  We show that QM can be represented as a natural projection of a classical
statistical model on the phase space $\Omega= H\times H,$ where $H$ is the real
Hilbert space. Statistical states are given by Gaussian measures on $\Omega$
having zero mean value and dispersion of very small magnitude $\alpha$ (which
is considered as a small parameter of the model). Such statistical states can
be interpreted as fluctuations of the background field, cf. with SED and
Nelson's mechanics. Physical variables (e.g., energy) are given by maps $f:
\Omega \to {\bf R}$ (functions of classical fields). The conventional quantum
representation of our prequantum classical statistical model is constructed on
the basis of the Taylor expansion (up to the terms of the second order at the
vacuum field point $\psi\_{\rm{vacuum}}\equiv 0)$ of variables $f: \Omega \to
{\bf R}$ with respect to the small parameter $\sqrt{\alpha}.$ The complex
structure of QM is induced by the symplectic structure on the
infinite-dimensional phase space $\Omega.$ A Gaussian measure (statistical
state) is represented in QM by its covariation operator. Equations of
Schr\"{o}dinger, Heisenberg and von Neumann are images of Hamiltonian dynamics
on $\Omega.$ The main experimental prediction of our prequantum model is that
experimental statistical averages can deviate from ones given by QM.
