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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Einstein's dream

Andrei Yu. Khrennikov

Published Aug 26, 2007• 10 citations
Publisher

Abstract

We show that (in contrast to rather common opinion) QM might be not a complete theory. We present a prequantum model with the infinite dimensional phase space such that QM can be considered as an algorithm for approximation of averages with respect to prequantum fluctuations. Such an approximation is based on the asymptotic expansion of classical statistical averages with respect to a small parameter &kgr;. Therefore statistical predictions of QM are only approximative and a better precision of measurements would induce deviations of experimental averages from quantum mechanical ones. In this paper we present a natural physical interpretation of &kgr; as the time scaling parameter (between quantum and prequantum times). We analyse consequences of various choices of the prequantum time scale. Possible connections with attophysics and Grand Unification model are discussed.

DOI
10.1117/12.733007

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AIP conference proceedings•Jan 1, 2007•4 citations
We show that (in contrast to rather common opinion) QM might be not a complete theory. We present a prequantum model with the infinite dimensional phase space such that QM can be considered as an algorithm for approximation of averages with respect to prequantum fluctuations. Such an approximation is based on the asymptotic expansion of classical statistical averages with respect to a small parameter &kgr;. Therefore statistical predictions of QM are only approximative and a better precision of measurements would induce deviations of experimental averages from quantum mechanical ones. In this paper we present a natural physical interpretation of &kgr; as the time scaling parameter (between quantum and prequantum times). We analyse consequences of various choices of the prequantum time scale. Possible connections with attophysics and Grand Unification model are discussed.
We show that the probabilistic formalism of QM can be obtained as a special projection of classical statistical mechanics for systems with an infinite number of degrees of freedom. Such systems can be interpreted as classical fields. Thus in our approach QM is a projection of (prequantum) classical statistical field theory (PCSFT). This projection is based on the Taylor expansion of classical physical variables – maps f: Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures on Ω having zero mean value and negligibly small dispersion. On one hand, the creation of such a prequantum model strongly supports attempts (first of all by Schrödinger and Einstein) to create purely field model of QM. On the other hand, it gives the possibility to go beyond QM. The main experimental prediction is that averages calculated in the mathematical formalism of QM (von Neumann’s trace formula) are only approximative averages. If predictions of PCSFT are correct then it would be possible to find deviations of experimental averages from quantum ones. 1
In Section 3.1, we showed that QM can be represented as a classical field model in which physical quantities are given by quadratic functionals of fields. In this chapter we shall show that in complete accordance with Einstein’s views QM is an approximation of CSM of fields, i.e., classical statistical mechanics with the infinite- dimensional phase space H = L2(R3;R) × L2(R3;R), where L2(R3;R) is the space of real-valued square-integrable functions. (In the previous considerations the symbol L2(R3) was used to denote the space of complex-valued square-integrable functions.) QM can be represented as the quadratic Taylor approximation of CSM of fields. By expanding classical quantities, functionals of the prequantum field, φ → f (φ), into the Taylor series up to the second term, we obtain quantum quantities. By averaging these expansions we establish a coupling between classical average (so to say, prequantum) and quantum average. The latter appears as the main term in the expansion of the classical phase space average with respect to a small parameter α, the dispersion of the prequantum fluctuations, α = σ 2(μ). Here μ is the probability distribution of a prequantum random field. In the limit α → 0 the classical prequantum and quantum averages coincide. Thus QM can be interpreted as the limit (as α → 0) of the theory of classical random fields, cf. Section 3.11 for an experimentwhichmay show a deviation from the prediction of QM, namely, violation of Born’s rule.
We show that QFT (as well as QM) is not a complete physical theory. We constructed a classical statistical model inducing quantum field averages. The phase space consists of square integrable functions, f(ϕ),f(ϕ),f(ϕ), of the classical bosonic field, ϕ(x).ϕ(x).ϕ(x). We call our model prequantum classical statistical field-functional theory -- PCSFFT. The correspondence between classical averages given by PCSFFT and quantum field averages given by QFT is asymptotic. The QFT-average gives the main term in the expansion of the PCSFFT-average with respect to the small parameter ααα -- dispersion of fluctuations of "vacuum field functionals.'' The Scrödinger equation of QFT is obtained as the Hamilton equation for functionals, F(f),F(f),F(f), of classical field functions, f(ϕ).f(ϕ).f(ϕ). The main experimental prediction of PCSFFT is that QFT gives only approximative statistical predictions that might be violated in future experiments.
We show that QM can be represented as a natural projection of a classical statistical model on the phase space Ω = H × H , where H is the real Hilbert space. Statistical states are given by Gaussian measures on Ω having zero mean value and dispersion of very small magnitude α (which is considered as a small parameter of the model). Such statistical states can be interpreted as fluctuations of the background field, cf. with SED and Nelson's mechanics. Physical variables (e.g., energy) are given by maps f : Ω → R (functions of classical fields). The conventional quantum representation of our prequantum classical statistical model is constructed on the basis of the Taylor expansion (up to the terms of the second order at the vacuum field point ψ vacuum ≡ 0) of variables f : Ω → R with respect to the small parameter √α The complex structure of QM is induced by the symplectic structure on the infinite-dimensional phase space Ω. A Gaussian measure (statistical state) is represented in QM by its covariation operator. Equations of Schrödinger, Heisenberg and von Neumann are images of Hamiltonian dynamics on Ω. The main experimental prediction of our prequantum model is that experimental statistical averages can deviate from ones given by QM.
Albert Einstein did not believe in completeness of QM. He dreamed of creation of prequantum classical statistical mechanics such that QM will be reproduced as its approximation. He also dreamed of total exclusion of corpuscules from the future model. Reality of Einstein's dream was pure fields' reality. Recently I made his dream come true in the form of so called prequantum classical statistical field theory (PCSFT). In this approach quantum systems are described by classical random fields, e.g., electromagnetic field (instead of photon), electron field or neutron field. In this paper we generalize PCSFT to composite quantum system. It is well known that in QM, unlike classical mechanics, the state of a composite system is described by the tensor product of state spaces for its subsystems. In PCSFT one can still use Cartesian product, but state spaces are spaces of classical fields (not particles). In particular, entanglement is nothing else than correlation of classical random fields, cf. again Einstein. Thus entanglement was finally demystified.