Phase Space Bounds for Quantum Mechanics on a Compact Lie Group

Abstract

Let K be a compact, connected Lie group and KC its complexification. I consider the Hilbert space HL 2 (KC ; t )of holomorphic functions introduced in (H1), where the parameter t is to be interpreted as Planck's constant. In light of (L-S), the complex group KC may be identified canonically with the cotangent bundle of K. Using this identification I associate to each F 2H L 2 ( K C ; t )a "phase space probability density." The main result of this paper is Theorem 1, which provides an upper bound on this density which holds uniformly over all F and all points in phase space. Specifically, the phase space probability density is at most at (2t) n , where n = dim K and at is a constant which tends to one exponentially fast as t tends to zero. At least for small t, this bound cannot be significantly improved. With t regarded as Planck's constant, the quantity (2t) n is precisely what is ex- pected on physical grounds. Theorem 1 should be interpreted as a form of the Heisenberg uncertainty principle for K, that is, a limit on the concentration of states in phase space. The theorem supports the interpretation of the Hilbert space HL 2 (KC ; t )as the phase space representation of quantum mechanics for a particle with configuration space K. The phase space bound is deduced from very sharp pointwise bounds on functions in HL 2 (KC ; t )(Theorem 2). The proofs rely on precise calculations involving the heat kernel on K and the heat kernel on KC=K.