Hidden parameters. The theory of hidden parameters Hidden parameters in quantum mechanics PDF

In quantum mechanics

The theory of hidden parameters (TSP) is a traditional, but not the only basis for the construction of various types of Bell theorem. The starting point may also recognize the existence of a positively defined probability distribution function. Based on this assumption, without resorting to additional assumptions, the work of a variety of species is formulated and proved in operation. On the specific example It is shown that the formal quantum calculation sometimes gives negative values Figured in the proof of joint probabilities. An attempt was made to find out the physical meaning of this result and proposed algorithm for measuring the negative joint probabilities of this type.

Since the laws of quantum theory predict the results of the experiment, generally speaking, only statistically, then, based on a classic point of view, it would be possible to assume that there are hidden parameters, which, being unobserved in any ordinary experiment, actually determine the result of the experiment as Always considered earlier in accordance with the principle of causality. Therefore, an attempt was made to invent such parameters within the framework of quantum mechanics.

In a narrow meaning, applicable in quantum mechanics and theoretical physics of the microman, where the determinism of the laws of macroscopic physics ceases to act, the theory of hidden parameters served as an important tool for knowledge.

But the value of the approach to the theory of hidden parameters undertaken in the framework of the study of the microworld and quantum mechanical paradoxes is not limited to this circle of phenomena. Perhaps a wider, truly philosophical interpretation of the reasons why this phenomenon takes place in our world.

In the philosophy of knowledge

However, the affected issue of hidden parameters is related not only to narrowphysical problems. It is related to the overall methodology of knowledge. A small excerpt from the treatise on the understanding written by A. M. Nikiforov, helps to understand the essence of this phenomenon:

First, let's try to understand what is an understanding on the usual household level. It can be said that understanding is the process of information incomprehensible to understandable. That is, by means of available logical manipulations, we build a presentation (model) of the ideas of ideas that it was not clear to us. [...] There is a different approach to understanding when the presence of a certain essence or substance with a substance necessary propertiesthat ensure the existence of the phenomena of interest to us ... It should be noted that this approach underlies the theory of relativity and quantum mechanics that declared, as, but do not explain why. [...] It must be said that if the first approach is more stringent and clear, then the second is more powerful, versatile and simple ... The first approach is widely used in science, and it can be considered dominant, but the second is also applied. An example of that "theory of hidden parameters" [Allocated by the author], in accordance with which the discrepancy between the theory with the experiment is removed by the introduction of a certain hypothetical object. The parameters of this object are substituted into the formula, and it starts to coincide with the experiment.

In quantum mechanics, this theory has a significant scope, although it is not generally accepted.

Historical example

Many centuries Geometry Euclidea was considered unshakable science. For a long time, before the start of physical oscillating of the microworld and astrophysical measurements there was no reason to consider it incomplete. However, the situation has changed in the first decade of the 20th century. In physics, the conceptual crisis was increased, which Albert Einstein was able to resolve. Together with the resolution of private tasks - coordination of observations with predictions of theories of that time ("salvation of phenomenon") - in works, together with Niels Bor Einstein, it was possible to bring a brilliant conclusion regarding the possibility of the mass of the masses on the geometry of the space and the speed of the moving object - at speeds, commensurate with light, - For local time for this object.

In geometry, this was an epochable theoretical discovery for cosmology, although he echoed with theoretical prerequisites made by German Minkowski, but who took a special place in modern cosmology.

The effect of the real influence of gravity on the geometry of space can be considered "hidden parameter" in the classical theory of Euclidean, disclosed however, in Einstein theory. The reasoning from the point of view of the methodology of knowledge: in one concept (theoretical) system, a certain parameter can be hidden, and in other - to become disclosed, in demand and theoretically reasonable. In the first case, his "non-expression" does not mean the lack of this parameter in nature as such. It was not that this parameter was not significant, and therefore not found, not introduced by any of the scientists in the "fabric" of this theory.

