#  Bild-conception for Scientific in Classical and Quantum Physics: from Hertz
and Boltzmann to Schrödinger and De Broglie

Andrei Khrennikov
Linnaeus University, International Center for Mathematical Modeling
in Physics and Cognitive Sciences Växjö, SE-351 95, Sweden
email<EMAIL_ADDRESS>
###### Abstract

We start with methodological analysis of the notion of scientific theory and
its interrelation with reality. This analysis is based on the works of
Helmholtz, Hertz, Boltzmann, and Schrödinger (and reviews of D’ Agostino).
Following Helmholtz, Hertz established the “Bild concept” for scientific
theories. Here “Bild” (“picture”) carries the meaning “model” (mathematical).
The main aim of natural sciences is construction of the causal theoretical
models (CTMs) of natural phenomena. Hertz claimed that CTM cannot be designed
solely on the basis of observational data; it typically contains hidden
quantities. Experimental data can be described by an observational model (OM),
often on the price of acausality. CTM-OM interrelation can be tricky.
Schrödinger used the Bild concept to create CTM for quantum mechanics (QM) and
QM was treated as OM. We follow him and suggest a special CTM for QM, so-
called prequantum classical statistical field theory (PCSFT). QM can be
considered as a PCSFT-image, but not as straightforward as in Bell’s model
with hidden variables. The common interpretation of the violation of the Bell
inequality is criticized from the perspective of the two level structuring of
scientific theories. Such critical analysis of von Neumann and Bell no-go
theorems for hidden variables was performed already by De Broglie (and Lochak)
in 1970s. The Bild-approach is applied to the two level CTM-OM modeling of
Brownian motion: the overdamped regime corresponds to OM. In classical
mechanics CTM=OM; on the one hand, this is very convenient, on the other hand,
this exceptional coincidence blurred the general CTM-OM structuring of
scientific theories. We briefly discuss ontic-epistemic structuring of
scientific theories (Primas-Atmanspacher) and its relation to the Bild
concept. Interestingly, Atmanspacher as well as Hertz claim that even
classical physical theories should be presented on the basic of two level
structuring.

keywords: Bild conception; scientific theory; Helmholtz; Hertz; Boltzmann;
Schrödinger; De Broglie; quantum mechanics; Brownian motion; Bell, von Neumann

##  1 Introduction

The Bild-conception of scientific theory was developed by Hertz [28, 29]
starting with Helmholtz analysis [65] of interrelation between physical
reality and scientific theory. This line of thinking was continued by
Boltzmann [16, 17] and in 1950’s by Schrödinger [57]-[63]. The articles of D’
Agostino [20]-[25] contain philosophically deep reviews on their works. The
German word “Bild” is translated to English as “picture”. But in relation to
the analysis of the meaning of a scientific theory it has the meaning of a
model, a mathematical model.

Helmholtz pointed out that a scientific theory does not describe reality as it
is. A scientific theory structures our sensations and perceptions within a
priori forms of intuition (cf. with Kant). Such structuring leads to models of
reality reflecting some features of the environment of observers. Therefore
the dream for creation of a “true theory” matching perfectly with natural
phenomena is in contradiction with Helmholtz’s philosophy of science.
Observational data should be taken with caution. Helmholtz highlighted
causality of the natural phenomena and for him the main task of a scientific
theory is to reflect this causality. Thus, from his viewpoint the main aim of
scientific studies is construction of the causal theoretical models (CTMs) of
natural phenomena. Theoretical causality is an image of natural causality. In
terms of cognition, causality of human reasoning reflects causality of natural
processes and it was developed in during biological evolution, from the
primitive forms of life to humans.

Hertz followed Helmholtz’ approach to scientific theory, but he claimed that
generally CTM can’t be designed solely on the basis of observational data and
it typically contains hidden quantities. (So, in physics hidden variables were
employed long before the quantum revolution.) Experimental data is described
by an observational model (OM) which is often acausal. The CTM-OM
interrelation can be tricky. This framework Hertz presented [28, 29] as a
Bild-conception (model-concept). He highlighted the role of mathematics and
treatment of a scientific model as a mathematical model (see also Plotnitsky
[54]). In particular, Hertz presented Maxwell’s theory as the system of the
Maxwell equations.

Later the Bild-conception resurrected in the foundational studies of
Schrödinger [57]-[57] (see especially [57]) who tried to create CTM for
quantum mechanics (QM) and QM was treated as OM. He advertized the two level
structuring of the description of microphenomena. We follow him and suggest a
special CTM for QM, so-called prequantum classical statistical field theory
(PCSFT) [39, 41, 42]. QM treated as OM can be considered as a PCSFT-image, but
not as straightforward as in Bell’s model with hidden variables [9, 10].

We analyze Bell’s model with hidden variables within the Bild-framework and
criticize identification of subquantum (hidden) quantities with quantum
observables and hidden probability distributions with quantum probability
distributions. The evident barrier for such identification is the Heisenberg
uncertainty principle (and the Bohr complementarity principle [15, 51, 52]).
The same viewpoint was presented long ago by De Broglie [27] (see also Lochak
[31, 32, 33]) who justified the legitimacy of his double solution theory [26,
8], in fact, within the Bild-conception (although it seems that he was not
aware about it). He pointed to inconsistency of the no-go interpretation of
the von Neumann [66] and Bell [9, 10] theorems. De Broglie double solution
model is an CTM for QM. (Its structuring within the Bild-conception deserves a
separate article as well as the Bild-conception presentation of Bohmian
mechanics.)111 This is the good place to note that one should not identify De
Broglie’s and Bohm’s theories, not consider the later as just an extension and
improvement of the former. De Broglie did not consider his theory as non-
local, he noted that its nonlocality is only apparent, this is nonlocality of
mathematical equations and not physical processes [26]. Deep foundational
studies on De Broglie’s double solution model are presented in works of
Bacciagaluppi,e.g., [6, 7], see also appendixes written by Bacciagaluppi and
Valentini in book [8].

We also use the Bild-approach for the two level CTM-OM modeling of Brownian
motion: the overdamped regime corresponds to OM [2]. Coarse gained velocities
are observable quantities. This example represents clearly the physical origin
of the two level structuring of the mathematical description of Brownian
motion. This is the time-scale separation technique. The evolution of the
momenta of the Brownian particles is very fast and cannot be resolved on the
time-scales available to the experiment. We notice that the OM model for the
Brownian motion shows some distinguished properties of QM, see, e.g., article
[2] for the corresponding uncertainty relations and Brownian entanglement
theory.

The idea of time-scale separations is one of the most pertinent ones in non-
equilibrium statistical physics. In a qualitative form it appears already in
good textbooks on this subject [46, 48], and has been since then formalized in
various contexts and on various levels of generality [47, 19, 50, 1, 64].

In classical mechanics CTM=OM; on the one hand, this is very convenient, on
the other hand, this exceptional coincidence blurred the general CTM-OM
structuring of scientific theories.

We also briefly discuss ontic-epistemic structuring of scientific theories
(Primas-Atmanspacher [55, 4], see also artciles [3, 5]) and its relation to
the Bild concept.

This paper is continuation of my works [41, 42]. I hope that in this paper the
Bild-conception and its implementation for quantum and classical mechanics are
presented clearer. Presentation of the two level CTM-OM description for
Brownian motion is a good complement to such description for quantum
phenomena. The CTM-OM viewpoint on the Bell inequality project clarifies the
difference in the positions of Schrödinger [57]-[63] and Bell [9, 10] on the
possibility to construct a subquantum model with hidden variables.

##  2 Two level structuring of scientific theories

We start by citing the article fo D’ Agostino [25]:

Hermann von Helmholtz (1821-1894) was one of the first scientists to criticize
the objective conception of physical theory by denying that theoretical
concepts describe real physical objects. He realized that Immanuel Kant’s a
priori forms of intuition should to be taken into account in analyzing
problems that were emerging at the end of the nineteenth century in the new
formulations of physics.

The objective conception of physical theory also was criticized by such
physicists as Heinrich Hertz (1857-1894) and Ludwig Boltzmann (1844-1906), who
adopted the Kantian term Bild 222Here the German word “Bild” (“picture”) is
used in the sense of a model. to designate the new conception of physical
theory, which they took to mean not a faithful image of nature but an
intellectual construct whose relationship to empirical phenomena was to be
analyzed.

The works of von Helmholtz, Hertz, and Boltzmann [65, 28, 29, 16, 17] played
the crucial role in development of a novel scientific methodology. Since the
time of Galileo and Newton, scientific theories varied essentially in their
content, but nobody questioned their “ontological significance”, their
adequacy to physical reality.

In 1878, von Helmholtz posed the following philosophical questions [65]:

What is true in our intuition and thought? In what sense do our
representations correspond to actuality?.

