Construction of Quantum Mechanics on Phase Space

Abstract

We now come to the mathematical foundation of quantum mechanics on phase space. In the two previous chapters, motivation for and some of the desired mathematical structure of quantum mechanics on phase space has emerged. In this chapter, we give a rigorous mathematical analysis of the origin, the variety, and the structure of quantum mechanics on phase space. To review, the discovered components of the structure of quantum mechanics on phase space are:

• (i) A symmetry group;

• (ii) A classical phase space, Γ;

• (iii) Representations of the symmetry group by unitary operators acting on the Hilbert space L²(Γ);

• (iv) Localization operators (P.V.M.’s) on these representations;

• (v) Harmonic decompositions of these representations into irreducible sub-representations;

• (vi) Intertwining operators between these irreducible sub-representations and the standard irreducible representations [as commonly used in physics and obtained by the Mackey-Frobenius method of induced representations.] The intertwined images of the (P.V.M.) localization operators become P.O.V.M.’s.

• (vii) Informational completeness criteria for these P.O.V.M.’s.