Phase Space Bounds for Quantum Mechanics on a Compact Lie Group

Let K be a compact, connected Lie group and its complexification. I consider the Hilbert space 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 may be identified canonically with the cotangent bundle of K. Using this identification I associate to each 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 , where and a _t 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 is precisely what is expected 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 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 (Theorem 2). The proofs rely on precise calculations involving the heat kernel on K and the heat kernel on . Received: 9 July 1996/Accepted: 9 September 1996