Coherent state triplets and their inner products

TL;DR

This paper develops a framework involving triplets of spaces related to group representations, enabling easier computation of inner products in coherent state and vector coherent state representations through integral expressions.

Contribution

It introduces a new triplet space construction that simplifies inner product calculations in coherent state representations, with numerous illustrative examples.

Findings

Inner products can be efficiently computed using integral expressions.

The triplet space construction is often more practical than algebraic methods.

The approach is applicable to many examples of group representations.

Abstract

It is shown that if H is a Hilbert space for a representation of a group G, then there are triplets of spaces F_H, H, F^H, in which F^H is a space of coherent state or vector coherent state wave functions and F_H is its dual relative to a conveniently defined measure. It is shown also that there is a sequence of maps F_H -> H -> F^H which facilitates the construction of the corresponding inner products. After completion if necessary, the F_H, H, and F^H, become isomorphic Hilbert spaces. It is shown that the inner product for H is often easier to evaluate in F_H than F^H. Thus, we obtain integral expressions for the inner products of coherent state and vector coherent state representations. These expressions are equivalent to the algebraic expressions of K-matrix theory, but they are frequently more efficient to apply. The construction is illustrated by many examples.