Quantization of bounded symplectic domains associated with compact Lie groups

Abstract

We present a systematic quantization scheme for bounded symplectic domains of the form $D \times G \subset T^\ast G$, where $D \subset \mathfrak{g}^\ast$ is a bounded region defined by algebraic inequalities and $G$ is a compact Lie group with Lie algebra $\mathfrak{g}$. The finiteness of the symplectic volume implies that quantization yields a finite-dimensional Hilbert space, with observables represented by Hermitian matrices, for which we provide an explicit realization. Boundary effects necessitate modifications of the standard von Neumann and Dirac conditions, which usually underlie the correspondence principle. Physically, the compact group $G$ plays the role of momentum space, while $\mathfrak{g}^\ast$ corresponds to the (noncommutative) position space of a particle. The assumption of compact momentum space has profound physical consequences, including the supertunneling phenomenon and the emergence of a maximal fermion density.