Anomalous Commutator Algebra for Conformal Quantum Mechanics

TL;DR

This paper investigates how renormalization introduces a scale in conformal quantum mechanics, leading to an anomaly that breaks the symmetry and affects the conservation of certain charges.

Contribution

It explicitly demonstrates the emergence of a conformal anomaly and symmetry breaking in the algebra of generators due to renormalization effects.

Findings

Renormalization induces a dimensional scale in conformal quantum mechanics.

The conformal algebra exhibits an anomaly at the quantum level.

Dilation and special conformal charges are not conserved due to the anomaly.

Abstract

The structure of the commutator algebra for conformal quantum mechanics is considered. Specifically, it is shown that the emergence of a dimensional scale by renormalization implies the existence of an anomaly or quantum-mechanical symmetry breaking, which is explicitly displayed at the level of the generators of the SO(2,1) conformal group. Correspondingly, the associated breakdown of the conservation of the dilation and special conformal charges is derived.