Phase Space Quantization I: Geometrical Ideas

This chapter introduces quantum mechanics in phase space. One of the guiding principles is the belief that the physical states of a system can be parametrized by finite dimensional smooth manifolds. Our motivating example is radar theory. Radar measurements take place on classical phase space and exhibit quantum features such as operator representations of the Heisenberg group. Classical physics is described by algebraic and geometrical structures in symplectic manifolds that serve as the arena of classical as well as quantum measurements. We treat the Kahler case as a special kind of symplectic manifold that paves the way to a presentation of quantum mechanics by deformation theory. Deformation quantization proceeds via the introduction of a star product, an associative but not commutative product between functions in phase space. The founders of this approach “suggest quantization be understood as a deformation of the structure of the algebra of classical observables rather than a radical change in the nature of the observables.” We show how the quantization of the spin 1/2 particle arises in deformation theory: it can be understood by a star product on the Bloch sphere autonomously, through the geometrical data. Hilbert space methods are introduced in this approach to make contact with conventional approaches.