Perturbative anomalies in quantum mechanics

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 $H_{CE}^1({\mathbb{R}}^2,\mathfrak{u}(V))$. In turn, the perturbative anomalies of the symmetry $\hat{S}$ are related to the second cohomology group $H_{CE}^2({\mathbb{R}}^2,\mathfrak{u}(V))$.

8 pages