial continuous representation, we also loose the
classical notion of locality which requires a continuous representation
precluded by the quantum formalism. This is a natural consequence of Planck’s
quantum postulate according to which energy must be considered as discrete,
and consequently also space and time. In fact, according to the formalism the
only distance that we could measure is the distance between basis elements
which is always $\sqrt{2}$ (by Pythagoras theorem). This mathematical distance
is purely abstract and bears no relation with a distance in Euclidean physical
space. For example, in Figure 5 we picture two screens, one with two detectors
and the other with three detectors. Given the possible combinations, we obtain
six different powers
$|11\rangle,|12\rangle,|13\rangle,|21\rangle,|22\rangle,|23\rangle$. The
powers $|11\rangle$ and $|23\rangle$ are highlighted,

Figure 5: The 6 powers of action that arise when considering
one screen with 2 detectors and another with 3.

According to this representation, a single power of action will produce non-
local effects in multiple screens which will be strictly determined in terms
of definite valued intensities. Let us stress the main point of our reference
to an ISA, namely, that QM only talks about intensities, never about single
outcomes. In particular, an intensity equal to 1 should not be confused with
the certainty of obtaining a single outcome; all predictions in QM provide
certain knowledge when considering intensive values —which is what the theory
talks about. This implies a shift from the binary certainty that was
dogmatically imposed by Dirac —through the ad hoc addition of the projection
postulate— to an intensive certainty theoretically imposed by the invariance
present in Heisenberg’s original quantum formalism itself, where the intensity
equal to 1 is just a particular case of the possible values of intensities
within the closed interval $[0,1]$. Let us also remark that the operationality
we are proposing here allows us to restore a causal representation —completely
analogous to classical theories— which not only describes the interaction and
evolution of the powers of action in a given situation but also connects them
with the intensities obtained in measurement procedures. This methodological
move allows us to abandon the Bohrian reference —established in SQM— to
“quantum particles” (represented by quantum states) as well as the non-causal
“collapse” evolution imposed by Dirac to single (binary) measurement outcomes.
As was pointed out earlier, when we have experiments where we are faced with a
single outcome at a time, we must consider these single outcomes not as the
expression of particles, and consequently as the main thing to explain, but as
a partial information provided within measurement procedures. A single outcome
must be then considered as a minimal operational expression of a power. And
thus, to fully measure the intensive physical element the theory talks about
in these experiments, it is necessary to repeat the experiment until the
number of clicks allows us to establish the intensity described by the theory.
Besides, the fact the we observe multiscreen effects is a supplementary proof
of the fact we are not talking about particles (this effect is
incomprehensible if we suppose particles) but about powers of action.

Given an ISA $\Psi$, a factorization
$\mathbb{C}^{i_{1}}\otimes\dots\otimes\mathbb{C}^{i_{n}}$ and a basis
$B=\\{|k_{1}\dots k_{n}\rangle\langle k_{1}\dots k_{n}|\\}$ of cardinality
$N=i_{1}\dots i_{n}$, we define an experimental arrangement denoted
$\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots i_{n}}$, as a specific choice of
screens with detectors together with the potentia of each power, that is,

$\operatorname{EA}_{\Psi,B}^{N,i_{1}\dots
i_{n}}=\sum_{k_{1},k_{1}^{\prime}=1}^{i_{1}}\dots\sum_{k_{n},k_{n}^{\prime}=1}^{i_{n}}\alpha_{k_{1},\dots,k_{n}}^{k_{1}^{\prime},\dots,k_{n}^{\prime}}|k_{1}\dots
k_{n}\rangle\langle k_{1}^{\prime}\dots k_{n}^{\prime}|.$

