Strict quantization of coadjoint orbits

TL;DR

This paper develops a rigorous framework for strict quantization of coadjoint orbits of compact Lie groups, constructing explicit positive quantization maps using coherent states and representation theory.

Contribution

It introduces a new strict quantization scheme for integral coadjoint orbits, utilizing coherent states and positive maps, extending previous deformation quantization approaches.

Findings

Constructed strict quantization for all integral coadjoint orbits

Quantization maps are positive and derived from coherent states

Established conditions linking physical and strict quantizations

Abstract

A strict quantization of a compact symplectic manifold $S$ on a subset $I\subseteq\R$, containing 0 as an accumulation point, is defined as a continuous field of $C^*$-algebras $\{A_{\hbar}\}_{\hbar\in I}$, with $A_0=C_0(S)$, and a set of continuous cross-sections $\{Q(f)\}_{f\in C^{\infty}(S)}$ for which $Q_0(f)=f$. Here $Q_{\hbar}(f^*)=Q_{\hbar}(f)^*$ for all $\hbar\in I$, whereas for $\hbar\to 0$ one requires that $i[Q_{\hbar}(f),Q_{\hbar}(g)]/\hbar\to Q_{\hbar}(\{f,g\})$ in norm. We discuss general conditions which guarantee that a (deformation) quantization in a more physical sense leads to one in the above sense. Using ideas of Berezin, Lieb, Simon, and others, we construct a strict quantization of an arbitrary integral coadjoint orbit $O_{\lm}$ of a compact connected Lie group $G$, associated to a highest weight $\lm$. Here $I=0\cup 1/\N$, so that $\hbar=1/k$, $k\in\N$, and $A_{1/k}$ is defined as the $C^*$-algebra of all matrices on the finite-dimensional Hilbert space $V_{k\lm}$ carrying the irreducible representation $U_{k\lm}(G)$ with highest weight $k\lm$. The quantization maps $Q_{1/k}(f)$ are constructed from coherent states in $V_{k\lm}$, and have the special feature of being positive maps.