Asymptotics of unitary matrix elements in canonical bases

TL;DR

This paper analyzes the asymptotic behavior of matrix elements in canonical bases of unitary group representations, linking them to symplectic geometry and Berezin-Toeplitz quantization, with applications to quantum mechanics.

Contribution

It provides a new asymptotic formula for matrix elements in irreducible unitary representations using symplectic geometry and quantization techniques.

Findings

Derived asymptotics of matrix elements as highest weight increases

Connected matrix element asymptotics to symplectic geometry of coadjoint orbits

Recovers known results for Wigner's d-matrix elements in quantum mechanics

Abstract

We compute the asymptotics of matrix elements in canonical bases of irreducible representations of the unitary group as the highest weight goes to infinity, in terms of the symplectic geometry of the associated coadjoint orbit. This uses tools of Berezin-Toeplitz quantization, and recovers as a special case the asymptotics of Wigner's d-matrix elements for the spin representations in quantum mechanics.