Von Helmholtz’ answers to these questions were based on his physiological,
especially visual, research that led him to the following conclusion [65]:

Inasmuch as the quality of our sensation gives us a report of what is peculiar
to the external influence by which it is excited, it may count as a symbol of
it, but not as an image. For from an image one requires some kind of alikeness
with the object of which it is an image … .  We point out that if the fathers
of QM would take this statement into account, then they surprise by “unusual
features” of the quantum mechanical description of micro-phenomena would not
be so strong. We also note that Bohr’s views on QM match with this conclusion
of Helmholtz. Surprisingly, it seems that Bohr had never referred to his
works.

Helmholtz’s viewpoint on interrelation of sensations and generally
observations and real objects led to the well known statement on the
parallelism of the laws of nature and science [65]:

Every law of nature asserts that upon preconditions alike in a certain
respect, there always follow consequences which are alike in a certain other
respect. Since like things are indicated in our world of sensations by like
signs, an equally regular sequence will also correspond in the domain of our
sensations to the sequence of like effects by the law of nature [that like
effects follow from] … like causes.

We point out that the statement  that upon preconditions alike in a certain
respect, there always follow consequences which are alike in a certain other
respect  is about ontic causality. So, for Helmholtz, nature is causal, i.e.,
laws in nature really exist and laws presented in scientific theories are
mental representations of laws of nature. The laws expressed by our sensation
and through them by our perception are “parallel” to natural laws, but only
parallel, not identical, since our mind operates not with precise images of
real objects, but only with symbols assigned to them.

Hertz questioned Helmholtz’s parallelism of laws. Hertz believed that
Helmholtz’s parallelism of laws not only was indeterminate but in general even
impossible if theory were limited to describing observable quantities [29]:

If we try to understand the motions of bodies around us, and to refer them to
simple and clear rules, paying attention only to what can be directly
observed, our attempt will in general fail. We soon become aware that the
totality of things visible and tangible do not form an universe conformable to
law, in which the same results always follow from the same conditions. We
become convinced that the manifold of the actual universe must be greater than
the manifold of the universe which is directly revealed to us by our senses.

By Hertz a causal theory cannot be based solely on observable quantities [29]
:  — do not form an universe conformable to law, in which the same results
always follow from the same conditions. Only by introducing hidden quantities
Helmholtz’s parallelism of laws can become a general principle in physical
theory. But such hidden quantities (concepts that correspond to no
perceptions) brings too much freedom in the choice of theoretical concepts. To
limit this freedom of choice, Hertz introduced special requirements for the
validation of a physical theory. Besides causality, the most important was
theory’s simplicity [29]:

It is true we cannot a priori demand from nature simplicity, nor can we judge
what in her opinion is simple. But with regard to images [Bilder] of our own
creation we can lay down requirements.We are justified in deciding that if our
images are well adapted to the things, the actual relations of the things must
be represented by simple relations between the images.

So, Helmholtz and Herz questioned the ontological status of scientific
theories, as describing reality as it is. Scientific theories are only
“Bilder”, models of reality. Outputs of sensations and observations are just
symbols encoding external phenomena. Hence, one should not sanctify
observational quantities and their role in scientific theories. Moreover, an
observational theory, i.e., operating with solely observables cannot be
causal. Causality demands introduction of hidden (unobservable) quantities. Of
course, a theory with hidden quantities should be coupled to observational
data. However, this coupling need not be straightforward.

According to Helmholtz a scientific theory should be causal. Hertz claimed
[29] that generally the causality constraint requires invention of hidden
quantities, a causal description cannot be done solely in terms of
observational quantities. This approach unties scientists’ hands, by
introducing hidden quantities they can generate a variety of theoretical
causal models coupled to the same observational quantities. How can one select
a “good” causal model? Hertz suggested to use model’s simplicity as a
criterion for such selection. We note that even a “good model” does not
describe reality as it is, it provides just a mathematical symbolic
representation involving a variety of elements having no direct relation with
observational quantities.

It is natural to search for such (causal) theoretical model that would
describe what nature really is, a “true model” (an ontic model). It is not
clear whether Hertz might hope to design such a model for the electromagnetic
phenomenon.333He tried to model it with systems of mechanical oscillators,
i.e., to go beyond the electromagnetic field representation [28]. But he did
not succeed with this project. His project was not meaningless. It has some
degree of similarity with the representation of the quantum electromagnetic
field as a system of quantum oscillators - photons. Schrödinger who later
contributed to development of the Bild concept of scientific theories,
especially in the relation to the quantum foundations claimed [58] that no
true model can be formulated on the basis of our large-scale experience,
because

we find nature behaving so entirely differently from what we observe in
visible and palpable bodies of our surroundings … . A completely satisfactory
model of this type is not only practically inaccessible, but not even
thinkable. Or, to be precise, we can, of course, think it, but however we
think it, it is wrong; not perhaps quite as meaningless as a “triangular
circle,” but much more so than a “winged lion.”

Creation of a causal theoretical model coupled to some observed natural
phenomena is a complex and long process. Moreover, there is always a chance
that such a model would be never found - due to intellectual incapacity of
humankind. Therefore it is natural to design models matching observations, but
not satisfying the causality constraint. We call such models observational
models.

Thus, we distinguish two classes of models, observational models (OMs) and
causal theoretical models (CTMs). We remark that both kinds of scientific
models are mental constructions, providing symbolic mathematical descriptions
of natural phenomena. One may say that any model is theoretical, so OM is also
theoretical. And he would be right. So, the main difference between OM and CTM
is in causality. If OM is causal by itself, then there is no need to go beyond
it with some CTM.

Interrelation between CTM and OM, ${\bf M}_{\rm{CTM}}$ and ${\bf
M}_{\rm{OM}},$ depends on the present stage of development of science. If
${\bf M}_{\rm{CTM}}$ rightly reflects the real physical processes, then
development of measurement technology can lead to novel observational
possibilities and some hidden quantities of ${\bf M}_{\rm{CTM}}$ can become
measurable. Hence, ${\bf M}_{\rm{CTM}}$ becomes OM, ${\bf M}_{\rm{CTM}}\to{\bf
M}_{\rm{OM}}^{\prime}.$ In principle, ${\bf M}_{\rm{OM}}^{\prime}$ need not
cover all observations described by the previous OM ${\bf M}_{\rm{OM}}.$ New
theoretical efforts might be needed to merge ${\bf M}_{\rm{OM}}$ and ${\bf
M}_{\rm{OM}}^{\prime}.$ This abstract discussion will be illustrated by the
concrete example from classical statistical physics - the two level modeling
of the Brownian motion (section 8.1).

The ideas of Helmholtz and Hertz were further developed (and modified) in the
works of Boltzmann [16, 17]. Then, 60 years later, Schrödinger [57]-[63]
contributed to development of the Bild viewpoint on quantum theories. He
confronted with the special case of the aforementioned problem.

OM for micro-phenomena was developed (in particular, due to his own efforts):
this is QM. But QM suffered from acausality. The impossibility to solve the
measurement problem (which was highlighted by von Neumann [66]) generates a
gap in the quantum description of micro-phenomena. Schrödinger came back to
this problem in 1950’s [57]\- [63]; this comeback was stimulated by
development of quantum field theory and the method of second quantization.

He saw in quantum field theory a possibility to justify his attempts of the
purely wave (continuous) approach to modeling of the micro-phenomena. In
complete agreement with the Bild concept, he considered QM as an observational
model. As well as von Neumann, Schrödinger highlighted its acausality. But it
was not treated as a property of nature as it is, i.e., quantum acausality (of
measurements and spontaneous quantum events) is not ontic. We notice that, for
von Neumann, it is ontic, he wrote about “irreducible quantum randomness”
[66]. Quantum acausality is just a property of special OM - QM. Schrödinger
claimed that quantum acausality is related to ignoring of the Bild concept and
assigning the ontological status to quantum particles, see his article “What
is an elementary particle?” [57]. We remark that Bohr did not question the
ontological status of quantum systems, atoms, electrons and may be even
photons [15, 51, 52]. Schrödinger considered indistinguishability of quantum
particles as a sign that they do not have the ontological status. Hence,
instead of OM (= QM), one can hope to develop CTM for microphenomena, by
liberating it from particles and operating solely with waves.

