PHASE-SPACE BOUNDS FOR QUANTUM-MECHANICS ON A COMPACT LIE GROUP

Let K be a compact, connected Lie group and K-C its complexification. I consider the Hilbert space HL(2)(K-C, v(t)) of holomorphic functions introduced in [H1], where the parameter t is to be interpreted as Pla nck's constant. In Light of CL-S], the complex group K-C may be identi fied canonically with the cotangent bundle of K. Using this identifica tion I associate to each F epsilon HL(2)(K-C, v(t)) a ''phase space pr obability density.'' The main result of this paper is Theorem 1, which provides an upper bound on this density which holds uniformly over al l F and all points in phase space. Specifically, the phase space proba bility density is at most a(t)(2 pi t)(-n), where n = dim K and a(t) i s a constant which tends to one exponentially fast as t tends to zero. At least for small t, this bound cannot be significantly improved. Wi th t regarded as Planck's constant, the quantity (2 pi t)(-n) is preci sely what is expected on physical grounds. Theorem 1 should be interpr eted as a form of the Heisenberg uncertainty principle for K, that is, a limit on the concentration of states in phase space. The theorem su pports the interpretation of the Hilbert space HL(2)(K-C, v(t)) as the phase space representation of quantum mechanics for a particle with c onfiguration space K. The phase space bound is deduced from very sharp pointwise bounds on functions in HL(2)(K-C, v(t)) (Theorem 2). The pr oofs rely on precise calculations involving the heat kernel on K and t he heat kernel on K-C/K.