It is not too surprising that for physicists, the formulation of 
relationship between classical and quantum mechanics belongs among
the most exciting entertainments and theoretical challenges.
Many years ago, the present reviewer encountered one of these
sophisticated exercises communicated privately by R. F. Bishop 
and performed in the context of coupled-cluster description of 
many-body QUANTUM bound states. There, the parameters in the 
variational wave functions obeyed CLASSICAL field equations. 

Khrennikov (cf. ref. [4]) offers something similar. In his approach the
role of the classical partner is played by a suitable statistical
system. The key idea of the correspondence relates the density
matrix of von Neuman to the covariance matrix of the classical
probability measure in phase space.

The Khrennikov's present work is motivated by his experience with the
presentation of his results in a series of conferences. He revealed that
the acceptance of his argumentation is perceivably hindered by
several more or less purely technical obstacles related, e.g., to
his use of functional integrals. This (plus, independently, an
implicit attention paid at present to the so called quantum
informatics) led him to his present study of a simplified version of
his construction where the states of the quantum system as assumed
represented just in a finite-dimensional, $m-$dimensional Hilbert
space. 

The most pedagogical part of his effort is represented in
section 3 where he chooses $m=1$ and where his key ideas about
correspondence between the classical and quantum mean values are 
explained. In section 4 the reader finds the basic theorem on the 
projection where the classical and quantum mean values are connected 
by a certain asymptotic identity. Certain unnecessary technical 
constraints (i.e., first of all, the ``unrealistic" use of the 
Hilbert space over reals) are finally removed in sections 5 and 6.

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MR2429280 Khrennikov, A. Yu. Quantum mechanics as the quadratic Taylor approximation of classical mechanics: the finite-dimensional case. (Russian) Teoret. Mat. Fiz. 152 (2007), no. 2, 278--291. 81Q99