We show that the mathematical formalism of quantum mechanics can be interpreted as a method for approximation of classical (measure-theoretic) averages of functions f : L2(R3) → R. These are classical physical variables in our model with hidden variables - Prequantum Classical Statistical Field Theory (PCSFT). In this paper we provide a simple stochastic picture of such a quantum approximation procedure. In the probabilistic terms this is nothing else than the approximative method for computation of averages for functions of random variables. Since in PCSFT the space of hidden variables is L2(R3), the role of a classical random variable is played by a random field. In PCSFT we consider Gaussian random fields representing random fluctuations at the prequantum length scale. Quantum mechanical expression for the average (given by the von Neumann trace formula) is obtained through moving from the prequantum length scale to the quantum one (the scale at that we are able to perform measurements).
We show that the mathematical formalism of quantum mechanics can be interpreted as a method for approximation of classical (measure-theoretic) averages of functions f : L2(R3) → R. These are classical physical variables in our model with hidden variables - Prequantum Classical Statistical Field Theory (PCSFT). In this paper we provide a simple stochastic picture of such a quantum approximation procedure. In the probabilistic terms this is nothing else than the approximative method for computation of averages for functions of random variables. Since in PCSFT the space of hidden variables is L2(R3), the role of a classical random variable is played by a random field. In PCSFT we consider Gaussian random fields representing random fluctuations at the prequantum length scale. Quantum mechanical expression for the average (given by the von Neumann trace formula) is obtained through moving from the prequantum length scale to the quantum one (the scale at that we are able to perform measurements).
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 study asymptotic expansions of Gaussian integrals of analytic functionals on infinite-dimensional spaces (Hilbert and nuclear Frechet). We obtain an asymptotic equality coupling the Gaussian integral and the trace of the composition of scaling of the covariation operator of a Gaussian measure and the second (Frechet) derivative of a functional. In this way we couple classical average (given by an infinite-dimensional Gaussian integral) and quantum average (given by the von Neumann trace formula). We can interpret this mathematical construction as a procedure of “dequantization” of quantum mechanics. We represent quantum mechanics as an asymptotic projection of classical statistical mechanics with infinite-dimensional phase space. This space can be represented as the space of classical fields, so quantum mechanics is represented as a projection of “prequantum classical statistical field theory.”
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 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 the Planck magnitude -- fluctuations of the ``vacuum field.'' Physical variables (e.g., energy) are given by maps f:Ω→R (functions of classical fields). The crucial point is that statistical states and variables are symplectically invariant. The conventional quantum representation of our prequantum classical statistical model is constructed on the basis of the Teylor expansion (up to the terms of the second order at the vacuum field point ω=0) of variables f:Ω→R with respect to the small parameter κ=h. A Gaussian symplectically invariant measure (statistical state) is represented by its covariation operator (von Neumann statistical operator). A symplectically invariant smooth function (variable) is represented by its second derivative at the vacuum field point ω=0. From the statistical viewpoint QM is a statistical approximation of the prequantum classical statistical field theory (PCSFT). Such an approximation is obtained through neglecting by statistical fluctuations of the magnitude o(h),h→0, in averages of physical variables. Equations of Schrödinger, Heisenberg and von Neumann are images of dynamics on Ω with a symplectically invariant Hamilton function.
Recently it was shown that the main distinguishing features of quantum mechanics (QM) can be reproduced by a model based on classical random fields, so called prequantum classical statistical field theory (PCSFT). This model provides a possibility to represent averages of quantum observables, including correlations of observables on subsystems of a composite system (e.g., entangled systems), as averages with respect to fluctuations of classical (Gaussian) random fields. In this note we consider some consequences of PCSFT for quantum information theory. They are based on the observation \cite{W} of two authors of this paper that classical Gaussian channels (important in classical signal theory) can be represented as quantum channels. Now we show that quantum channels can be represented as classical linear transformations of classical Gaussian signal