Perturbative anomalies in quantum mechanics

TL;DR

This paper introduces a cohomological framework to analyze perturbative anomalies in quantum mechanics, linking anomalies to specific Chevalley-Eilenberg cohomology groups of symmetry representations.

Contribution

It develops a novel cohomological approach connecting perturbations and anomalies in quantum systems to Chevalley-Eilenberg cohomology groups.

Findings

Perturbations relate to the first cohomology group.

Anomalies relate to the second cohomology group.

Provides a mathematical framework for anomaly classification.

Abstract

In this work, we propose a cohomological approach to studying perturbative anomalies in quantum mechanics. The Hamiltonian $\hat{H}$ together with the symmetry generator $\hat{S}$ forms a unitary representation of the two-dimensional Abelian Lie algebra $\mathfrak{g}\cong \mathbb{R}^{2}$ on the Hilbert space $V$. We show that perturbations of such a system are related to the first Chevalley-Eilenberg cohomology group $H^{1}_{CE}(\mathbb{R}^{2},\mathfrak{u}(V))$. In turn, the perturbative anomalies of the symmetry $\hat{S}$ are related to the second cohomology group $H^{2}_{CE}(\mathbb{R}^{2},\mathfrak{u}(V))$.