Strongly continuous representations in the Hilbert space: a far-reaching concept

TL;DR

This paper explores the concept of strong continuity in Hilbert space representations, emphasizing its importance in quantum physics and the analytic foundations of representation theory, including local groups and projective representations.

Contribution

It provides a comprehensive analysis of strong continuity in representations, linking it to quantum physics, local groups, and projective representations, and revisits fundamental theorems in this context.

Findings

Continuity is crucial in the analytic foundations of quantum physics.

Strong continuity relates to the possibility of defining local groups.

Connections between continuity and projective representations are established.

Abstract

We revisit the fundamental notion of continuity in representation theory, with special attention to the study of quantum physics. After studying the main theorem in the context of representation theory, we draw attention to the significant aspect of continuity in the analytic foundations of Wigner work. We conclude the paper by reviewing the connection between continuity, the possibility of defining certain local groups, and their relation to projective representations.