This situation clearly reveals the property of such "hidden parameters". This is not the negation of the predecessor theory, but finding objective restrictions for its predictions. In the above case, the physical space is indeed high accuracy in the case of not strong gravitational fields operating within the framework of this space (what is the earthly field), but more and more ceases to be with a huge strengthening of gravitational potential. The last in the observed nature can be manifested only in extraterrestrial space objects such as black holes and some other "exotic" space objects.

Notes

Links

I. Z. Tekhmistro, V. I. Standko et al. "Concept of integrity" - Chapter 3 Concept of integrity and experiment: causality and nonlocality in quantum physics (l. E. Pargamin)

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It is possible to experimentally determine whether unaccounted hidden parameters are available in quantum mechanics.

"God does not play bones from the Universe."

Albert Einstein challenged to colleagues who developed a new theory - quantum mechanics. In his opinion, the principle of the uncertainty of Heisenberg and the Schrödinger equation made an unhealthy uncertainty into a micromir. He was confident that the creator could not allow the world of electrons so much distinguished from the usual world of Newtonian billiard balls. In fact during for long years Einstein played the role of a devil's lawyer with regard to quantum mechanics, inventing ingenious paradoxes, designed to make the creators of the new theory in a dead end. Thus, however, he did a good deed, seriously puzzling the theorists of the opposite camp with his paradoxes and forcing deeply thinking about how to solve them, which is always useful when being developed new Region. Knowledge.

There is a strange irony of fate in that Einstein entered the story as a principled opponent of quantum mechanics, although he initially stood at its origins. In particular, Nobel Prize In 1921, he was not at all for the theory of relativity at 1921, but for the explanation of the photovoltaic effect on the basis of new quantum ideas, literally overwhelmed scientific peace at the beginning of the twentieth century.

Most of all Einstein protested against the need to describe the phenomena of the micromyr in terms of probabilities and wave functions ( cm. Quantum mechanics), and not with the usual position of the coordinates and velocities of particles. That's what he meant under the game in the bone. He recognized that the description of the movement of electrons through their speeds and the coordinates contradicts the principle of uncertainty. But, Einstein argued, there must be some more variables or parameters, taking into account the quantum-mechanical picture of the microworld will return to the path of integrity and determinism. That is, he insisted, it only seems to us that God plays with us in the bone, because we do not understand everything. Thus, he first formulated hypothesis hidden variable In quantum mechanics equations. It is that in fact the electrons have fixed coordinates and speed, like Newtonian billiard balls, and the principle of uncertainty and a probabilistic approach to their definition within quantum mechanics - the result of the incomplete theory itself, which is why it does not allow them to be kind Determine.

The theory of the hidden variable can be visually imagined about this: the physical justification of the principle of uncertainty is that it is possible to measure the characteristics of a quantum object, such as an electron, only through its interaction with another quantum object; In this case, the state of the measured object will change. But, perhaps, there is some other way to measure using unknown tools tools yet. These tools (let's call them "subelectricons") may interact with quantum objects, without changing their properties, and the principle of uncertainty will not be applicable to such measurements. Although no actual data in favor of the hypothesis of this kind was there, they ghostly loomed on the side of the main way of development of quantum mechanics - basically, I suppose, because of the psychological discomfort experienced by many scientists because of the need to refuse the well-established newtonian ideas about the universe device.

And in 1964, John Bell received a new and unexpected theoretical result for many. He proved that a certain experiment can be held (the details later), the results of which will allow determine whether quantum-mechanical objects are really described by the wave functions of probability distribution, as they are, or there is a hidden parameter that allows you to accurately describe their position and impulse, as at the Newtonian ball. Bella Theorem, as it is now called, shows that as if there is a hidden parameter in the quantum-mechanical theory of a hidden parameter affecting anyone The physical characteristic of the quantum particle and in the absence of such a serial experiment can be carried out, statistical results which will be confirmed or refuted the presence of hidden parameters in quantum-mechanical theory. Signally speaking, in one case, the statistical ratio will be no more than 2: 3, and in the other - at least 3: 4.

