Prequantum classical statistical model with infinite dimensional phase-space-2: complex representation of symplectic phase-space model

Abstract

We show that QM can be represented as a natural projection of a classical statistical model on the phase space $Ω= H\times H,$ 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 $f: Ω\to {\bf R}$ (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 $f: Ω\to {\bf R}$ with respect to the small parameter $κ= \sqrt{h}.$ 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 $o(h), h\to 0,$ in averages of physical variables. Equations of Schrödinger, Heisenberg and von Neumann are images of dynamics on $Ω$ with a symplectically invariant Hamilton function.