The number that accompanies the power $|k_{1}\dots k_{n}\rangle\langle
k_{1}\dots k_{n}|$ is its potentia (or intensity) and the basis $B$ is the set
of powers defined by the specific choice of screens with detectors. The number
$N$ which is the cardinal of $B$ is called the degree of complexity (or
degree) of the experimental arrangement. Notice that in the case where we have
an experimental arrangement with one screen, then
$\operatorname{EA}_{\Psi,B}^{N;n}$ is a rank 1 matrix which can be understood
as a vector —like in Dirac’s formulation—, with two screens,
$\operatorname{EA}_{\Psi,B}^{N;i.j}$ is a density matrix —which are orthodoxly
interpreted as mixtures888See for a detailed analysis [21]., namely, as the
convex sum of pure states— and in general, for many screens,
$\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots i_{n}}$ is a (multi)-tensor product.
In contraposition to the orthodox account, a product state (or factorization)
is not the separation of a system (defined atomistically) into many parts
(sub-atomic particles or sub-systems) but simply the way in which a single
power is non-locally expressed on different screens.

Finally, we use quantum laboratory999We use the term quantum lab in order to
stress the fact that a laboratory is a concept that need not be necessarily
linked to the substantialist classical representation. (or quantum lab or
Q-Lab) as a synonim of ISA, in other words, a Q-Lab is the same as an ISA.
Now, given an $\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots i_{n}}$ and some of
the screens $k_{1},\dots,k_{s}$, we define the shadow of $\Psi$ in the screens
$k_{1},\dots,k_{s}$ as the partial trace in the factors $k_{1},\dots,k_{s}$ of
$\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots i_{n}}$,

$\operatorname{EA}_{\Psi,B^{\prime}}^{K;k_{1}\dots
k_{s}}:=Tr_{k_{1},\dots,k_{s}}(\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots
i_{n}}),\quad K=k_{1}\dots k_{s}.$

To sum up, a Q-Lab is denoted as $\Psi$ and is defined as a graph of powers
(all the powers) together with its intensities (see [19] for the precise
definition) containing all possible experimental set ups. An equivalent class
of experimental arrangements produced within the same lab
$\operatorname{EA}_{\Psi}^{N}$ consists of all the specific experimental
arrangements produced by choosing different configurations of screens and
detectors such that the total number of powers is $N$; i.e.,
$\operatorname{EA}_{\Psi,B}^{N},\operatorname{EA}_{\Psi,B^{\prime}}^{N},\operatorname{EA}_{\Psi,B^{\prime\prime}}^{N}$.
Notice that the factorization is already included in the information contained
within each basis. Let us notice that our notation allows to include the
information of the particular arrangement, namely, the number of screens and
detectors that are placed in each of the screens. Let us also remark that, as
explained in [19], choosing an $\operatorname{EA}_{\Psi,B}^{N}$ is the same as
choosing a specific context within the graph of powers:

Figure 6: An $\operatorname{EA}_{\Psi,B}^{4}$ chosen from
the net of powers given by the Q-Lab $\Psi$.

We are now in conditions to address more in detail the essential equivalence
relations we have reached for QM through the formal account of bases and
factorizations as well as their conceptual representation.