(Here I want to notice in brackets that in that year, when Bell proved his theorem, I was a senior student in Stanford. A redhead, with a strong Irish accent Bella was difficult not to notice. I remember, I stood in the Corridor of the Scientific Corps of the Stanford Linear Accelerator , and then he came out of his office in a state of extreme excitement and he stated that he had just found a truly important and interesting thing. And, although I have no evidence to this account, I would really like to hope that I am in That day became an invalid witness to his discovery.)

However, the experience offered by Bella turned out to be simple only on paper and at first seemed practically impossible. The experiment should have looked like this: under external influence The atom was to synchronously emit two particles, for example, two photons, and in opposite directions. After that, it was necessary to catch these particles and toolically determine the direction of the back of each and to make it a thousand times to accumulate sufficient statistics to confirm or refute the existence of a hidden parameter on the Bella Theorem (expressing the language mathematical statistics, it was necessary to calculate correlation coefficients).

The most unpleasant surprise for everyone after the publication the Bella Theorem was just the need for a colossal series of experiments, which at that time seemed practically impossible to obtain a statistically reliable picture. However, there were no decades as experimentenary scientists not only developed and built the necessary equipment, but also accumulated a sufficient data array for statistical processing. Without going into technical details, I will only say that then in the middle of the sixties, the complexity of this task seemed so monstrous that the probability of its implementation was equal to however if someone had planned to plant for writing machines Million trained monkeys from the proverb in the hope of finding Among the fruits of their collective labor creation, equal to Shakespeare.

When, in the early 1970s, the results of the experiments were generalized, everything became extremely clear. The wave function of the probability distribution completely unmistakably describes the movement of particles from the source to the sensor. Consequently, the equations of wave quantum mechanics do not contain hidden variables. This is the only known case in the history of science, when a brilliant theoretical proved opportunity experimental verification of the hypothesis and gave justification method Such an inspection, brilliant experimenters with titanic efforts conducted a complex, expensive and protracted experiment, which eventually confirmed the already dominant theory and did not even introduce anything new to it, as a result of which everyone felt severely deceived in expectations!

However, not all the works disappeared in vain. Most recently, scientists and engineers to a considerable surprise have found the Bella Theorem very decent practical use. Two particles emitted by the source at the Bell Installation are coherent (have the same wave phase), since they are emitted synchronously. And this property is now going to use in cryptography to encrypt particularly secret messages sent by two separate channels. When interception and attempt to decrypt the message according to one of the channels, coherence is instantly violated (again, by virtue of the principle of uncertainty), and the message is inevitably and instantly self-suited at the time of the connection between particles.

And Einstein, it seems, was wrong: God still plays in the bone from the universe. Perhaps Einstein still followed to listen to the advice of her old friend and Colleague Niels Bora, who once again hearing the old chorus about the "game in the bone", exclaimed: "Albert, you finally point to God, what to do ! "

John Stewart Bell, 1928-91

Physicist from Northern Ireland. Born in Belfast, in the poor family. In 1949 he graduated from the Belfast Royal University, after which the short time worked there as an assistant of the physical laboratory. After several years of work at the Institute of Atomic Energy in Harwell (Harwell) in 1960, Bell was invited to the European Center for Nuclear Research (CERN) in Geneva and worked there the remaining part of life. Scientist's wife, Mary Bell, was also a physicist and an employee of CERN. He brought his fame Theorem Bell formulated during short-term internships in the United States.

The principle of sufficient basis is the key in the program of expansion of physics on the scale of the Universe: it seeks a rational explanation of any choice that nature does. Free, unprecedented behavior of quantum systems is contrary to this principle.

