Representation of quantum field theory as classical statistical mechanics for field functionals

The representation of quantum field theory as classical statistical mechanics for the scalar boson field is discussed. Its generalization to other quantum field requires more complicated considerations but is possible in principle. The suggested model uses the phase of classical prequantum field model where Schwartz space of distributions and the Gaussian measure correspond to the free boson field. The most important operators of quantum field theory, such as the free Hamiltonian and the operator N of number of particles, are constructed by secondary quantization. They are approximated by the bounded operators corresponding to the approximation of the kernels by smooth functions. The quantum field model with continuous operators is the space of density operators and the space of self-adjoint linear continuous operators. The quantum field theory can be represented as the classical statistical mechanics of Gaussian ensembles of harmonic oscillators in the space of the field functional.