##  6 Equivalence Relations in Quantum Mechanics

While in section 4 we presented the basis and factorization invariance
theorems, in section 5 we introduced a series of operational physical notions
capable to express the mathematical formalism of the theory. We are now in
conditions to think of both theorems, in truly physical terms, as equivalence
relations of the theory, something essential for QM, understood in physical
terms, as providing a relation between different reference frames when
discussing about the same state of affairs. These relations might be
considered as analogous to Poincare’s relativity principle101010As stated by
Einstein [25, p. 98]: “Special principle of relativity: If a system of
coordinates K is chosen so that, in relation to it, physical laws hold good in
their simplest form, the same laws hold good in relation to any other system
of coordinates K’ moving in uniform translation relatively to K.” which guided
Einstein in his search to develop the Special Theory of Relativity beyond the
ontological picture —transformed in “common sense” within modernity— of
physical reality in terms of entities within absolute space and time.111111It
is important to emphasize that this reference to “relativity” cannot be
equated with the perspectival relativism that can be found in Dirac’s re-
formulation of the notion of (quantum) state in which the different basis
dependent representations cannot be globally considered. Einstein stressed the
need to hold fast to the equivalence relation between different reference
frames —as stated by Poincare’s principle— as a necessary condition for a
physical theory, and to the empirical finding of the invariance of the speed
of light, even if this meant doing away with one of the fundamental aspects of
the classical representation and replacing the Galilean transformations
—grounded on our “commonsensical” classical Newtonian spatiotemporal
representation— by the newly developed Lorentz transformations. In our case we
also hold fast to the need of providing an objective-invariant relation
between the different basis-dependent representations of QM, which is clearly
possible, but it entails rejecting the atomist worldview of the classical
representation. Given that physics is the search for unity within change, the
invariant-objective relations between different bases and factorizations which
allow to describe the same becomes essential. Indeed, physics attempts to
define moments of unity not only in order to consider a multiplicity of
different phenomena in a consistent manner, but also to account for the
equivalence between representations, dependent on empirical and formal
perspectives (i.e., reference frames). There must always exist an equivalence
relation which consistently connects the (different) possible descriptions
provided by (different) reference frames. This is, in fact, the condition
which allowed Galileo and Newton to unite the apparently distinct celestial
and terrestrial phenomena. This is called in physics the principle of
relativity, namely, the requirement that the equations describing the order of
physical relations have the same form in different yet equivalent frames of
reference. The relativity principle has played an essential role within
physical theories, from classical mechanics and electromagnetism up to
relativity theory.121212One decade later, Einstein would then extend it to the
general principle of relativity which states that all systems of reference are
equivalent with respect to the formulation of the fundamental laws of physics
[37, p. 220]. The reason is obvious: without such a principle it becomes
simply impossible to talk about the same experience in different yet
equivalent situations, and consequently, the production of a meaningful
physical discourse about the possible experiences in different labs becomes
precluded. Without such a principle it becomes simply impossible to say we are
performing the same experiment in different labs and physical theories become
operationally impracticable.

Going back to SQM, it becomes then clear the radical subversion of the
discipline that took place in the 20th century through Bohr and Dirac’s re-
definition of the notion of (quantum) state in (complementary) basis dependent
terms (section 2). As we discussed above, the restriction to binary values
(‘clicks’ in detectors) involves the abolition of operational-invariance;
i.e., the consistent translation between the experience found in different
reference frames of the same state of affairs. The idea that the state of a
system is dependent on a particular basis (i.e., a reference frame) breaks
down the very meaning of the notion of state as referring to something which
remains the same independently of reference frames. For if a state is
different in different bases it becomes nothing but different to itself, a
self-contradiction. This is, of course, the disintegration of the possibility
of considering any meaningful moment of unity, any common reference within the
theory. Of course, for many, this has not been understood as “the end of
physics”, but rather, as the production of a new (anti-realist)
“instrumentalist” account of the discipline where the reference to the world
and reality was replaced by the “more productive” accurate prediction of
measurement outcomes. As described by Arthur Fine:

> “[The] instrumentalist moves, away from a realist construal of the emerging
> quantum theory, were given particular force by Bohr’s so-called ‘philosophy
> of complementarity’; and this nonrealist position was consolidated at the
> time of the famous Solvay conference, in October of 1927, and is firmly in
> place today. Such quantum nonrealism is part of what every graduate
> physicist learns and practices. It is the conceptual backdrop to all the
> brilliant success in atomic, nuclear, and particle physics over the past
> fifty years. Physicists have learned to think about their theory in a highly
> nonrealist way, and doing just that has brought about the most marvelous
> predictive success in the history of science.” [11, p. 1195]

Even though we agree with Fine’s historical account of the transformation of
physics, we strongly disagree with his conclusions. It is certainly true that
physicists learned through the work of Bohr to think about theories in a
highly nonrealist way, but this new praxis did not produce “the most marvelous
predictive success in the history of science”, but —on the very contrary— the
destruction of the guiding principles of physics as well as the the
prohibition of essential lines of research. A very good example is the banning
of Einstein’s critical work in QM which led to the prohibition of
investigating quantum entanglement, a notion which remained erased from
physics for almost half a century. On the contrary, going back to the realist
program of science, the logos approach re-connects with the search for
theoretical (formal-conceptual) unity within experience through the guidance
of the principles of operational-invariance and objectivity. Thus, escaping
complementarity and the relativist account of states, objects and outcomes, we
will now consider the basis and factorization invariance theorems (section 4)
as general equivalence relations of the theory of quanta.

