Prequantum classical statistical model with infinite dimensional phase-space

Abstract

We show that quantum mechanics can be represented as an asymptotic projection of statistical mechanics of classical fields. Thus our approach does not contradict to a rather common opinion that quantum mechanics could not be reduced to statistical mechanics of classical particles. Notions of a system and causality can be reestablished on the prequantum level, but the price is sufficiently high -- the infinite dimension of the phase space. In our approach quantum observables, symmetric operators in the Hilbert space, are obtained as derivatives of the second order of functionals of classical fields. Statistical states are given by Gaussian ensembles of classical fields with zero mean value (so these are vacuum fluctuations) and dispersion $\alpha$ which plays the role of a small parameter of the model (so these are small vacuum fluctuations). Our approach might be called {\it Prequantum Classical Statistical Field Theory} - PCSFT. Our model is well established on the mathematical level. However, to obtain concrete experimental predictions -- deviations of real experimental averages from averages given by the von Neumann trace formula - we should find the energy scale $\alpha$ of prequantum classical fields.