Bild Conception of Scientific Theory Structuring in Classical and Quantum Physics: From Hertz and Boltzmann to Schrödinger and De Broglie

We start with a 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 conception” 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 a 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 based on the price of acausality. CTM-OM interrelation can be tricky. Schrödinger used the Bild concept to create a 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 the 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.


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.) 1e 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.
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.

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 (1844Boltzmann ( -1906)), who adopted the Kantian term Bild2 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 quan-tities, 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. 3Schrödinger 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, M CTM and M OM , depends on the present stage of development of science.If M CTM rightly reflects the real physical processes, then development of measurement technology can lead to novel observational possibilities and some hidden quantities of M CTM can become measurable.Hence, M CTM becomes OM, M CTM → M ′ OM .In principle, M ′ OM need not cover all observations described by the previous OM M OM .New theoretical efforts might be needed to merge M OM and M ′ OM .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 multidimensional space R 3n .In 1920s the fact that the multi-particle Schrödinger equation describes the waves not on "the physical space" R 3 , but on "the mathematical space" 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 R 3n has the same descriptive status as the use of 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]. 4However, 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 describethat 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"5 , 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 onticepistemic 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] 6 : 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 thingsmetaphors 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.

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 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 M OM and its causal theoretical completion M CT M .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 CT M to states from S OM .Causality can be transformed into acausality through consideration of probability distributions.So, consider some space of probability distributions P CT M on the state space S CT M and construct a map from P CT M 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 M CT M and M OM .Thus, we need to define two physically natural maps: Since J S is not defined for states of CTM, but only for probability distributions, "physically natural" means coupling between the probability structures of M CT M and M OM ; the minimal coupling is the equality of averages between variables and correlations Generally the correlation need not be defined, so (3) should hold for variables f, g ∈ V CT M 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 CT M can be represented as a space of functions f : S CT M → R. Such model is causal, the state φ uniquely determines the values of all variables belonging V CT M : φ → f (φ).The state space S CT M can be endowed with a σ-algebra of subsets F .Elements of P CT M are probability measures on F .The minimal mathematical restriction on elements of V CT M is that they are measurable functions, f : S CT M → R. In such a framework, if the integrals exist, e.g., if CTM-variables are square integrable: Since in M OM quantities have the experimental statistical verification, we establish some degree of experimental verification for M CT M through mapping of M CT M to M OM .But such verification is only indirect, one should not expect direct coupling between quantities of M CT M 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 CT M can be mapped into the same state from S OM .
• J S (P CT M ) need not coincide with S OM .
• A cluster of elements of V CT M can be mapped into a single variable (observable) from V OM .
• J V (V CT M ) need not coincide with V OM .Moreover, the model-correspondence maps J S , J V need not be defined on whole spaces P CT M and V CT M .They have their domains of definition, D J S ⊂ P CT M and D J V ⊂ V CT M .(In principle, one can reduce P CT M to P ′ CT M = D J S and V CT M to V ′ CT M = 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 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 κ (in the deformation quantization this is action, roughly speaking the Planck constant h).Thus, M CT M = M CT M (κ).In such more general framework, the correspondence maps also depend on κ, i.e., J S = J S (κ), J V = J V (κ).The probabilistic coupling constraints (2), (3) can be weakened: (see [?]).The problem of identification of the parameter κ 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 M OM and PCSFT as M CT M .Re-denote these model with the symbols M QM and M PCSFT .We briefly recall the basic elements of PCSFT (see [39,41,42] for details).
In M QM states are given by density operators acting in complex Hilbert space H (with scalar product •|• ) and observables are represented by Hermitian operators in H. Denote the space of density operators by S QM and the space of Hermitian operators by V QM . In for any vestor a ∈ H, and finite second momentum, i.e., Denote the space of such probability measures by the symbol P PCSFT .We can start not with probability measures, but with H-valued random vectors with zero mean value and finite second moment: φ = φ(ω), such that E[φ] = 0 and E[ φ 2 ] < ∞. 7 The space of such random vectors is denoted by the symbol R PCSFT .In the finite-dimensional case, these are complex vectorvalued random variables; if H is infinite-dimensional, then the elements of R PCSFT are random fields.
An example of random fields is given by selection H = L 2 (R n ; C) of square integrable complex valued functions.Each M CT M state φ is an L 2function, φ : R n → C. Random fields belonging to R PCSFT are functions of two variables, φ = φ(x; ω) : chance parameter ω and space coordinates x.
We remark that, for the state space H = L 2 (R n ; C), the quantity E p can be represented as 7 Random vectors are defined on some Kolmogorov probability space (Ω, F , P ), these are functions φ : Ω → H which are measurable w.r.t. to the Borel σ-algebra of H, i.e., for any Borel subset B of H, φ −1 (B) ∈ F .A map is measurable iff, for any c > 0, the set where is the energy of the field.The quantity 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 φ = φ(x; ω) be a random field.Then its energy is the random variable and E p is its average.
For any p ∈ P PCSFT , its (complex) covariance operator Bp is defined by its bilinear (Hermitian) form: or, for a random field φ, we have: We note that E p = H φ 2 dp(φ) = Tr Bp (10) or in terms of a random field: Thus, the average energy of a random field φ = φ(ω, x) can be expressed via its covariance operator.Generally a probability measure (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 Bp is • Hermitian, • positively semidefinite, • trace class.
A "physically natural coupling" of the models M QM and M PCSFT is based on the following formula coupling mathematically the averages for these models.For a probability measure p ∈ P PCSFT and a variable f ∈ V PCSFT , we have where f (φ) = φ| Âf |φ .This formula is obtained through expansion of the quadratic form φ| Âf |φ w.r.t. the basis of eigenvectors of the Hermitian operator Âf .
Let us consider the following maps J S : P PCSFT → S QM and J V : This correspondence connects the averages given by the causal theoretical and observational models: 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 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: If random fields have low energy, i.e., E p << 1, the quantity 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.
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.