Given two ISAs $\Psi_{1}$, $\Psi_{2}$ and two bases $B_{1}$ and $B_{2}$ of the
same space we can define four different EAs,
$\operatorname{EA}_{\Psi_{1},B_{1}}^{N;i_{1}\dots
i_{s}},\operatorname{EA}_{\Psi_{1},B_{2}}^{N;i_{1}\dots
i_{s}},\operatorname{EA}_{\Psi_{2},B_{1}}^{N;i_{1}\dots i_{s}}$ and
$\operatorname{EA}_{\Psi_{2},B_{2}}^{N;i_{1}\dots i_{s}}$. Notice that all
four EAs are different, but in different ways. The EAs
$\operatorname{EA}_{\Psi_{1},B_{1}}^{N;i_{1}\dots i_{s}}$ and
$\operatorname{EA}_{\Psi_{1},B_{2}}^{N;i_{1}\dots i_{s}}$ are different but
can be converted from one to another by a change of basis (rearranging the
screens and detectors). This means they can be understood as making reference
to the same (intensive) state of affairs. The same counts for
$\operatorname{EA}_{\Psi_{2},B_{1}}^{N;i_{1}\dots i_{s}}$ and
$\operatorname{EA}_{\Psi_{2},B_{2}}^{N;i_{1}\dots i_{s}}$. However, the
experimental arrangements $\operatorname{EA}_{\Psi_{1},B_{1}}^{N;i_{1}\dots
i_{s}}$ and $\operatorname{EA}_{\Psi_{2},B_{1}}^{N;i_{1}\dots i_{s}}$ even
though have the same basis, are related to different states, this means there
is no obvious way of converting one to the other through a transformation
(same for $\operatorname{EA}_{\Psi_{1},B_{2}}^{N;i_{1}\dots i_{s}}$ and
$\operatorname{EA}_{\Psi_{2},B_{2}}^{N;i_{1}\dots i_{s}}$).

$\textstyle{\operatorname{EA}_{\Psi_{1},B_{1}}^{N;i_{1}\dots
i_{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\equiv}$$\scriptstyle{\not\equiv}$$\textstyle{\operatorname{EA}_{\Psi_{1},B_{2}}^{N;i_{1}\dots
i_{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\not\equiv}$$\textstyle{\operatorname{EA}_{\Psi_{2},B_{1}}^{N;i_{1}\dots
i_{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\equiv}$$\textstyle{\operatorname{EA}_{\Psi_{2},B_{2}}^{N;i_{1}\dots
i_{s}}}$

Now, in the light of this new conceptualization of the mathematical formalism,
we can rephrase the basis and factorization invariance theorems in the
following intuitive terms:

######  Theorem 6.1.

(Basis Invariance Theorem) Given a specific quantum laboratory, all
experimental arrangements of the same degree are equivalent.

######  Theorem 6.2.

(Factorization Invariance Theorem) The experiments performed with an
experimental arrangement of degree $N$ can also be performed with an
experimental arrangement of degree $N+M$.

