Quantum mechanics as an approximation of statistical mechanics for classical fields

We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the term of the second order. To escape technical difficulties related to the infinite dimension of phase space for quantum mechanics, we start with a detailed presentation of our approach for the finite dimensional quantum mechanics. We also separate real and complex cases, because the reproduction of the complex structure of quantum mechanics is a special problem which is not related to approximation of classical averages. In our approach quantum mechanics is an approximative theory. It predicts statistical averages only with some precision. In principle, there might be found deviations of averages calculated within the quantum formalism from experimental averages (which are supposed to be equal to classical averages given by our model).