Central extensions and quantum physics

TL;DR

This paper discusses how classical symmetries in physical systems are modified in quantum theory through central extensions, leading to new quantum degrees of freedom and affecting various physical phenomena.

Contribution

It elucidates the role of central extensions in quantum symmetry groups and their impact on quantum phenomena like the Hall effect and conformal invariance.

Findings

Central extensions introduce phase factors in quantum symmetries.

Quantum effects generate additional degrees of freedom.

Symmetries in classical field theories are altered in quantum regimes.

Abstract

The unitary implementation of a symmetry group $G$ of a classical system in the corresponding quantum theory entails unavoidable deformations $\TG$ of $G$, namely, central extensions by the typical phase invariance group U(1). The appearance of central charges in the corresponding Lie-algebra quantum commutators, as a consequence of non-trivial responses of the phase of the wave function under symmetry transformations, lead to a quantum generation of extra degrees of freedom with regard to the classical counterpart. In particular, symmetries of the Hall effect, Yang-Mills and conformally invariant classical field theories are affected when passing to the quantum realm.