Phase Space Bounds for Quantum Mechanics on a Compact Lie Group

Abstract:

Let K be a compact, connected Lie group and $K_{\Bbb{C}}$ its complexification. I consider the Hilbert space ${\cal{H}}L^2\left(K_{\Bbb{C}},\nu _t\right)$ 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 $K_{\Bbb{C}}$ may be identified canonically with the cotangent bundle of K. Using this identification I associate to each $F\in {\cal{H}}L^2\left( K_{\Bbb{C}},\nu _t\right)$ 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 $a_t\left( 2\pi t\right)^{-n}$, where $n=\dim K$ 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 $\left( 2\pi t\right)^{-n}$ 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 ${\cal{H}}L^2\left( K_{\Bbb{C}},\nu _t\right)$ 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 ${\cal{H}}L^2\left( K_{\Bbb{C}},\nu _t\right)$ (Theorem 2). The proofs rely on precise calculations involving the heat kernel on K and the heat kernel on $K_{\Bbb{C}}/K$.