Since waves propagate in space, for Schrödinger causality (in fact, the wave
causality) is coupled to continuity in the space, so the waves should be
continuous (see Plotnitsky [54] on analysis of continuity vs. discontinuity in
physics). We remark that he considered continuity of waves on multi-
dimensional space $\mathbb{R}^{3n}.$ In 1920s the fact that the multi-particle
Schrödinger equation describes the waves not on “the physical space”
$\mathbb{R}^{3},$ but on “the mathematical space” $\mathbb{R}^{3n},$ was
disturbing for him. This was the main reason for Schrödinger to accept the
probabilistic interpretation of the wave function. At that time he did not use
the Bild concept for scientific theories (was not aware about the works of
Helmholtz, Herz, and Boltzmann?). By the Bild concept the wave representation
of QM is just a symbolic mathematical representation of the micro-phenomena.
The use of the multi-dimensional space $\mathbb{R}^{3n}$ has the same
descriptive status as the use of $\mathbb{R}^{3}.$

Schrödinger dreamed for creation of CTM for micro-phenomena, his concrete
intention was towards a wave-type model. He also highlighted the principle fo
continuity for “quantum waves”, but he suspected that it would be valid only
at the microlevel. He pointed to quantum field theory as a good candidate to
proceed in this direction. Since he coupled causality and continuity, it
became possible to relax the causality-continuity constraint and restrict this
constraint to the level of infinitesimals. In a theoretical model completing
QM (an observational model) for which Schrödinger dreamed, causality need not
be global.

Schrödinger’s continuous wave completion project for QM has some degree of
similarity with Einstein’s project on designing a classical field model of
micro-phenomena which he announced with Infeld in a popular form in book
[36].444Einstein’s intention was that a complete theory beyond QM should be
non-linear field theory. Later Infeld contributed a lot into this project. In
contrast tyo Einstein, Schrödinger dreamed for a linear model. However, in
contrast to Schrödinger, Einstein did not appeal to the Bild concept on the
two level modeling of natural phenomena, observational and causal theoretical
(OM and CTM), and a possible gap between these two models. The presence of
such gap, in particular, implies that CTM need not describe the observational
data straightforwardly.

Einstein’s project on reconsideration of quantum foundations starting with the
EPR-paper [35] was not directed to the two level structuring of the
mathematical description of microphenomena. He dreamed for CTM which would
match perfectly with quantum observations. This dream was later formalized by
Bell in his hidden variables model [9, 10].

Schrödinger understood [58] that CTM of microphenomena of the wave type is not
the observed or observable facts; and still less do we claim that we thus
describe what nature (matter, radiation, etc.) really is. In fact we use this
picture (the so-called wave picture) in full knowledge that it is neither.
This statement expresses the extreme view on the Bild concept; Schrödinger
[58] also pointed out that  observed facts … appear to be repugnant to the
classical ideal of a continuous description in space and time. Such
highlighting of decoupling of theory and observations was too provocative and
played the negative role. The idea of using the Bild concept in quantum
foundations was rejected by the majority of experts in quantum foundations.

However, the Bild concept did not disappear completely and its trace can be
found in the philosophy of the ontic-epistemic structuring of physical
theories that was developed by Primas and Atmanspacher [4] (see also, e.g.,
[3, 5]). They tried to find an answer [3] to the old question:  Can nature be
observed and described as it is in itself independent of those who observe and
describe - that is to say, nature as it is “when nobody looks”?

As well as Helmholtz, Hertz, Boltzmann, and Schrödinger, they pointed out that
observations give to observers only some knowledge about systems, this
knowledge is incomplete. This knowledge is mathematically structured within an
epistemic (=observational) model . For them, QM is such a model, i.e., w.r.t.
QM the views of Schrödinger and Primas-Atmanspacher coincide. Then, in the
same way as Schrödinger, they want to have a complete model of microphenomena.
The crucial difference from the Bild concept is that Primas and Atmanspacher
were seeking for an ontic model, a model of reality as it is, the “true model”
in terms of Schrödinger. Generally Primas and Atmanspacher also supported the
idea of the two level structure of scientific theories: epistemic
(observational) and ontic. As well as Schrödinger, they pointed out that the
connection between epistemic and ontic models is not straightforward.
Causality is the basic property of the ontic model. So, if one would ignore
the term “ontic” 555Its use would be very disturbing for Helmholtz, Hertz,
Boltzmann, and Schrödinger., then formally (and mathematically) Primas-
Atmanspacher structuring of the scientific description of nature is similar to
the Bild concept. (In contrast to Schrödinger, they did not emphasize the
continuous wave structure of an ontic model beyond QM.)

However, by pointing to formal mathematical similarity of the ontic-epistemic
and Bild approaches, one should remember that they differ crucially from the
foundational perspective. We recall [3] that  Ontological questions refer to
the structure and behavior of a system as such, whereas epistemological
questions refer to the knowledge of information gathering and using systems,
such as human beings.

From the Bild perspective, it is totally meaningless even to  refer to the
structure and behavior of a system as such …  The essence of the ontic-
epistemic approach is expressed in the following quote from Atmanspacher [3]
(for more details, the reader is referred to Primas [55] ):

Ontic states describe all properties of a physical system exhaustively.
(“Exhaustive” in this context means that an ontic state is “precisely the way
it is”, without any reference to epistemic knowledge or ignorance.) Ontic
states are the referents of individual descriptions, the properties of the
system are treated as intrinsic properties.Their temporal evolution (dynamics)
is reversible and follows universal, deterministic laws. As a rule, ontic
states in this sense are empirically inaccessible. Epistemic states describe
our (usually non-exhaustive) knowledge of the proper- ties of a physical
system, i.e. based on a finite partition of the relevant phase space. The
referents of statistical descriptions are epistemic states, the properties of
the system are treated as con- textual properties. Their temporal evolution
(dynamics) typically follows phenomenological, irreversible laws. Epistemic
states are, at least in principle, empirically accessible

From the Bild perspective, the statement:  Ontic states are the referents of
individual descriptions, the properties of the system are treated as intrinsic
properties, is meaningless, since systems do not have intrinsic properties, a
theoretical causal model beyond the quantum observational (epistemic) model
still describes not the properties of the systems, but our mental pictures.

And we conclude this section by the quote from Nietzsche (written in 1873, but
published later); his statement is very similar similar to Helmholtz’s
statements, but it is more passionate or even poetic! It seems that Nietzsche
was influenced by Helmholtz, especially on nerve stimulus. Nietzsche wrote
about language, but the point is more general [34]666I would like to thank
Arkady Plotnitsky for mentioning this quote in our discussion on the works of
Helmholtz, Hertz, Boltzmann, and Schrödinger and especially for sending to me
this reference to Nietzsche.:

The various languages placed side by side show that with words it is never a
question of truth, never a question of adequate expression; otherwise, there
would not be so many languages. The “thing in itself” (which is precisely what
the pure truth, apart from any of its consequences, would be) is likewise
something quite incomprehensible to the creator of language and something not
in the least worth striving for. This creator only designates the relations of
things to men, and for expressing these relations he lays hold of the boldest
metaphors. To begin with, a nerve stimulus is transferred into an image: first
metaphor. The image, in turn, is imitated in a sound: second metaphor. And
each time there is a complete overleaping of one sphere, right into the middle
of an entirely new and different one. One can imagine a man who is totally
deaf and has never had a sensation of sound and music. Perhaps such a person
will gaze with astonishment at Chladni’s sound figures; perhaps he will
discover their causes in the vibrations of the string and will now swear that
he must know what men mean by “sound.” It is this way with all of us
concerning language; we believe that we know something about the things
themselves when we speak of trees, colors, snow, and flowers; and yet we
possess nothing but metaphors for things-metaphors which correspond in no way
to the original entities. In the same way that the sound appears as a sand
figure, so the mysterious of the thing in itself first appears as a nerve
stimulus, then as an image, and finally as a sound. Thus the genesis of
language does not proceed logically in any case, and all the material within
and with which the man of truth, the scientist, and the philosopher later work
and build, if not derived from never-never land, is a least not derived from
the essence of things.

##  3 Coupling of theoretical and observational models

Models considered in natural science are mainly mathematical. Therefore
coupling between CTM and OM corresponding to the same natural phenomena is a
mapping of one mathematical structure to another.

Consider some mathematical model ${\bf M},$ either CTM or OM. It is typically
based on two spaces, the space of states $S$ and the space of variables
(quantities) $V.$ For OM, $V$ is the space of observables, instead of states
one can consider measurement contexts.

Consider OM model ${\bf M}_{OM}$ and its causal theoretical completion ${\bf
M}_{CTM}.$ It is natural to have a mathematical rule establishing
correspondence between them. We recall that CTMs are causal and OMs are often
acausal; if it happens that OM is causal, then there is no need for a finer
description given by some CTM. Thus, the task is to establish correspondence
between causal and acausal models. It is clear that such correspondence cannot
be straightforward. We cannot map directly states from $S_{CTM}$ to states
from $S_{OM}.$ Causality can be transformed into acausality through
consideration of probability distributions. So, consider some space of
probability distributions $P_{CTM}$ on the state space $S_{CTM}$ and construct
a map from $P_{CTM}$ to $S_{OM},$ the state space of OM. This approach
immediately implies that the states of OM are interpreted statistically. We
also should establish correspondence between variables (quantities) of ${\bf
M}_{CTM}$ and ${\bf M}_{OM}.$ Thus, we need to define two physically natural
maps:

$J_{S}:P_{CTM}\to S_{OM},\;J_{V}:V_{CTM}\to V_{OM}.$ (1)

Since $J_{S}$ is not defined for states of CTM, but only for probability
distributions, “physically natural” means coupling between the probability
structures of ${\bf M}_{CTM}$ and ${\bf M}_{OM};$ the minimal coupling is the
equality of averages between variables

$\langle J_{V}(f)\rangle_{J_{S}(P)}=\langle f\rangle_{P}$ (2)

and correlations

$\langle J_{V}(f)J_{V}(g)\rangle_{J_{S}(P)}=\langle fg\rangle_{P}.$ (3)

Generally the correlation need not be defined, so (3) should hold for
variables $f,g\in V_{CTM}$ and observables $A_{f}=J_{V}(f)$ and
$A_{g}=J_{V}(g)$ for which the correlations in the states $P$ and $J_{S}(P)$
are defined.

