Applications of the Capelli identities in physics and representation theory

TL;DR

This paper explores how Capelli identities can be used to construct representations of Heisenberg and Lie algebras in quantum mechanics and representation theory, highlighting their mathematical and physical applications.

Contribution

It demonstrates the application of Capelli identities and their generalizations in constructing algebra representations relevant to physics and mathematics.

Findings

Capelli identities facilitate the construction of Heisenberg algebra representations.

Generalizations of Capelli identities extend their applicability in Lie algebra representations.

The methods enable holomorphic representations via Vector Coherent State techniques.

Abstract

Capelli identities are shown to facilitate the construction of representations of various Heisenberg algebras that arise in many-particle quantum mechanics and the construction of holomorphic representations of many Lie algebras by Vector Coherent State methods. We consider the original Capelli identity and its generalizations by Turnbull and by Howe and Umeda.