Symplectic geometry on an infinite-dimensional phase space and an asymptotic representation of quantum averages by Gaussian functional integrals

We study the relation between the mathematical structures of statistical mechanics on an infinite-dimensional phase space (denoted by $\Omega$) and quantum mechanics. It is shown that quantum averages (given by the von Neumann trace formula) can be obtained as the main term of the asymptotic expansion of Gaussian functional integrals with respect to a small parameter $\alpha$. Here $\alpha$ is the dispersion of the Gaussian measure. The symplectic structure on the infinite-dimensional phase space plays a crucial role in our considerations. In particular, the Gaussian measures that induce quantum averages must be consistent with the symplectic structure. The equations of Schrödinger, Heisenberg and von Neumann are images of the Hamiltonian dynamics on $\Omega$.