Is it possible to observe him in quantum physics? It depends on whether it is possible to extend the quantum mechanics for the entire universe and propose the most fundamental description of nature from possible - either a quantum mechanic serves as an approach to another cosmological theory. If we can spread quantum theory to the universe, the freedom of liberty theorem will be applicable in a cosmological scale. Since we assume that there is no theory of fundamental quantum, we mean that nature is truly free. Freedom of quantum systems in a cosmological scale would mean the limitation of the principle of a sufficient basis, because there can be no rational or sufficient basis for many cases of free behavior of quantum systems.

But, offering the expansion of quantum mechanics, we make a cosmological error: apply the theory beyond the boundaries of the area in which it can be checked. A more careful step would be a consideration of the hypothesis that quantum physics is an approximation, valid only for small subsystems. To determine whether the quantum system is present somewhere else in the universe or is it possible to apply a quantum description in the theory of the whole universe, additional information is needed.

Can there be a deterministic cosmological theory that comes down to quantum physics when we are insulating the subsystem and neglect everyone else in the world? Yes. But it is given a high price. According to such a theory, the probability of quantum theory arises only because of the influence of the entire universe. The probabilities will inflate the place to certain predictions at the level of the universe. In the cosmological theory, quantum uncertainties manifest while trying to describe a small part of the universe.

The theory received the name of the theory of hidden parameters, since quantum uncertainties are eliminated by such information about the universe, which is hidden from the experimenter operating with a closed quantum system. Theories of this kind serve to obtain predictions for quantum phenomena, consistent with the predictions of the traditional quantum Physics. So, such a solution to the problem of quantum mechanics is possible. In addition, if determinism is restored by distributing quantum theory to the entire universe, hidden parameters are not associated with a refined description of individual elements of the quantum system, but with the interaction of the system with the rest of the universe. We can call them hidden relational parameters. According to the principle of maximum freedom described in the previous chapter, the quantum theory is probabilistic and internal uncertainty in it is maximal. In other words, information on the state of the atom that we need is needed to restore determinism, and which is encoded in the relationship of this atom from the whole universe, the maximum. That is, the properties of each particle are maximally encoded using hidden connections from the Universe as a whole. The task of clarifying the meaning of quantum theory in search of a new cosmological theory is key.

What is the price of a "entrance ticket"? Refusal to the principle of relativity of simultaneity and return to the picture of the world, in which the absolute definition of simultaneity is fair in the entire Universe.

We must act carefully because we do not want to conflict with the theory of relativity that have many successful applications. Among them, the quantum field theory is a successful unification of the special theory of relativity (service station) and quantum theory. It is it that underlies the standard particle physics model and allows to obtain many exact predictions confirmed by experiments.

But in the quantum field theory, it is not necessary without any problems. Among them are complex manipulation with infinite values, which must be done before getting the prediction. Moreover, the quantum field theory inherited all the conceptual problems of quantum theory and does not offer anything new to solve them. Old problems together with new infinity problems show that the quantum field theory is approaching a deeper theory.

Many physicists starting with Einstein, dreamed of going beyond the quantum field theory and find the theory that gives full description Each experiment (which, as we saw, impossible within the framework of quantum theory). This led to a fatal contradiction between quantum mechanics and a hundred. Before proceeding to return time to physics, we need to figure out what this contradiction consists.

It is believed that the inability of quantum theory to present a picture of what is happening in a particular experiment is one of its advantages, and not at all defect. Niels Bor argued (see chapter 7) that the purpose of physics is to create a language on which we can tell each other about how we conducted experiments with atomic systems and what results received.

I find it unconvincing. I have the same feelings, by the way, in relation to some modern theorists convinced that the quantum mechanics do not care about the physical world, but with information about him. They argue that quantum states do not correspond to physical reality, and simply encode information about the system that we can get. These are smart people, and I love to argue with them, but I am afraid that they underestimate science. If a quantum mechanic is only a probabilities prediction algorithm, can we come up with something better? In the end, something happens in a concrete experiment, and only this is a reality, called an electron or photon. Is we able to describe the existence of individual electrons on the mathematical language? Perhaps there is no principle that guarantees that the reality of each subatomic process should be understood by the person and can be formulated on human Language Or with the help of mathematics. But should we not try? Here I am on the side of Einstein. I believe that there is an objective physical reality and something that dedicated to the description occurs when the electron jumps from one energy level to another. I will try to build a theory capable of giveing \u200b\u200bsuch a description.

