Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics

TL;DR

This paper explores how Lie group representations can be formulated within rigged Hilbert spaces, providing theoretical insights, examples, and criteria for operator integrability in quantum mechanics.

Contribution

It introduces a method to obtain differentiable representations as projective limits and provides criteria for operator Lie algebra integrability in rigged Hilbert spaces.

Findings

Differentiable representations are projective limits of continuous ones.

Constructed illustrative examples of group representations in rigged Hilbert spaces.

Established a criterion for operator Lie algebra integrability.

Abstract

We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous representations in a nested scale of Hilbert spaces. We also construct a couple of examples illustrative of the key features of group representations in rigged Hilbert spaces. Finally, we establish a simple criterion for the integrability of an operator Lie algebra in a rigged Hilbert space.