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Fundamentals of Prequantum Classical Statistical Field

Theory

Published Mar 19, 2014• 0 citations
Publisher

Abstract

Quantum mechanics is a statistical theory (see, e.g., Margenau [239, 240], Ballentine [25, 26], Klyshko [216], Holevo [115, 116], Khrennikov [192]). It cannot tell us anything about an individual quantum system, e.g., an electron or a photon. It predicts only probabilities for results of measurements for large ensembles of quantum systems. Classical statistical mechanics (CSM) does the same.Why are QM and CSM based on different probabilistic models?

DOI
10.1201/b15540-6

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Quantum mechanics is a statistical theory (see, e.g., Margenau [239, 240], Ballentine [25, 26], Klyshko [216], Holevo [115, 116], Khrennikov [192]). It cannot tell us anything about an individual quantum system, e.g., an electron or a photon. It predicts only probabilities for results of measurements for large ensembles of quantum systems. Classical statistical mechanics (CSM) does the same.Why are QM and CSM based on different probabilistic models?
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
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.
Standard quantum mechanics—the theory that is made use of in practice and is, because of this, taught in university courses— is a theory that is weakly objective only. For this reason it is most conveniently and neatly expressed by means of a set of basic rules of calculation that enable physicists to make predictions concerning what will be observed. Classical and quantum physics both make considerable use of the expressions "physical system" and "state of a physical system." Quantum mechanics is essentially a probabilistic theory. Except in special cases it makes no predictions that bear on individual systems. Rather, it predicts statistical frequencies. In other words, it predicts, as a rule, the number n of times that a given event will be observed when a large number N of physical systems of the same type and satisfying specified conditions are subjected to a measurement process.