Asymptotic expansion of Gaussian integrals of analytic functionals on infinite-dimensional spaces and quantum averages

Abstract

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''.