While the Basis Invariance Theorem implies that the knowledge we obtain from
the intensities in one experimental arrangement is equivalent to the knowledge
obtainable in any other experimental arrangement of the same degree, the
Factorization Invariance Theorem tells us that reducing the complexity of an
experimental arrangement will not increase our knowledge in any way. In the
light of these Theorems, let us make some comments regarding the notation
$\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots i_{n}}$. First of all, choosing a
different basis $B^{\prime}$ in the same factorization, by the Basis
Invariance Theorem, we obtain an equivalent experimental arrangement,
$\operatorname{EA}_{\Psi,B^{\prime}}^{N;i_{1}\dots i_{n}}$,

$\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots
i_{n}}\equiv\operatorname{EA}_{\Psi,B^{\prime}}^{N;i_{1}\dots i_{n}}.$

Hence, we denote $\operatorname{EA}_{\Psi}^{N;i_{1}\dots i_{n}}$ to an
experimental arrangement with a fixed number of screens and detectors but
without specifying which detectors are chosen. Another possibility regarding
the notation is to choose a different factorization of the same vector space
$N=j_{1}\dots j_{n^{\prime}}$ (and consequently another set of powers
$B^{\prime\prime}$). Then, in this situation we also have by the Basis
Invariance Theorem

$\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots
i_{n}}\equiv\operatorname{EA}_{\Psi,B^{\prime\prime}}^{N;j_{1}\dots
j_{n^{\prime}}}.$

Hence, we denote $\operatorname{EA}_{\Psi}^{N}$ to an experimental arrangement
of degree $N$ without specifying which screens and which detectors are chosen.
The last possibility to consider comes from the relation between experimental
arrangements of different degrees of complexity. By the Factorization
Invariance Theorem, we have the following relation

$\operatorname{EA}_{\Psi}^{N}\Rrightarrow\operatorname{EA}_{\Psi}^{N+M}.$

Hence, we denote $\operatorname{EA}_{\Psi}$ to an experimental arrangement of
any degree of complexity.

######  Remark 6.3.

Given a factorization there are multiple bases that can be chosen in each
factor. However, a basis of the whole space determines a specific
factorization uniquely.

$\text{Factorization}\not\Rightarrow\text{Basis}$
$\text{Basis}\Rightarrow\text{Factorization}\quad$

In conceptual terms, given an $\operatorname{EA}_{\Psi}^{N}$ of some quantum
laboratory $\Psi$ there are different possible experimental arrangements that
can be constructed but none which is uniquely determined. However, given an
$\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots i_{n}}$ it is possible to determine
all other experimental arrangements $\operatorname{EA}_{\Psi}^{N}$ that can be
constructed.

$\operatorname{EA}_{\Psi}^{N}\not\Rightarrow\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots
i_{n}}$ $\operatorname{EA}_{\Psi,B}^{N;i_{1}\dots
i_{n}}\Rightarrow\operatorname{EA}_{\Psi}^{N}\quad$

######  Corollary 6.4.

Given a Quantum Lab, all experimental arrangements are Globally Consistent.

###### Proof.

Follows from the previous theorems. ∎

We reach in QM a new form of equivalence determined by the role of bases as
reference frames. While in the classical case a reference frame must be
understood as a formal perspective providing the specific values of all the
properties which determine a given state of affairs and these values can be
translated via Galilean transformation to any other reference frame, what is
truly new in the quantum case is that while a basis provides the intensive
values only of a specific set of powers the transformation to a new basis
allows to find out the specific intensive values of a different set of powers
of the same (intensive) state of affairs.

## Conclusions

In this paper we provided a consistent and coherent objective-invariant
account of bases and factorizations through the reference to an intensive
—rather than binary— state of affairs. We did so through the addition of new
physical concepts such as power of action, quantum laboratory, experimental
arrangement, etc. With the aid of these concepts we provided a new, more
intuitive, account of the basis and factorization invariance theorems.
Furthermore, we provided a deeper understanding of the complex relation
between factorizations and bases and between a quantum lab and the different
experimental arrangements that can be built within it.

## Acknowledgements

This work was partially supported by the following grants: ANID-FONDECYT,
Project number: 3240436.