Mathematically causality can be realized as functional representation of
variables (see monograph of Wagner [69] on such representation of causality).
Therefore we assume that $V_{CTM}$ can be represented as a space of functions
$f:S_{CTM}\to\mathbb{R}.$ Such model is causal, the state $\phi$ uniquely
determines the values of all variables belonging $V_{CTM}:\phi\to f(\phi).$
The state space $S_{CTM}$ can be endowed with a $\sigma$-algebra of subsets
${\cal F}.$ Elements of $P_{CTM}$ are probability measures on ${\cal F}.$ The
minimal mathematical restriction on elements of $V_{CTM}$ is that they are
measurable functions, $f:S_{CTM}\to\mathbb{R}.$ In such a framework,

$\langle f\rangle_{P}=\int_{S_{CTM}}f(\lambda)P(d\lambda),\langle
fg\rangle_{P}=\int_{S_{CTM}}f(\lambda)g(\lambda)P(d\lambda),$ (4)

if the integrals exist, e.g., if CTM-variables are square integrable:

$\int_{S_{CTM}}|f(\lambda)|^{2}P(d\lambda)<\infty.$

Since in ${\bf M}_{OM}$ quantities have the experimental statistical
verification, we establish some degree of experimental verification for ${\bf
M}_{CTM}$ through mapping of ${\bf M}_{CTM}$ to ${\bf M}_{OM}.$ But such
verification is only indirect, one should not expect direct coupling between
quantities of ${\bf M}_{CTM}$ and experiment (as Einstein, Bell and all their
followers wanted to get). Generally these maps are neither one-to-one nor
onto.

  * •

A cluster of probability distributions on $S_{CTM}$ can be mapped into the
same state from $S_{OM}.$

  * •

$J_{S}(P_{CTM})$ need not coincide with $S_{OM}.$

  * •

A cluster of elements of $V_{CTM}$ can be mapped into a single variable
(observable) from $V_{OM}.$

  * •

$J_{V}(V_{CTM})$ need not coincide with $V_{OM}.$

Moreover, the model-correspondence maps $J_{S},J_{V}$ need not be defined on
whole spaces $P_{CTM}$ and $V_{CTM}.$ They have their domains of definition,
${\cal D}_{J_{S}}\subset P_{CTM}$ and ${\cal D}_{J_{V}}\subset V_{CTM}.$ (In
principle, one can reduce $P_{CTM}$ to $P_{CTM}^{\prime}={\cal D}_{J_{S}}$ and
$V_{CTM}$ to $V_{CTM}^{\prime}={\cal D}_{J_{V}}$ and operate with maps
$J_{S},J_{V}$ which are defined everywhere on these reduced spaces of CTM’s
states and variables).

We remark that the same ${\bf M}_{OM}$ can be coupled to a variety of CTMs. We
also remark that the same observational data can be mathematically described
by a variety of OMs.

We also remark that similarly to the deformation quantization (here we discuss
just the mathematical similarity) CTM may depend on some small parameter
$\kappa$ (in the deformation quantization this is action, roughly speaking the
Planck constant $h).$ Thus, ${\bf M}_{CTM}={\bf M}_{CTM}(\kappa).$ In such
more general framework, the correspondence maps also depend on $\kappa,$ i.e.,
$J_{S}=J_{S}(\kappa),J_{V}=J_{V}(\kappa).$ The probabilistic coupling
constraints (2), (3) can be weakened:

$\langle J_{V}(\kappa;f)\rangle_{J_{S}(\kappa;P)}=\langle
f\rangle_{P}+o(\kappa),\kappa\to 0,$ (5) $\langle
J_{V}(\kappa;f)J_{V}(\kappa;g)\rangle_{J_{S}(\kappa;P)}=\langle
fg\rangle_{P}+o(\kappa),\kappa\to 0$ (6)

(see ). The problem of identification of the parameter $\kappa$ with some
physical scale is complex (see, e.g., [37, 38] for an attempt of such
identification within PCSFT).

##  4 Prequantum classical statistical field theory as a causal theoretical
model for quantum mechanics

We illustrate the general scheme of CTM-OM correspondence by two theories of
micro-phenomena, QM as ${\bf M}_{OM}$ and PCSFT as ${\bf M}_{CTM}.$ Re-denote
these model with the symbols ${\bf M}_{\rm{QM}}$ and ${\bf M}_{\rm{PCSFT}}.$
We briefly recall the basic elements of PCSFT (see [39, 41, 42] for details).

In ${\bf M}_{QM}$ states are given by density operators acting in complex
Hilbert space ${\cal H}$ (with scalar product $\langle\cdot|\cdot\rangle)$ and
observables are represented by Hermitian operators in ${\cal H}.$ Denote the
space of density operators by $S_{\rm{QM}}$ and the space of Hermitian
operators by $V_{\rm{QM}}.$

In ${\bf M}_{\rm{PCSFT}}$ states are vectors of ${\cal H},$ i.e.,
$S_{\rm{PCSFT}}={\cal H}.$ Physical variables are quadratic forms

$\phi\to f(\phi)=\langle\phi|\hat{A}|\phi\rangle,$

where $\hat{A}\equiv\hat{A}_{f}$ is a Hermitian operator. The space of
quadratic forms is denoted by the symbol $V_{\rm{PCSFT}}.$ Consider
probability measures on the $\sigma$-algebra of Borel subsets of ${\cal H}$
(i.e., generated by balls in this space) having zero first momentum, i.e.,

$\int_{{\cal H}}\langle\phi|a\rangle dp(\phi)=0$ (7)

for any vestor $a\in H,$ and finite second momentum, i.e.,

${\cal E}_{p}\equiv\int_{{\cal H}}\|\phi\|^{2}dp(\phi)<\infty.$ (8)

Denote the space of such probability measures by the symbol $P_{\rm{PCSFT}}.$

We can start not with probability measures, but with ${\cal H}$-valued random
vectors with zero mean value and finite second moment: $\phi=\phi(\omega),$
such that $E[\phi]=0$ and $E[\|\phi\|^{2}]<\infty.$777Random vectors are
defined on some Kolmogorov probability space $(\Omega,{\cal F},P),$ these are
functions $\phi:\Omega\to{\cal H}$ which are measurable w.r.t. to the Borel
$\sigma$-algebra of ${\cal H},$ i.e., for any Borel subset $B$ of ${\cal H},$
$\phi^{-1}(B)\in{\cal F}.$ A map is measurable iff, for any $c>0,$ the set
$\Omega_{\phi,c}=\\{\omega\in\Omega:||\phi(\omega)||<c\\}\in{\cal F}.$ The
space of such random vectors is denoted by the symbol $R_{\rm{PCSFT}}.$ In the
finite-dimensional case, these are complex vector-valued random variables; if
${\cal H}$ is infinite-dimensional, then the elements of $R_{\rm{PCSFT}}$ are
random fields.

An example of random fields is given by selection ${\cal
H}=L_{2}(\mathbb{R}^{n};\mathbb{C})$ of square integrable complex valued
functions. Each ${\bf M}_{CTM}$ state $\phi$ is an $L_{2}$-function,
$\phi:\mathbb{R}^{n}\mapsto\mathbb{C}.$ Random fields belonging to
$R_{\rm{PCSFT}}$ are functions of two variables, $\phi=\phi(x;\omega):$ chance
parameter $\omega$ and space coordinates $x.$

We remark that, for the state space ${\cal
H}=L_{2}(\mathbb{R}^{n};\mathbb{C}),$ the quantity ${\cal E}_{p}$ can be
represented as

${\cal E}_{p}=\int_{{\cal H}}{\cal E}(\phi)dp(\phi),$

where

${\cal E}(\phi)=\|\phi\|^{2}=\int_{\mathbb{R}^{n}}|\phi(x)|^{2}dx$

is the energy of the field. The quantity ${\cal E}_{p}$ can be interpreted as
the average of the field energy with respect to the probability distribution
$p$ on the space of fields. We can also use the random field representation.
Let $\phi=\phi(x;\omega)$ be a random field. Then its energy is the random
variable

${\cal E}_{\phi}(\omega)=\int_{\mathbb{R}^{n}}|\phi(x;\omega)|^{2}dx$

and ${\cal E}_{p}$ is its average.

