DEFORMATIONS OF ALGEBRAS OF OBSERVABLES AND THE CLASSICAL LIMIT OF QUANTUM MECHANICS

Abstract

The quantum algebra of observables of a particle moving on a homogeneous configuration space Q = G/H, the transformation group C*-algebra C* (G, G/H), is deformed into its classical counterpart C0 ((T*G)/H). The Poisson structure of the latter is obtained as the classical limit of the quantum commutator. The superselection sectors of both algebras describe the particle moving in an external Yang–Mills field. Analytical aspects of deformation theory, such as the nature of the limit ħ → 0, are studied in detail. A physically motivated convergence criterion in ħ is introduced. The Weyl–Moyal quantization formalism, and the associated use of Wigner distribution functions, is generalized from flat phase spaces T*ℝn to Poisson manifolds of the form (T*G)/H. The classical limit of quantum states as well as of superselection sectors is investigated. The former is handled by introducing the notion of a classical germ, generalizing coherent states. The latter is analyzed by studying the Jacobson topology on the primitive ideal space of a certain continuous field of C*-algebras, constructed from the classical and the quantum algebras of observables. The symplectic leaves of (T*G)/H are confirmed to be the correct classical analogue of the quantum superselection sectors of C*(G, G/H).