## References

  * [1] Aerts, D., 1981, The one and the many: towards a unification of the quantum a classical description of one and many physical entities, Doctoral dissertation, Brussels Free University, Brussels.
  * [2] Aerts, D., 1984, “How do we have to change quantum mechanics in order to describe separated systems”, in The Wave-Particle Dualism pp. 419-431, S. Diner, D. Fargue, G. Lochak and F. Selerri (Eds.), Springer, Dordrecht.
  * [3] Birkhoff, G. $\&$ von Neumann, J., 1936, “The logic of quantum mechanics”, Annals of Mathematics, 37, 823-843.
  * [4] Bohr, N., 1935, “Can Quantum Mechanical Description of Physical Reality be Considered Complete?”, Physical Review, 48, 696-702.
  * [5] Bohr, N., 1960, The Unity of Human Knowledge, In Philosophical writings of Neils Bohr, vol. 3., Ox Bow Press, Woodbridge.
  * [6] Bokulich, A., 2006, “Heisenberg Meets Kuhn: Closed Theories and Paradigms”, Philosophy of Science, 73, 90-107.
  * [7] Bokulich, A. $\&$ Bokulich, P., 2020, “Bohr’s Correspondence Principle”, The Stanford Encyclopedia of Philosophy (Fall 2020 Edition), E.N. Zalta (ed.), forthcoming. https://plato.stanford.edu/archives/fall2020/entries/bohr-correspondence/.
  * [8] Butterfild, J., 2017, interview: Jeremy Butterfield: What is contextuality?, (published 3rd of March, 2017) URL: https://www.youtube.com/watch?v=ZJAnixX8T4U
  * [9] Clifton, R.K., 1995, “Why Modal Interpretations of Quantum Mechanics must Abandon Classical Reasoning About Physical Properties”, International Journal of Theoretical Physics, 34, 1302-1312.
  * [10] Clifton, R.K., 1996, “The Properties of Modal Interpretations of Quantum Mechanics”, British Journal for the Philosophy of Science, 47, 371-398.
  * [11] Curd, M. and Cover, J. A., 1998, Philosophy of Science. The central issues, Norton and Company (Eds.), Cambridge University Press, Cambridge.
  * [12] de la Torre, A.C., Goyeneche, D. $\&$ Leitao, L., 2010, “Entanglement for all quantum states”, European Journal of Physics, 31, 325-332.
  * [13] de Ronde, C., 2016, “Probabilistic Knowledge as Objective Knowledge in Quantum Mechanics: Potential Immanent Powers instead of Actual Properties”, in Probing the Meaning of Quantum Mechanics: Superpositions, Semantics, Dynamics and Identity, pp. 141-178, D. Aerts, C. de Ronde, H. Freytes and R. Giuntini (Eds.), World Scientific, Singapore.
  * [14] de Ronde, C., 2020, “Unscrambling the Omelette of Quantum Contextuality (Part I): Preexistent Properties or Measurement Outcomes?”, Foundations of Science, 25, 55-76.
  * [15] de Ronde, C. $\&$ Fernández Mouján, R., 2018, “Epistemological vs. Ontological Relationalism in Quantum Mechanics: Relativism or Realism?”, in Probing the Meaning of Quantum Mechanics, pp. 277-318, D. Aerts, M.L. Dalla Chiara, C. de Ronde and D. Krause (Eds.), World Scientific, Singapore.
  * [16] de Ronde, C. $\&$ Fernández-Mouján, R., 2021, “Are ‘Particles’ in Quantum Mechanics “Just a Way of Talking”?”, preprint (http://philsci-archive.pitt.edu/19968/).
  * [17] de Ronde, C., Domenech, G. $\&$ Freytes, H., 2018, “Quantum Logic in Historical and Philosophical Perspective”, Internet Encyclopedia of Philosophy, URL: https://iep.utm.edu/qu-logic/
  * [18] de Ronde, C. $\&$ Massri, C., 2019, “The Logos Categorical Approach to Quantum Mechanics: II. Quantum Superpositions and Measurement Outcomes.”, International Journal of Theoretical Physics, 58, 1968-1988.