For any $p\in P_{\rm{PCSFT}},$ its (complex) covariance operator $\hat{B}_{p}$
is defined by its bilinear (Hermitian) form:

$\langle a|\hat{B}_{p}|b\rangle=\int_{{\cal H}}\;\langle
a|\phi\rangle\langle\phi|b\rangle\;dp(\phi),\;a,b\in{\cal H},$ (9)

or, for a random field $\phi,$ we have:

$\langle a|\hat{B}_{\phi}|b\rangle=E[\langle
a|\phi\rangle\langle\phi|b\rangle].$

We note that

${\cal E}_{p}=\int_{{\cal H}}\|\phi\|^{2}dp(\phi)=\rm{Tr}\hat{B}_{p}$ (10)

or in terms of a random field:

${\cal
E}_{p}=E[||\phi||^{2}]=E[\int_{\mathbb{R}^{n}}|\phi(x;\omega)|^{2}dx]=\rm{Tr}\hat{B}_{p}.$
(11)

Thus, the average energy of a random field $\phi=\phi(\omega,x)$ can be
expressed via its covariance operator.

Generally a probability measure (${\cal H}$-valued random variable ) is not
determined by its covariance operator (even under the constraint given by zero
average).

A complex covariance operator has the same properties as a density operator,
besides normalization by the trace one; a covariance operator $\hat{B}_{p}$ is

  * •

Hermitian,

  * •

positively semidefinite,

  * •

trace class.

A “physically natural coupling” of the models ${\bf M}_{\rm{QM}}$ and ${\bf
M}_{\rm{PCSFT}}$ is based on the following formula coupling mathematically the
averages for these models. For a probability measure $p\in P_{\rm{PCSFT}}$ and
a variable $f\in V_{\rm{PCSFT}},$ we have

$\langle f\rangle_{p}=\int_{{\cal
H}}f(\phi)dp(\phi)=\rm{Tr}\hat{A}_{f}\hat{B}_{p},$ (12)

where $f(\phi)=\langle\phi|\hat{A}_{f}|\phi\rangle.$ This formula is obtained
through expansion of the quadratic form $\langle\phi|\hat{A}_{f}|\phi\rangle$
w.r.t. the basis of eigenvectors of the Hermitian operator $\hat{A}_{f}.$

Let us consider the following maps $J_{S}:P_{\rm{PCSFT}}\to S_{\rm{QM}}$ and
$J_{V}:V_{\rm{PCSFT}}\to V_{\rm{QM}}$,

$J_{S}(p)=\hat{\rho}_{p}=\hat{B}_{p}/\rm{Tr}B_{p},\;J_{V}(f)=\hat{A}_{f}.$
(13)

This correspondence connects the averages given by the causal theoretical and
observational models:

$\frac{1}{{\cal E}_{p}}\langle f\rangle_{p}=\rm{Tr}\hat{\rho}_{p}\hat{A}_{f},$
(14)

i.e., the QM and PCSFT averages are coupled with the scaling factor which is
equal to the inverse of the average energy of the random field (for ${\cal
H}=L_{2}).$

Thus, density operators representing quantum states correspond to covariance
operators of random fields normalized by the average energy of a random field
and the Hermitian operators representing quantum observables correspond to
quadratic forms of fields.

Let us rewrite (14) in the form:

$\langle\frac{f}{{\cal E}_{p}}\rangle_{p}=\rm{Tr}\hat{\rho}_{p}\hat{A}_{f}.$

If random fields have low energy, i.e., ${\cal E}_{p}<<1,$ the quantity

$g_{p}(\phi)\equiv\frac{f(\phi)}{{\cal E}_{p}}$

can be interpreted as an amplification of the PCSFT physical variable $f.$
Hence, by connecting QM with PCSFT, QM can be interpreted as an observational
theory describing averages of amplified ‘subquantum’ physical variables -
quadratic forms of random fields. The subquantum random fields are
unobservable and they can be experimental verified only indirectly, via
coupling with the observational model - QM.

In contrast to QM, PCSFT is causal: selection of a vector (‘field’)
$\phi\in{\cal H}$ determines the values of all PCSFT-variables, quadratic
forms of classical fields: $\phi\to\langle\phi|\hat{A}|\phi\rangle.$

For physical variables, the correspondence map $J_{V}$ is one-to-one, but the
map $J_{S}$ is not one-to-one. But it is a surjecion, i.e., it is on-to map.

##  5 On usefulness of causal theoretical models

The above presentation of the possible two level description of the
microphenomena, QM vs. PCSFT, can be used as the initial point for the
discussion on usefulness of CTMs. To be provocative, we start by noting that
for Bohr and other fellows of the orthodox Copenhagen interpretation of QM
attempts to construct CTM for QM is meaningless [14, 15, 51, 52]. At the same
time Bohr never claimed that such CTM can’t be constructed [51, 52]; he was
not interested in no-go theorems. In his writings I did not find any word
about the von Neumann no-go theorem. I am sure that he would ignore the Bell
no-go theorem [10] and be surprised by interest to it in the modern quantum
foundational community. Bohr highlighted the observational status of QM, but
for him any kind of CTM is a metaphysical. For “real physicist”, it is
meaningless to spend time by trying to design prequantum CTM. This position is
very common among “real physicists”.888 At the beginning of this century
academician Kamil Akhmetovich Valiev (one of the engines of the quantum
computer project in Russia) told me: “Andrei, why do you consume your days for
such totally meaningless activity? You are talented and can contribute so much
to real physics - quantum computing!” He pointed to my attempts to find a
prequantum CTM. For Bohr, it is impossible to complete QM in the causal way by
operating with quantities which have direct connection to observations. And he
is completely right: the complementarity principle and Heisenberg uncertainty
relation block searching of a finer OM for QM. It seems that in contrast to
von Neumann, Bohr was not disturbed by acausality, observational acausality is
a consequence of contextuality and complementarity of quantum observations. We
also repeat that Bohr did not deny the possibility to construct CTMs beyond
QM, but for him the introduction of hidden variables was a metaphysical and
totally meaningless exercise.

Bohr’s position can be questioned and on questioners’ side are Helmholtz,
Herz, Boltzmann, Schrödinger. As we have seen, Schrödinger agreed with Bohr
that QM is a good OM for microphenomena. He did not think that acausal
structure of QM prevents construction of corresponding CTM. For him causality
is closely coupled to continuity and hence to his original wave approach to
microphenomena.

As Helmholtz, Herz, Boltzmann, Schrödinger, I think that consideration of
acusality as the property of nature (at least at the micro-level) destroys
completely the methodology of science. If Helmholtz did mistake by saying [65]
that

every law of nature asserts that upon preconditions alike in a certain
respect, there always follow consequences which are alike in a certain other
respect,  then physics becomes a science about gambling (as, e.g., QBists
claim). It is difficult (at least for me) to accept this position. Thus, the
main impact of creation of CTMs is reestablishing of causality that might be
violated in OM.

Now we turn to QM. Reestablishing of causality of microphenomena (even without
the direct coupling to observations) would demystify quantum theory. We do not
claim that PCSFT is the “true CTM” for QM; as Schrödinger claimed [58] it is
meaningless and even dangerous for science development to search for such a
model. But PCSFT can be used as a causal Bild of quantum processes. One of the
main advantages of this Bild is that it is local. PCSFT reproduces not only QM
averages, but event its correlations [39]; hence, the Bell inequalities can be
violated for PCSFT-variables (hidden variables from the observational
viewpoint). PCSFT demystifies quantum entanglement by connecting it with
correlations of subquantum (classical) random fields (cf. [39, 41, 42]). PCSFT
can be considered as a step towards merging of QM with general relativity, but
within some CTM.

Can one earn from CTM something that might be lifted to the observational
level? In our concrete case, can some theoretical elements of PCSFT be
realized experimentally (may be in future)? The basic element of PCSFT is a
random field $\phi=\phi(x;\omega).$ Measurement of such a subquantum field
would be the real success of PCFT. However, it seems that one cannot expect
this. As was pointed out by Bohr (in 1930s), even the classical
electromagnetic field cannot be measured in a fixed point.

Another component of PCSFT which can be connected with real physics is the
need in the background field. This component was not discussed in the above
brief presentation, so see [39, 40] for details. Such random background field
$\phi_{\rm{bground}}(x,\omega)$ is the necessary element of the mathematical
model ${\bf M}_{\rm{PCSFT}}$ for generation of entangled states in
$S_{\rm{QM}}.$ In this way PCSFT is related to stochastic electrodynamics and
supports it. Unfortunately, the background (zero point field) is not a
component of conventional QM; stochastic electrodynamics is commonly
considered as unconventional model of microphenomena. From the Bild-viewpoint
this model should be treated as one of possible CTM for QM, this viewpoint
would clarify the interrelation between these two models. But in this paper we
do not plan to go deeper in this issue. We note the background field carries
long-distance correlations which contribute into violation of the Bell type
inequalities. However, these are classical field correlations having nothing
to do with the spooky action at a distance.

