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 $g\cong \mathbb{R}^{2}$ 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 $H^{2}_{CE}(\mathbb{R}^{2},\mathfrak{u}(V))$.

Theorems & Definitions (18)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Example 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Lemma 2.7
• proof