  * [19] de Ronde, C. $\&$ Massri, C., 2021, “The Logos Categorical Approach to Quantum Mechanics: I. Kochen-Specker Contextuality and Global Intensive Valuations.”, International Journal of Theoretical Physics, 60, 429-456.
  * [20] de Ronde, C. $\&$ Massri, C., 2022, “Against the Tyranny of Pure States in Quantum Theory”, Foundations of Science, 27, 27-41.
  * [21] de Ronde, C. $\&$ Massri, C., 2022, “The Many Inconsistencies of the Purity-Mixture Distinction in Standard Quantum Mechanics”, preprint (quant-ph:2208.10574).
  * [22] de Ronde, C. $\&$ Massri, C., 2023, “Relational quantum entanglement beyond non-separable and contextual relativism”, Studies in History and Philosophy of Science, 97, 68-78.
  * [23] Dieks, D., 2022, “Perspectival Quantum Realism”, Foundations of Physics, 52.
  * [24] Dirac, P.A.M., 1974, The Principles of Quantum Mechanics, 4th Edition, Oxford University Press, London.
  * [25] Einstein, A., 1920, Relativity. The Special and General Theory, Henry Holt $\&$ Company, New York.
  * [26] Einstein, A., 1949, “Remarks concerning the essays brought together in this co-operative volume”, in Albert Einstein. Philosopher-Scientist, P.A. Schlipp (Eds.), pp. 665-689, MJF Books, New York.
  * [27] Heisenberg, W., 1925, “Über Quantentheoretische Umdeutung Kinematischer und Mechanischer Beziehungen”, Zeitschrift für Physik 33, 879-893. English translation in _Sources of Quantum Mechanics_ , B.L. van der Waerden, ed., Dover, 1968.
  * [28] Heisenberg, W., 1958, Physics and Philosophy, World perspectives, George Allen and Unwin Ltd., London.
  * [29] Heisenberg, W., 1971, Physics and Beyond, Harper & Row, New York.
  * [30] Heisenberg, W., 1973, “Development of Concepts in the History of Quantum Theory”, in The Physicist’s Conception of Nature, pp. 264-275, J. Mehra (Ed.), Reidel, Dordrecht.
  * [31] Hilgevoord, J. $\&$ Uffink, J., 2001, “The Uncertainty Principle”, The Stanford Encyclopedia of Philosophy (Winter 2001 Edition), E. N. Zalta (Ed.), http://plato.stanford.edu/archives/win2001/entries/qt-uncertainty/.
  * [32] Howard, D., 1994, “What makes a classical concept classical? Toward a reconstruction of Niels Bohr’s philosophy of physics”, Niels Bohr and Contemporary Philosophy, pp. 201-230, Kluwer Academic Publishers, Amsterdam.
  * [33] Kochen, S. $\&$ Specker, E., 1967, “On the problem of Hidden Variables in Quantum Mechanics”, Journal of Mathematics and Mechanics, 17, 59-87. Reprinted in Hooker, 1975, 293-328.
  * [34] Landau, L., 1927, “The Damping Problem in Wave Mechanics”, Collected Papers of L.D. Landau, pp. 8-18, 1965.
  * [35] Laudan, L., 1981, “Ernst Mach’s Opposition to Atomism”, in Science and Hypothesis, Springer, Dordrecht.
  * [36] Maudlin, T., 2019, Philosophy of Physics. Quantum Theory, Princeton University Press, Princeton.
  * [37] Moller, C., 1952, The Theory of Relativity 2nd Edition, Oxford University Press, Oxford.
  * [38] Rovelli, C., 1996, “Relational Quantum Mechanics”, International Journal of Theoretical Physics, 35, 1637-1678.
  * [39] Schrödinger, E., 1950, “What is an elementary particle?”, Endeavor, VolIX, N35, July 1950.
  * [40] Wright, V.J. $\&$ Kunjwal, R., “Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem.”, preprint (quant-ph:2109.13594).