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". 8For 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 φ = φ(x; ω).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 φ bground (x, ω) is the necessary element of the mathematical model M PCSFT for generation of entangled states in S 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, M Bell , with OM -M QM .The subquantum quantities, functions of hidden variables, A = A(λ), were identified with quantum observables.In particular the ranges of values of quantities from M 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], M Bell , confronts the complementarity principle.The later point will be clarified below.
We recall the mathematical structure of M Bell by connecting it with the framework of section 3. Bell considered [10] an arbitrary set of hidden variables Λ; this is the set of states of his CTM, i.e., S Bell = Λ.To put this model in the mathematical framework of probability theory, Λ should be endowed with some σ-algebra of its subsets, say F .Denote by P Bell the space of all probability measures on (Λ, F ).The space of Bell-variables consists of all measurable functions A : Λ → R, A = A(λ), i.e., random variables in terms of the Kolmogorov probability theory.We stress that the correspondence map J S : P Bell → 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(λ) 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 M Bell with QM observables means that the range of values of A ∈ V Bell coincides with the spectrum of the corresponding Hermitian operator Â.This is the important mathematical constraint on the map J V : V Bell → 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 ( Â1 , Â2 ) and ( B1 , B2 ).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, 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 [ Â1 , Â2 ], [ B1 , B2 ] equals to zero), then the CHSH inequality can't be violated [43].Bell considered variables of his CTM M Bell as representing physical observables, hence all observables can be represented as functions and their values are identified with outcomes of observations.Besides the pairs (A i (λ), B j (λ)) of compatible observables, one can consider the pairs (A i (λ), A j (λ)) and (B i (λ), B j (λ)).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 M Bell as representing physical reality, one confronts with treating of complementarity as the fundamental property of (observational) microphenomena.At the level of correlations, where (A ik ), (B jk ) are observables' outcomes.At the same time, for the probability distribution P ρ such that ρ = J S (P ρ ), we have where A ′ ik and B ′ jk are outcomes of random variables A i = A i (λ) and B j = B j (λ).However, since these outcomes can be identified with the outcomes of the quantum observables, A ′ ik = A ik , B ′ jk = B jk , we can write 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. 9As 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 λ ∈ Λ, but even variables A = A(λ) 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(λ), 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.

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.

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 τ x is much larger than that of the momentum τ p , τ x ≫ τ p (overdamped regime).On times much larger than τ p one is interested in the change of the coordinate and defines the coarse-grained velocity as v = ∆x/∆t for τ x ≫ ∆t ≫ τ 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 timescale 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 M CT M B with the phase space (x, p) and OM M 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], M 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 M OMB and M 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 M CTMB is getting the OM-status.However, new theoretical efforts are needed to merge M CTMB and M OMB treated as OMs.The osmotic velocity u (an element of M OMB ) is not straightforwardly derived within M 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 vari-ables 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. The system under analysis consists of N identical Brownian particles with coordinates 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 ẋi is essentially less than the characteristic relaxation time of the coordinates: • Dynamics is considered in the time-range: The conditional probability P (x, t|x ′ , t ′ ) satisfies the Fokker-Planck equation (the special case of the Kolmogorov equation for diffusion): [56]: with the initial condition (corresponding to the definition of conditional probability) Now consider an ensemble Σ(x, t) of all realizations of the N-particle system having at time t the fixed coordinate vector x.This ensemble of systems is chosen out of all possible realizations for measuring particles' coordinates.For Σ(x, t), the average coarse-grained velocity for the particle with index j might be heuristically defined as v j (x, t) = lim ǫ→0 y .y j − x j ǫ P (y, t + ǫ|x, t).
However, irregularity of the Brownian trajectories implies non-existence of this limit; so one should to define the velocities for different directions of time [?]: v +,j (x, t) = lim ǫ→+0 y .j y j − x j ǫ P (y j , t + ǫ|x, t), v −,j (x, t) = lim ǫ→+0 y .j x j − y j ǫ P (y j , t − ǫ|x, t).(27) M PCSFT states are vectors of H, i.e., S PCSFT = H.Physical variables are quadratic forms φ → f (φ) = φ| Â|φ , where Â ≡ Âf is a Hermitian operator.The space of quadratic forms is denoted by the symbol V PCSFT .Consider probability measures on the σalgebra of Borel subsets of H (i.e., generated by balls in this space) having zero first momentum, i.e., Hφ|a dp(φ) = 0