ИСТИНА

Under the standard Hilbert symplectic space we understand the orthogonal sum $E=Q\oplus P\cong Q\times P$ of the two mutually isomorphic real Hilbert spaces $Q$ and $P$ (each often regarded as a dual to the another, $P\cong Q'$ and vice versa) endowed with the unitary operator $I:E'\to E$ ($E'\cong P\times Q$) defined by $I{p \choose q}={q\choose-p}$. We call the space $\mathcal E(E)$ of all infinitely (Hadamard) smooth real functions on the symplectic space $(E,I)$ the Poisson algebra if it is endowed with the ``Poisson braces'' bilinear operation $ \{\cdot,\cdot\}: \mathcal E(E)\times \mathcal E(E)\to \mathcal E(E)$ defined by $\{f,g\} (z)= f'(z)(Ig'(z))$ where $f'(z)\in E'$ is the derivative of $f$ at $z\in E$~. We call a function $f\in \mathcal E(E)$ quadratic, if $f'''\equiv 0$, and we call a subalgebra of the Poisson algebra non-trivial Poisson subalgebra, if it contains: a) all constants and continuous linear functionals, b) dence, w.r.t.\ the topology of the pointwise convergence, subspace of non-linear quadratic functions and c) infinite dimensional set of non-quadratic functions. Under the canonical $\hbar$-quantization of a non-trivial Poisson subalgebra $\mathcal A$ (elements of the algebra are called the classical observables, and real number $\hbar>0$ plays here the role of the Planck constant $h$ divided by $2\pi$) we understand any linear operator $\hat{\phantom.}:f\mapsto\hat f$ from the Poisson algebra into the some complex Lie algebra of self-adjoint operators (acting in some auxiliary complex Hilbert space $H$ and having common dense invariant subspace $D_0\subset H$ in their domains) such that $\hat 1\psi=\psi$ and on the quadratic functions subalgebra $ \mathcal A_2=\{f\in\mathcal A: f'''\equiv 0\}$ we have the homomorphic ``canonical commutation relations'': $i\hbar\widehat{\{f,g\}}=[\hat f,\hat g] \equiv \hat f\hat g-\hat g\hat f)$. We refer to the canonical $\hbar$-quantization as to the Schroedinger one if the auxiliary complex Hilbert space $H$ is a completion of an invariant, w.r.t.\ all isometric changes of variables, space $S(Q)$ of smooth complex functions, square-integrable w.r.t.\ some generalized measure on $Q$. If $Q$ is infinite dimensional, such quantizations are called second quantization. We construct a Schroedinger canonical second $\hbar$-quantization of a non-trivial Poisson subalgebra $\mathcal A$ for each real $\hbar>0$ using infinite dimensional pseudo-differential operators.