For the first time, the theory of hidden parameters presented the Duke of Louis de Broglie at the famous V Solveyevsky Congress in 1927, shortly after the quantum mechanic acquired its final formulation. De Broglie inspired the idea of \u200b\u200bEinstein on the duality of wave and corpuscular properties (See chapter 7). The theory of Debriel was allowed a mystery of a particle wave in the simplest way. He argued that physically there are particles and a wave. Earlier, in dissertation of 1924, he wrote that the corpuscular wave dualism is universal, so that particles such as electrons are also a wave. In 1927, de Broglov said that these waves apply as on the surface of the water, interfering with each other. The particle corresponds to the wave. Besides electrostatic, magnetic and gravitational forces, Quantum force acts on the particles. It attracts particles to the ridge of the waves. Consequently, on average particle, most likely, will be there, but this connection is probabilistic. Why? Because we do not know where the particle was first. And if so, we can not predict where it will be after. The hidden variable in this case is the exact position of the particle.

Later, John Bell proposed to call the theory of de Brogly theory of real variables (Beables), in contrast to the quantum theory of the observed variables. Real variables are always present, unlike the observed: the latter arise as a result of the experiment. According to de Brogle, and the particles, and the waves are real. The particle always occupies a certain position in space, even if a quantum theory cannot predict it.

The theory of de Broglie, in which the particles and the waves are real, did not receive wide recognition. In 1932, the great mathematician John Von Neuman posted a book in which he argued that the existence of hidden parameters is impossible. A few years later, Greta Hermann, a young German mathematician, pointed out the vulnerability of proof von Neuman. Apparently, he made a mistake, initially believing proven what he wanted to prove (that is, he made an assumption for Axiom and deceived himself and others). But the work of Herman ignored.

Two decades have passed before the error has discovered again. In the early 50s, American physicist David Bom wrote a textbook of quantum mechanics. Bom independently of de Broglie opened the theory of hidden parameters, but when he sent an article to the editorial board of the magazine, he received a refusal: his calculations were contrary to the well-known proof of the background of Nymanan in the impossibility of hidden parameters. Bom quickly found a mistake at Nymanan. Since then, the approach of de Broglya - Boma to quantum mechanics used few in their works. This is one of the views on the basics of the quantum theory, which is discussed today.

Thanks to the theory of de Broglie - Boma, we understand that the theories of hidden parameters are an option to resolve the paradoxes of quantum theory. Many features of this theory were inherent in any theories of hidden parameters.

The theory of De Brogly - Boma has a dual attitude towards the theory of relativity. Its statistical predictions are consistent with quantum mechanics and do not contradict the special theory of relativity (for example, the principle of relativity of simultaneity). But, unlike quantum mechanics, de Broglie's theory - Boma offers more, rather than statistical predictions: it gives a detailed physical picture of what is happening in each experiment. A wave that changes in time affects the movement of particles and violates the relativity of simultaneity: the law according to which the wave affects the movement of the particle can be faithful in one of the reference systems associated with the observer. Thus, if we adopt the theory of hidden parameters de Broglya - Boma as an explanation of quantum phenomena, we must take on faith that there is a dedicated observer whose watches show a dedicated physical time.

Such an attitude to the theory of relativity applies to any theories of hidden parameters. Statistical predictions that are consistent with quantum mechanics are consistent with the theory of relativity. But any detailed picture of phenomena violates the principle of relativity and will have an interpretation in the system with only one observer.

The theory of De Brogil - Boma is not suitable for the role of cosmological: it does not meet our criteria, namely the requirement that actions are mutual for both parties. The wave affects particles, but the particle has no effect on the wave. However, there exists and alternative theory Hidden parameters in which this problem is eliminated.