The most close to experimental verification is PCSFT representation of Born’s
rule as an approximate rule for calculation of probabilities, the standard
Born rule is perturbed by additional terms which can be in principle verified
(see [39] and especially article [40] suggesting the concrete experimental
test). By totally different reason this prediction was tested by the research
group of prof. Weihs [67, 68] \- testing Sorkin’s inequality in the triple
slit experiment. Surprisingly, transition from two slits to three slits is not
trivial and Weih’s group confronted difficulties related to nonlinearity of
detection processes. For the moment, no deviations from the Born rule were
observed.

For me, the main message of PCSFT as one of possible CTMs for QM is that
“quantum nonlocality” is an artifact of OM (=QM). The presence of such
artifacts in OMs is natural from the Bild-viewpoint. This is one of the
reasons to construct CYMs.

##  6 Bell’s project from the Bild-viewpoint

Unfortunately, Bell (as well as other ikons of modern quantum foundational
studies) did not read the works on the Bild concept for scientific theories.
By introducing hidden variables he suggested some CTM for OM=QM. However, he
considered too naive coupling of his CTM, ${\bf M}_{\rm{Bell}},$ with OM -
${\bf M}_{\rm{QM}}.$ The subquantum quantities, functions of hidden variables,
$A=A(\lambda),$ were identified with quantum observables. In particular the
ranges of values of quantities from ${\bf M}_{\rm{Bell}}$ coincide with the
ranges of values of quantum observables (and this is not the case e.g. in
PCSFT). As was pointed out in article [43], ${\bf M}_{\rm{Bell}},$ confronts
the complementarity principle. The later point will be clarified below.

We recall the mathematical structure of ${\bf M}_{\rm{Bell}}$ by connecting it
with the framework of section 3. Bell considered [10] an arbitrary set of
hidden variables $\Lambda;$ this is the set of states of his CTM, i.e.,
$S_{Bell}=\Lambda.$ To put this model in the mathematical framework of
probability theory, $\Lambda$ should be endowed with some $\sigma$-algebra of
its subsets, say ${\cal F}.$ Denote by $P_{Bell}$ the space of all probability
measures on $(\Lambda,{\cal F}).$ The space of Bell-variables consists of all
measurable functions $A:\Lambda\to\mathbb{R},A=A(\lambda),$ i.e., random
variables in terms of the Kolmogorov probability theory. We stress that the
correspondence map $J_{S}:P_{Bell}\to S_{QM}$ is not specified, it is just
assumed that such map exists. For Bell’s reasoning [10], this map need not be
onto $S_{QM}$ (so it need not be a surjection).

To make his model with hidden variables straightforwardly experimentally
verifiable, Bell identified the values of CTM quantities $A=A(\lambda)$ with
the values of QM quantities, the outcomes of quantum observables. First we
discuss the mathematical side of this assumption and then its foundational
side.

Mathematically identification of quantities of ${\bf M}_{\rm{Bell}}$ with QM
observables means that the range of values of $A\in V_{Bell}$ coincides with
the spectrum of the corresponding Hermitian operator $\hat{A}.$ This is the
important mathematical constraint on the map $J_{V}:V_{Bell}\to V_{QM}$ (we
recall that $V_{QM}$ is the set of density operators). Purely mathematical
relaxation of this assumption destroys the Bell inequality argument, e.g., as
in PCSFT.

However, Bell should proceed with this assumption on the coincidence of ranges
of values of the subquantum quantities and quantum observables, since he
dreamed for straightforward experimental verification of his model with hidden
variables [9, 10]. He was not accustomed with the Bild concept of a scientific
theory. In particular, Herz’s (and Schrödinger’s) statement on hidden
quantities which could not be observed directly was totally foreign for Bell.
For him, as well as for Bohr, a theory which quantities can’t be directly
verified is a part of metaphysics, not physics [14, 15, 51, 52].

However, by identifying the outcomes of subquantum quantities with the
outcomes of quantum observables Bell confronts the complementarity principle.
This can be clearly seen in the CHSH-framework [18]. There are two pairs of
observables: $(A_{1},A_{2}),$ in “Alice’s lab”, and $(B_{1},B_{2}),$ in “Bob’s
lab” represented by Hermitian operators $(\hat{A}_{1},\hat{A}_{2})$ and
$(\hat{B}_{1},\hat{B}_{2}).$ Observables corresponding to cross measurements
for Alice-Bob are compatible, i.e., they can be jointly measurable, but local
observables of Alice as well as of Bob are incompatible, i.e., they cannot be
jointly measurable. In the operator terms,

$[\hat{A}_{i},\hat{B}_{j}]=0,\;[\hat{A}_{1},\hat{A}_{2}]\not=0,\;[\hat{B}_{1},\hat{B}_{2}]\not=0.$
(15)

This is the quantum mechanical description of the CHSH experimental context.
We note that if local observables are compatible for at least one lab (in the
operator terms at least one of commutators
$[\hat{A}_{1},\hat{A}_{2}],[\hat{B}_{1},\hat{B}_{2}]$ equals to zero), then
the CHSH inequality can’t be violated [43]. Bell considered variables of his
CTM ${\bf M}_{\rm{Bell}}$ as representing physical observables, hence all
observables can be represented as functions
$A_{i}=A_{i}(\lambda),B_{j}=A_{i}(\lambda)$ and their values are identified
with outcomes of observations. Besides the pairs
$(A_{i}(\lambda),B_{j}(\lambda))$ of compatible observables, one can consider
the pairs $(A_{i}(\lambda),A_{j}(\lambda))$ and
$(B_{i}(\lambda),B_{j}(\lambda)).$ By treating the later two pairs as
representing the outcomes of physical observables it is natural to assume the
possibility of their joint measurability, may be not nowadays, but in future,
when the measurement technologies would be improved. So, the complementarity
principle loses its fundamental value. By keeping Bell’s model ${\bf
M}_{\rm{Bell}}$ as representing physical reality, one confronts with treating
of complementarity as the fundamental property of (observational)
microphenomena.

At the level of correlations,

$\langle
A_{i}B_{j}\rangle_{\rho}=\rm{Tr}\hat{\rho}\hat{A}_{i}\hat{B}_{j}=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}A_{ik}B_{jk},$
(16)

where $(A_{ik}),(B_{jk})$ are observables’ outcomes. At the same time, for the
probability distribution $P_{\rho}$ such that $\hat{\rho}=J_{S}(P_{\rho}),$ we
have

$\langle
A_{i}B_{j}\rangle_{P_{\rho}}=\int_{\Lambda}A_{i}(\lambda)B_{j}(\lambda)P_{\rho}(d\lambda)=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}A^{\prime}_{ik}B^{\prime}_{jk}$
(17)

where $A^{\prime}_{ik}$ and $B^{\prime}_{jk}$ are outcomes of random variables
$A_{i}=A_{i}(\lambda)$ and $B_{j}=B_{j}(\lambda).$ However, since these
outcomes can be identified with the outcomes of the quantum observables,
$A^{\prime}_{ik}=A_{ik},B^{\prime}_{jk}=B_{jk},$ we can write

$\langle
A_{i}B_{j}\rangle_{P_{\rho}}=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}A_{ik}B_{jk}.$
(18)

But the same reasoning is applicable to the subquantum random variables
$A_{1}=A_{1}(\lambda),A_{2}=A_{2}(\lambda)$ and
$B_{1}=B_{1}(\lambda),B_{2}=B_{2}(\lambda)$ representing the incompatible
quantum observables:

$\langle
A_{1}A_{2}\rangle_{P_{\rho}}=\int_{\Lambda}A_{1}(\lambda)A_{2}(\lambda)P_{\rho}(d\lambda)==\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}A^{\prime}_{1k}A^{\prime}_{2k},$
(19) $\langle
B_{1}B_{2}\rangle_{P_{\rho}}=\int_{\Lambda}B_{1}(\lambda)B_{2}(\lambda)P_{\rho}(d\lambda)=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}B^{\prime}_{1k}B^{\prime}_{2k}.$
(20)

Again by identifying the values of subquantum and quantum observables, we
obrtain:

$\langle
A_{1}A_{2}\rangle_{P_{\rho}}=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}A_{1k}A_{2k},$
(21) $\langle
B_{1}B_{2}\rangle_{P_{\rho}}=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}B_{1k}B_{2k}.$
(22)

Representations (21), (22) of subquantum correlations (within CTM ${\bf
M}_{\rm{Bell}})$ via outcomes of observables supports the assertion that the
subquantum correlations $\langle A_{1}A_{2}\rangle_{P_{\rho}},\langle
B_{1}B_{2}\rangle_{P_{\rho}}$ should be measurable (at least in principle and
in future). It is not clear how Bell would treat this objection to his
argument. I guess that he would agree that his model with hidden variables,
${\bf M}_{\rm{Bell}},$ collides with the complementarity principle. However,
he might choose to move between Scylla and Charybdis:

  * •

S identification the values of subquantum random variables with the values of
quantum observables;

  * •

Ch: the complementarity principle,

and claim that S and Ch can peacefully coexist. He might say that (18) is
legal and hence the experimental verification of ${\bf M}_{\rm{Bell}}$ is
possible, but (21), (22) are illegal and treatment of these correlations as
experimentally verifiable is forbidden. For me the later position is
inconsistent (although logically possible). This inconsistency was the basis
of De Broglie’s critique of Bell’s argument [27] (section 7).

