Representation Space

Abstract

In quantum mechanics one can represent the Hilbert space of states as the space of square integrable complex functions on the spectrum of any given complete set of commuting observables. In the process of quantization, however, one has only the classical phase space (X, ω) to work with, and one has to find a suitable classical counterpart of the notion of a complete set of commuting observables. A natural choice is a set of n = 1/2dim X functions f₁,…, f_n on X, independent at all points of X, satisfying

$$ [{{f}_{i}},{{f}_{j}}] = 0,\quad i,j \in \{ 1,2,...,n\} $$

((4.1))

such that their Hamiltonian vector fields ${\xi_{{f_1}}},\,...,{\xi_{{f_n}}}$ are complete. However, for many phase spaces of interest there does not exist such a set. If one drops the assumption that the f_i be real and globally defined, one is led to the notion of a “polarization” of (X, ω). Note that the Hamiltonian vector fields ${\xi_{{f_i}}}$, i ∈ {1,…, n} span over C an involutive distribution F on X such that

$$ {\dim _{\text{C}}}{\text{F }} ={\text{}}\frac{1}{2}\dim {\mkern 1mu} {\text{X,}} $$

((4.2))

and also that ω restricted to F vanishes identically,

$$ \omega \left| {FxF = 0.} \right. $$

((4.3))

.