Being convinced, like Einstein, in the existence of a quantum theory of other, deeper theory, I have invented the theories of hidden parameters from time to study. Every few years I postponed all the work and tried to solve this crucial problem. For many years, I developed an approach based on the theory of hidden parameters, which Princeton Mathematician Edward Nelson offered. This approach was working, but an artificial element was present in it: to reproduce the prediction of quantum mechanics, certain forces had to be accurately balanced. In 2006, I wrote an article by explaining the unnaturality of the theory of technical reasons, and refused this approach.

Once in the evening (it was at the beginning of the autumn of 2010) I went to the cafe, opened a notebook and thought about my many unsuccessful attempts to go beyond quantum mechanics. And he remembered the statistical interpretation of quantum mechanics. Instead of trying to describe what is happening in a concrete experiment, it describes the imaginary collection of everything that should happen. Einstein expressed this as follows: "Attempt to submit a quantum-theoretical description as a complete description of individual systems leads to unnatural theoretical interpretations that become not needed, if we accept the fact that the description refers to ensembles (or collections) of systems, and not to individual systems."

Consider a lonely electron rotating around the proton in the hydrogen atom. According to the authors of the statistical interpretation, the wave is associated not with a separate atom, but with an imaginary collection of copies of the atom. In different samples in the collection, electrons have a different position in space. And if you observe the hydrogen atom, the result will turn out to be as if you accidentally chose an atom from the imaginary collection. The wave gives the likelihood of finding an electron in all different positions.

I liked this idea for a long time, but now it seemed crazy. How can an imaginary set of atoms influence measurements against one real atom? It would be contrary to the principle that nothing outside the universe can affect what is inside it. And I wondered: can I replace the imaginary collection of real atoms? Being real, they must exist somewhere. In the universe, the great set of hydrogen atoms. Can they make a "collection", about which the static interpretation of quantum mechanics treats?

Imagine that all hydrogen atoms in the universe play the game. Each atom admits that others are in a similar situation and have a similar story. Under the "similar" I mean that they will be described probably, with the help of the same quantum state. Two particles in the quantum world may have the same history and are described by the same quantum state, but differ in the exact values \u200b\u200bof real variables, for example, in its position. When two atoms have a similar story, one copies the properties of the other, including the exact values \u200b\u200bof real variables. To copy properties, atoms are not necessarily located nearby.

This is a nonlocal game, but any theory of hidden parameters is obliged to express the fact that the laws of quantum physics are notocal. Although the idea may seem ravous, it is less crazy than the idea of \u200b\u200ban imaginary collection of atoms affecting atoms in real world. I took it to develop this thought.

One of the copies is the position of the electron relative to the proton. Therefore, the position of the electron in a particular atom will change as it copies the position of electrons in other atoms in the universe. As a result of these jumps, the measurement of the position of the electron in a particular atom will be equivalent to how if I chose atom at random from the collection of all such atoms replacing the quantum state. To work, I came up with the copying rules, which lead to predictions for an atom, exactly consistent with the predictions of quantum mechanics.

And then I understood something that immensely deliberately. What if the system has no analogues in the universe? Copying cannot continue, and the results of quantum mechanics will not be reproduced. It would explain why quantum mechanics not applicable to complex systems like us, people, or cats: we are unique. This made it possible to resolve the long-standing paradoxes arising from the use of quantum mechanics to large objects, such as cats and observers. The strange properties of quantum systems are limited to atomic systems, because the latter are found in the universe in a great set. Quantum uncertainties arise because these systems constantly copy the properties of each other.

I call it a real statistical interpretation of quantum mechanics (or the "interpretation of white proteins" - in honor of albino protein, occasionally found in Toronto parks). Imagine that all gray proteins are similar to each other sufficiently and quantum mechanics will apply to them. Find one gray squirrel, and you will probably meet again. But the flashed white protein seems to have no copy, and, therefore, it is not a quantum-mechanical protein. Her (as me or you) can be considered as possessing unique properties and not having analogues in the universe.

