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Cambridge University Press eBooks

Prequantum Classical Statistical Field Theory (PCSFT)

Andrei Yu. Khrennikov

Published Nov 28, 2024• 0 citations
Publisher

Abstract

This chapter presents the basics of the mathematical formalism and methodologyof the prequantum classical statistical field theory (PCSFT). In theBild-conception framework, PCSFT gives an example of acausal theoretical model (CTM) beyond QM, considered as observationalmodel (OM). Generally CTM-OM correspondence is not as straightforwardas in Bell’s model with hidden variables. In PCSFT hidden variables are randomfields fluctuating at spatial and temporal scales which are essentiallyfiner than those approached by the present measurement technology. Thekey element of the PCSFT-QM correspondence is mapping of the complexcovariance operator of a subquantum random field to the density operator.For compound systems, the situation is more complicated. Here PCSFT providestwo descriptions of compound systems with random fields valued intensor vs. Cartesian product of the Hilbert spaces of subsystems. The lattermodel matches representation of compound systems in classical statisticalmechanics. Both approaches are used for measure-theoretic representationof the correlations violating the Bell inequalities.

DOI
10.1017/9781009313469.017

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This chapter presents the basics of the mathematical formalism and methodologyof the prequantum classical statistical field theory (PCSFT). In theBild-conception framework, PCSFT gives an example of acausal theoretical model (CTM) beyond QM, considered as observationalmodel (OM). Generally CTM-OM correspondence is not as straightforwardas in Bell’s model with hidden variables. In PCSFT hidden variables are randomfields fluctuating at spatial and temporal scales which are essentiallyfiner than those approached by the present measurement technology. Thekey element of the PCSFT-QM correspondence is mapping of the complexcovariance operator of a subquantum random field to the density operator.For compound systems, the situation is more complicated. Here PCSFT providestwo descriptions of compound systems with random fields valued intensor vs. Cartesian product of the Hilbert spaces of subsystems. The lattermodel matches representation of compound systems in classical statisticalmechanics. Both approaches are used for measure-theoretic representationof the correlations violating the Bell inequalities.
In a series of previous papers we developed a purely field model of microphenomena, so called prequantum classical statistical field theory (PCSFT). This model not only reproduces important probabilistic predictions of QM including correlations for entangled systems, but it also gives a possibility to go beyond quantum mechanics (QM), i.e., to make predictions of phenomena which could be observed at the subquantum level. In this paper we discuss one of such predictions - existence of nonlocal correlations between prequantum random fields corresponding to {\it all} quantum systems. (And by PCSFT quantum systems are represented by classical Gaussian random fields and quantum observables by quadratic forms of these fields.) The source of these correlations is the common background field. Thus all prequantum random fields are "entangled", but in the sense of classical signal theory. On one hand, PCSFT demystifies quantum nonlocality by reducing it to nonlocal classical correlations based on the common random background. On the other hand, it demonstrates total generality of such correlations. They exist even for distinguishable quantum systems in factorizable states (by PCSFT terminology - for Gaussian random fields with covariance operators corresponding to factorizable quantum states).
Prequantum classical statistical field theory (PCSFT) is a new attempt to consider quantum mechanics (QM) as an emergent phenomenon, cf. with De Broglie’s “double solution” approach, Bohmian mechanics, stochastic electrodynamics (SED), Nelson’s stochastic QM and its generalization by Davidson, ‘t Hooft’s models and their development by Elze. PCSFT is a comeback to a purely wave viewpoint on QM, cf. with early Schrödinger. There is no quantum particles at all, only waves. In particular, photons are simply wave‐pulses of the classical electromagnetic field, cf. SED. Moreover, even massive particles are special “prequantum fields”: the electron field, the neutron field and so on. PCSFT claims that (soon or later) people will be able to measure components of these fields: components of the “photonic field” (the classical electromagnetic field of low intensity), electronic field, neutronic field and so on. However, at the moment (in this paper) we restrict our efforts to reproduce “simply” predictions of QM in the classical field framework. We will show that correlations of entangled systems can be obtained from fluctuations of the prequantum field. We consider the most general case: in QM the state is given by the density operator.
Recently a new attempt to go beyond quantum mechanics (QM) was done in the form of so called prequantum classical statistical field theory (PCSFT). In this approach QM is considered as incomplete theory – in total accordance with Einstein [1]. PCSFT is (one of possible) completions of QM. Hidden variables are given by classical fields. (It is again in accordance with Einstein.) These prequantum fields fluctuate on time and space scales which are essentially finer than scales of QM, cf. 't Hooft's attempt to go beyond QM [2, 3]. However opposite to Einstein, PCSFT does not assign results of quantum observations directly to hidden variables. Bohr's idea that the whole experimental arrangement should be taken into account is basic for PCSFT-based measurement theory. This theory for measurements of momentum of quantum particles is presented in this paper.
We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. ``A quantum system in a stationary state ψψψ'' in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of {\it stability of hydrogen atom.
We developed a purely field model of microphenomena—prequantum classical statistical field theory (PCSFT). This model not only reproduces important probabilistic predictions of quantum mechanics (QM) including correlations for entangled systems, but also gives a possibility to go beyond QM, i.e. to make predictions of phenomena that could be observed at the subquantum level. In this paper, we discuss one such prediction—the existence of nonlocal correlations between prequantum random fields corresponding to all quantum systems. (And by PCSFT, quantum systems are represented by classical Gaussian random fields and quantum observables by quadratic forms of these fields.) The source of these correlations is the common background field. Thus all prequantum random fields are 'entangled', but in the sense of classical signal theory. On the one hand, PCSFT demystifies quantum nonlocality by reducing it to nonlocal classical correlations based on the common random background. On the other hand, it demonstrates total generality of such correlations. They exist even for distinguishable quantum systems in factorizable states (by PCSFT terminology—for Gaussian random fields with covariance operators corresponding to factorizable quantum states).