This is the good place to recall that the physical seed of the complementarity
principle is in Bohr’s quantum postulate on the existence of indivisible
quantum of action given by the Planck constant $h:$ incompatible observables
exist due to the existence in nature of the minimal action [11]-[13] (see
[45]). Thus, Bell’s conflict with the complementarity principle is in fact the
conflict with the quantum postulate - the existence of $h.$ Hence, this is the
conflict with the very foundation of quantum physics, e.g., the quantum model
of the black body radiation and processes of spontaneous and stimulated
emissions.

##  7 De Broglie’s critique of no-go theorems of von Neumann and Bell

Nowadays it is practically forgotten that De Broglie considered all no-go
theorems by which hidden variables for QM do not exist as totally misleading
[27]. His double solution model [26] can be considered as a model with hidden
variables of the field type. The pilot wave is a hidden variable. 999 In this
aspect the double solution model and PCSFT are similar. Schrödinger’s attempt
to find a subquantum model of the wave type as well as Einstein’s search for a
subquantum nonlinear field model were also steps in the same direction,
although, as was noted, methodologically their positions are different.
Schrödinger tried to realize the Bild approach for QM, treated as OM, but
Einstein dreamed for a model carrying jointly the features of CTM and OM. As
we shall see, De Broglie followed the Bild-conception without knowing about
it.

To justify his double solution model and its peaceful coexistence with QM, De
Broglie criticized the most famous no-go theorems, the von Neumann and Bell
theorems [66, 10]. He did not criticize the mathematical derivation of these
theorems, but their interpretation and straightforward identification of
hidden and observational quantities. His interpretation of these theorems was
presented in very detail by Lochak [31, 32, 33]. Paper [32] is available for
free reading via Google books; we cite it:

Von Neumann proved a theorem which claims that there are no pure states
without statistical dispersion. This result is indeed intuitively obvious
because the absence of dispersion in a pure state mean that it would be
possible to measure simultaneously the physical quantities attached to a
system described by this state. But in fact we know that this is impossible
for non-commuting quantities. In this sense, the theorem is nothing but a
consequence of Heisenberg’s uncertainties.

From this one can conclude that a pure state of QM can’t be considered as
representing an ensemble of systems following the laws of classical
probability theory (see [32] for detailed discussion) and quantum observables
as classical random variables. From this, von Neumann made the conclusion that
generally it is impossible to create any model with hidden variables behind
QM. As pointed out in [32],

De Broglie’s answer consists essentially in asserting that if any hidden
parameters do exist, they cannot obey quantum mechanics because if you try to
imagine hidden parameters it is of course in order to restore the classical
scheme of probabilities. Now if you need a classical scheme of probabilities
for objective(but hidden) values of physical quantities which are introduced
in quantity of hidden parameters, these probabilities cannot be probabilities
observed on the result of a measurement: simply because the observed
probabilities do obey the quantum scheme and not the classical one!

Hence, not only hidden parameters $\lambda\in\Lambda,$ but even variables
$A=A(\lambda)$ are hidden and the probability distributions of these variables
$P_{A}$ should not be identified with quantum probability distributions.

For De Broglie, it was evident that classical and quantum probability calculi
differ crucially, by attempting to apply the former for quantum observables
one immediately confronts with the Heisenberg uncertainty principle (and
Bohr’s complementarity principle[15, 51, 52]). This is precisely my viewpoint
which was presented in article [44] (see section 6).

Hence, De Broglie’s viewpoint on the interrelation of subquantum and quantum
models matches perfectly the CTM-OM approach. In fact, it matches the ontic-
epistemic framework, since De Broglie considered hidden variables and physical
quantities, $A=A(\lambda),$ as objective entities. But, as was already noted,
schematically CTM-OM and the ontic-epistemic frameworks are similar.

De Broglie’s statement that quantities of a subquantum theory are hidden and
their probability distributions should not be identified with the probability
distributions of quantum observables matches the PCSFT-QM coupling considered
in section 4. PCFT-quantities are quadratic forms of fields playing the role
of hidden variables. Such quantities have the continuous range of values, but
say the quantum spin observables have the discrete spectra. Of course, they
cannot have the same probability distribution, even without consideration of
correlations. The correspondence between classical probability calculus of
PCSFT and the quantum probability calculus is fuzzy, classical covariance
operators are mapped to density operators, see (13).

De Broglie and Lochak used the same argument for the critical analysis of the
Bell theorem: one should sharply distinguish subquantum and quantum quantities
and not identify their outcomes and probability distributions. Not only hidden
parameters are hidden, but also quantities dependent on them and their
probability distributions. So, Bell’s model is a very special CTM for QM. Yes,
it should be rejected, as follows from the quantum formalism and experiments.
But its rejection does not prevent search for other CTMs for QM with more
complicated connection between subquantum and quantum quantities and their
probability distributions. From this viewpoint, the foundational value of the
Bell theorem is overestimated.

I again repeat that it is a pity that the fathers of QM, including De Broglie,
were not aware about the works Helmholtz, Hertz, and Boltzmann. The Bild-
conception would provide the rigid philosophic basis for establishing proper
interrelation between subquantum models with hidden variables and QM.

##  8 Classical mechanics

For classical mechanics, CTM and OM coincide. On one hand, this was fortunate
for development of physics, since it simplified so much its philosophic basis
and highlighted the role of observation. On the other hand, identification of
one special mathematical model, Newtonian mechanics, with reality supported
similar ontic treatment of all physical models. The ontic viewpoint of a
scientific theory dominated during a few hundreds years, up to works of
Helmholtz, Hertz, Boltzmann, and Schrödinger. However, these works did not
revolutionized the philosophy of science. For example, acausality of QM is
still considered as the property of nature, so to say irreducible quantum
randomness is ontic.

We note that it seems that Hertz didn’t consider classical mechanics as OM,
see again [29]:  If we try to understand the motions of bodies around us, and
to refer them to simple and clear rules, paying attention only to what can be
directly observed, our attempt will in general fail.

This statement definitely refers to classical mechanics. Similarly to Hertz,
Atmanspacher [3] also considered the two level description even for classical
physics and suggested the corresponding mathematical examples.

The need of separate OM model for classical mechanics became clear in the
process of creation of the mathematical description of the Brownian motion
which will be considered in the next section and CTM-OM structured in
accordance with the Bild-conception ion.

###  8.1 Brownian motion: two levels of description from the time-scale
separation

Here we follow the article of Allahverdyan, Khrennikov, and Nieuwenhuizen [2]:

The dynamics of a Brownian particle can be observed at two levels [56]. Within
the first, more fundamental level the Brownian particle coupled to a thermal
bath at temperature $T$ is described via definite coordinate $x$ and momentum
$p$ and moves under influence of external potential, friction force, and an
external random force. The latter two forces are generated by the bath. The
second, overdamped regime applies when the characteristic relaxation time of
the coordinate $\tau_{x}$ is much larger than that of the momentum $\tau_{p}$,

$\tau_{x}\gg\tau_{p}$

(overdamped regime). On times much larger than $\tau_{p}$ one is interested in
the change of the coordinate and defines the coarse-grained velocity as
$v=\Delta x/\Delta t$ for $\tau_{x}\gg\Delta t\gg\tau_{p}$. This definition of
$v$ is the only operationally meaningful one for the (effective) velocity
within the overdamped regime. It appears that the coarse-grained velocity,
though pertaining to single particles, is defined in the context of the whole
systems of coupled Brownian particles.

The evolution of the momenta of the Brownian particles is very fast and cannot
be resolved on the time-scales available to the experiment. To obtain
experimentally accessible quantities, one employ the technique of the time-
scale separation and measurement of the coarse grained velocity and osmotic
velocity. These quantities can be measured. They are assigned not to an
individual Brownian particle, but to an ensemble of particles coupled to bath,
so these are statistical quantities.

