  Second quantization of a classical nonrelativistic one-particle system as a
deformation quantization of the Schrodinger spinless field is considered. Under
the assumption that the phase space of the Schrodinger field is $C^{\infty}$,
both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed
and compared. Then the geometric quantum mechanics is also quantized using the
Berezin method under the assumption that the phase space is $CP^{\infty}$
endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function
for an arbitrary particle state and its evolution equation are obtained. As is
shown this new "second quantization" leads to essentially different results
than the former one. For instance, each state is an eigenstate of the total
number particle operator and the corresponding eigenvalue is always ${1 \over
\hbar}$.
