Enlargement of symmetry groups in physics: a practitioner's guide

TL;DR

This paper provides a detailed, step-by-step guide for physicists and mathematicians to understand and construct projective unitary representations by enlarging symmetry groups, clarifying the role of topological and algebraic obstructions.

Contribution

It bridges physics and mathematics by explicitly describing how to enlarge symmetry groups for projective representations, including an algorithm and applications to physical cases.

Findings

Classifies Lie group central extensions via smooth cocycles.

Provides a practical algorithm for enlarging symmetry groups in quantum theories.

Connects group cohomology with physical symmetry group extensions.

Abstract

Wigner's classification has led to the insight that projective unitary representations play a prominent role in quantum mechanics. The physics literature often states that the theory of projective unitary representations can be reduced to the theory of ordinary unitary representations by enlarging the group of physical symmetries. Nevertheless, the enlargement process is not always described explicitly: it is unclear in which cases the enlargement has to be done to the universal cover, a central extension, or to a central extension of the universal cover. On the other hand, in the mathematical literature, projective unitary representations were extensively studied, and famous theorems such as the theorems of Bargmann and Cassinelli have been achieved. The present article bridges the two: we provide a precise, step-by-step guide on describing projective unitary representations as unitary representations of the enlarged group. Particular focus is paid to the difference between algebraic and topological obstructions. To build the bridge mentioned above, we present a detailed review of the difference between group cohomology and Lie group cohomology. This culminates in classifying Lie group central extensions by smooth cocycles around the identity. Finally, the take-away message is a hands-on algorithm that takes the symmetry group of a given quantum theory as input and provides the enlarged group as output. This algorithm is applied to several cases of physical interest. We also briefly outline a generalization of Bargmann's theory to time-dependent phases using Hilbert bundles.