In terms of the present article, Brownian motion is described by CTM ${\bf
M}_{CTMB}$ with the phase space $(x,p)$ and OM ${\bf M}_{\rm{OMB}}$ with the
coarse grained velocities $v_{+},v_{-}$ or the osmotic velocity
$u=v_{+}-v_{-},$ The later description is based on observational quantities
$(x,u).$ As was shown in article [2], ${\bf M}_{\rm{OMB}}$ shows some
properties of QM, e.g., there are analogs of the Heisenberg uncertainty
relations and entanglement; in particular, for a pair of Brownian particles
the joint probability distribution $P(t,x_{1},u_{1},x_{2},u_{2})$ does not
exist. Of course, the OMs ${\bf M}_{\rm{OMB}}$ and ${\bf M}_{\rm{QM}}$ differs
essentially. For example, for a single particle the probability distribution
$P(t,x,u)$ is well defined, incompatibility appears only in compound systems.

Nowadays the above two level structuring of scientific theory of the Brownian
motion is shaken by the novel experimental possibilities for the measurement
of momentum $p$ of a Brownian particle. A variety of experiments was performed
during the last years (see,e.g., [30]). In spite of some diversity in
experimental outputs, it is clear that experimental science is on the way to
establishing the robust procedures for measurement of the Brownian momentum
$p$. Through experimental research, CTM ${\bf M}_{\rm{CTMB}}$ is getting the
OM-status. However, new theoretical efforts are needed to merge ${\bf
M}_{\rm{CTMB}}$ and ${\bf M}_{\rm{OMB}}$ treated as OMs. The osmotic velocity
$u$ (an element of ${\bf M}_{\rm{OMB}})$ is not straightforwardly derived
within ${\bf M}_{\rm{CTMB}}.$ At least for me, connection between the velocity
and coarse grained velocity is not clear. How is the latter derived from the
former?

This special example supports the search for CTMs for QM (see the discussion
at the end of section 5). Some hidden quantities of such models can serve as
the candidates for the future experimental verification. One of the problems
of such project is that, since creation of QM, physicists (mathematicians and
philosophers) created too many subquantum models operating with a variety of
hidden quantities, as say the quantum potential in Bohmian mechanics or the
random field in PCSFT. What are the most probable candidates for future
experimental verification? The Bell hidden variable model [9, 10] is one of
CTMs for QM, it can be directly tested experimentally. It was tested and
rejected.

##  9 Discussion on the Bild-conception and its role in foundations of science

My aim is to recall to physicists and especially to experts in foundations
(not only of quantum physics, but also classical mechanics and field theory,
statistical mechanics, and thermodynamics) about works of Helmholtz, Hertz,
and Boltzmann [65, 28, 29, 16, 17] on the meaning of a scientific theory which
led to the Bild-conception - the mathematical model concept of a scientific
theory. By appealing to the two level description of natural phenomena, CTM-OM
description it is possible to resolve many foundational problems, including
acausality of QM. Moreover, the Bild-conception demystify quantum foundations.
The “genuine quantum foundational problems” such as the possibility to
introduce hidden variables were discussed long ago. The latter problem was
analyzed by Hertz who tried to reduce the classical electromagnetic field to
an ensemble of mechanical oscillators [28]. From the viewpoint of the Bild-
conception , Bell’s attempt to invent hidden variables for QM is very naive;
if such variables were existed their coupling with quantum observables might
be not as straightforward as in the Bell model. Within the Bild-conception ,
it becomes clear why Schrödinger did not consider acausality of quantum
observations as the barrier on the way towards a causal description of quantum
phenomena [57]-[58]. It seems that similarly to Bell [10], von Neumann was
neither aware about development of the philosophy of science by the German
school of physicists in 19s century. He treated the quantum measurement
problem too straightforward and acausality and irreducible quantum randomness
appeared as consequences of such treatment [66]. He did not appeal to the two
level CTM-OM description of microphenomena.

In a series of works [37]-[41], Krennikov et al. developed PCSFT, CTM with
classical random fields, reproducing QM interpreted as OM for microphenomena.
However, PSCFT-QM coupling is not so simple as in the Bell framework.

The two level description of physical phenomena is in fact widely used in
statistical physics and it sis based on time-scales separation technique and
consideration of coarse quantities. All such descriptions are well
accommodated within the Bild-conception . The Brownian motion in the
overdamped regime is described by OM which is not directly coupled to CTM
based on the classical mechanical description.

Finally, we remark that Primas-Atmanspacher [55, 4] ontic-epistemic approach
to physical theories (see also, e.g., [3, 5] is formally similar to the Bild-
conception . But in accordance with the Bild concept, no model describes
reality as it is.

## Acknowledgments

This work was partially supported by COST EU-network DYNALIFE, Information,
Coding, and Biological Function: the Dynamics of Life, CA211 69.

## Appendix:

Here we follow the article of Allahverdyan, Khrennikov, and Nieuwenhuizen [2].

The system under analysis consists of $N$ identical Brownian particles with
coordinates ${\bf x}=(x_{1},...,x_{N})$ and mass $m;$ particles interact with
thermal baths at temperatures $T_{i}$ and coupled via a potential
$U(x_{1},...,x_{N})$. We consider so.called overdamped limit [56]:

  * •

The characteristic relaxation time of particles’ momenta $p_{i}=m\dot{x}_{i}$
is essentially less than the characteristic relaxation time of the
coordinates: $\tau_{x}\gg\tau_{p}.$

  * •

Dynamics is considered in the time-range:

$\tau_{p}<<t\leq\tau_{x}.$

The conditional probability $P({\bf x},t|{\bf x}^{\prime},t^{\prime})$
satisfies the Fokker-Planck equation (the special case of the Kolmogorov
equation for diffusion): [56]:

$\partial_{t}P({\bf x},t|{\bf
x}^{\prime},t^{\prime})=-\sum_{i}\partial_{x_{i}}\,[\,f_{i}({\bf x})\,P({\bf
x},t|{\bf
x}^{\prime},t^{\prime})\,]+\sum_{i}T_{i}\,\partial^{2}_{x_{i}x_{i}}P({\bf
x},t|{\bf x}^{\prime},t^{\prime}),\qquad t\geq t^{\prime},$ (23)

with the initial condition (corresponding to the definition of conditional
probability)

$P({\bf x},t|{\bf x}^{\prime},t)=\delta({\bf x}-{\bf
x}^{\prime})\equiv\prod_{i=1}^{N}\delta(x_{i}-x^{\prime}_{i}).$ (24)

Now consider an ensemble $\Sigma({\bf x},t)$ of all realizations of the
$N$-particle system having at time $t$ the fixed coordinate vector ${\bf x}$.
This ensemble of systems is chosen out of all possible realizations for
measuring particles’ coordinates. For $\Sigma({\bf x},t),$ the average coarse-
grained velocity for the particle with index $j$ might be heuristically
defined as

$v_{j}({\bf x},t)={\rm lim}_{\epsilon\to 0}\,\int\d{\bf
y}\,\frac{y_{j}-x_{j}}{\epsilon}\,P({\bf y},t+\epsilon|{\bf x},t).$ (25)

However, irregularity of the Brownian trajectories implies non-existence of
this limit; so one should to define the velocities for different directions of
time [nelson]:

$v_{+,j}({\bf x},t)={\rm
lim}_{\epsilon\to+0}\,\int\d{y}_{j}\,\frac{y_{j}-x_{j}}{\epsilon}\,P(y_{j},t+\epsilon|{\bf
x},t),$ (26) $v_{-,j}({\bf x},t)={\rm
lim}_{\epsilon\to+0}\,\int\d{y}_{j}\,\frac{x_{j}-y_{j}}{\epsilon}\,P(y_{j},t-\epsilon|{\bf
x},t).$ (27)

What is the physical meaning of these expressions? The directional coarse
grained velocity $v_{+,j}({\bf x},t)$ is the average velocity to move anywhere
starting from $({\bf x},t)$, whereas $v_{-,j}({\bf x},t)$ is the average
velocity to come from anywhere and to arrive at ${\bf x}$ at the moment $t$.

For the overdamped Brownian motion, almost all trajectories are not smooth and
this is the reason for

$v_{+,j}({\bf x},t)\not=v_{-,j}({\bf x},t).$ (28)

The difference

$u({\bf x},t)=v_{+,j}({\bf x},t)-v_{-,j}({\bf x},t)$ (29)

characterizes the degree of non-smoothness; it is called osmotic velocity;
analitically it can be represented in the form

$u_{j}({\bf x},t)=\frac{v_{-,j}({\bf x},t)-v_{+,j}({\bf
x},t)}{2}=-T_{j}\partial_{x_{j}}\ln P({\bf x},t).$ (30)

If we consider $\epsilon$ much smaller than the characteristic relaxation time
of the momentum ( apply definitions (26) and (27) to a smoother trajectory),
then $v_{+,j}({\bf x},t)$ and $v_{-,j}({\bf x},t)$ will be equal to each other
and equal to the average momentum.

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