Star product on $L^2(S^n)$, $n = 2, 3, 5$

TL;DR

The work constructs a Berezin-type symbolic calculus on $L^2(S^n)$ for $n=2,3,5$ using Villegas-Blas coherent states and the transformation $\mathbf B_{S^n}$ to invariant Bargmann spaces. It yields an explicit formula for the composition of Berezin symbols, defining a noncommutative star product $*_m$ on the symbol algebra $A_{(n,m)}^{(\hbar)}$ with a well-defined semiclassical limit. The star product is shown to be associative, unital, and invariant under the groups $SU(2)$, $SU(2)\times SU(2)$, and $SU(4)$ acting on $\mathbb C^2$, $\mathbb C^4$, and $\mathbb C^8$ respectively. These results connect deformation quantization on spheres with higher-dimensional group actions and provide explicit, symmetry-respecting tools for analyzing $L^2(S^n)$ in quantum-mechanical contexts on curved spaces.

Abstract

We consider the bounded linear operators with domain in the Hilbert space $L^2(S^n)$, $n=2,3,5$ and describe its symbolic calculus defined by the Berezin quantization. In particular, we derive an explicit formula for the composition of Berezin's symbols and thus a noncommutative invariant star product, which in turn is invariant under the action of the group $SU(2)$, $SU(2)\times SU(2)$ and $SU(4)$ on $\mathbb C^2$ , $\mathbb C^4$ and $\mathbb C^8$ respectively.

Theorems & Definitions (31)

• Proposition 2.1
• proof
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Definition 3.1
• Remark 3.2
• Proposition 3.3