The game with jumping electrons violates the principles of special theory of relativity. Instant jumps across arbitrarily long distances require the concept of simultaneous events separated by large distances. This, in turn, implies the transmission of information at a speed greater than the speed of light. Nevertheless, statistical predictions are consistent with quantum theory and can be aligned with the theory of relativity. And yet in this picture there is a highlighted simultaneity - and, therefore, a dedicated time scale, as in the theory of de Broglya - Boma.

In both the theories described above, the theories of hidden parameters are followed by the principle of sufficient basis. There is a detailed picture of what is happening in individual events, and it explains what is considered uncertain in quantum mechanics. But the price of this is a violation of the principles of the theory of relativity. This is a high price.

Can there be the theory of hidden parameters compatible with the principles of the theory of relativity? Not. It would violate the theorem about the freedom of the will, from which it follows that while its conditions are performed, it is impossible to determine what will happen with a quantum system (and therefore there is no hidden parameters). One of these conditions is the relativity of simultaneity. Bella Theorem also eliminates local hidden parameters (local in the sense that they are causally connected and exchange information with a transmission rate of less than the speed of light). But the theory of hidden parameters is possible if it violates the principle of relativity.

While we are only checking the predictions of quantum mechanics at the statistical level, there is no need to be interested in what is actually a correlation. But if we try to describe the transfer of information within each confusing pair, the notion of instant communications will be required. And if we try to go beyond the statistical predictions of quantum theory and go to the theory of hidden parameters, we enter into conflict with the principle of relativity of simultaneity.

To describe correlations, the theory of hidden parameters should take the definition of simultaneity from the point of view of one selected observer. This, in turn, means that there is a dedicated concept of peace of rest and, therefore, that the movement is absolutely. It acquires absolute meaning as you can argue who is moving with regarding whom (let's call this character Aristotle). Aristotle is at rest, and everything he sees as a moving body is a really moving body. That's the whole conversation.

In other words, Einstein was wrong. And Newton. And Galilee. There is no relativity in motion.

This is our choice. Either quantum mechanics is the final theory and there is no possibility to penetrate its statistical curtain in order to achieve a deeper level of nature description, or Aristotle was right and dedicated motion and rest systems exist.

See: Bacciagaluppi, Guido, and Antony Valentini Quantum Theory At The Crossroads: Reconsidering The 1927 Solvay Conference. New York: Cambridge University Press, 2009.

See: Bell, John S. Speakable and unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy. NEW YORK: Cambridge University Press, 2004.

Neumann, John Von Mathematische Grundlagen der Quantenmechanik. BERLIN, JULIUS SPRINGER VERLAG, 1932, PP. 167 FF.; Neumann, John Von Mathematic Foundations of Quantum Mechanics. PrinceTon, NJ: Princeton University Press, 1996.

Hermann, Grete Die Naturphilosophischen Grundlagen der Quantenmechanik // Abhandlungen Der Fries'schen Schule (1935).

Bohm, David Quantum Theory. New York: Prentice Hall, 1951.

Bohm, David A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. II // Phys. Rev., 85: 2, 180-193 (1952).

Valentini, Antony Hidden Variables and the Large-Scale Structures of Space \u003d Time / in: Einstein, Relativity and Absolute Simultaneity. EDS. Craig, W. L., and Q. Smith. London: ROUTLEDGE, 2008. PP. 125-155.

Smolin, Lee Could Quantum Mechanics BE An Approximation to Another Theory? // Arxiv: Quant-pH / 0609109v1 (2006).

Einstein, Albert Remarks to the Essays Appearing in This Collective Volume / in: Albert Einstein: Philosopher-Scientist. ED. P. A. Schilpp. NEW YORK: TUDOR, 1951. P. 671.

See: Smolin, Lee A Real Ensemble Interpretation of Quantum Mechanics // ARXIV: 1104.2822V1 (2011).