Origin of the anomalies: the modified Heisenberg equation

TL;DR

This paper explains anomalies as arising from symmetry generators not preserving the Hamiltonian's domain, leading to extra terms in the Heisenberg equation, with applications to conformal symmetry breaking in 2D quantum mechanics.

Contribution

It provides a new explanation for anomalies based on domain issues of symmetry generators, linking to Fujikawa's path integral approach and applied to conformal symmetry breaking.

Findings

Anomalies originate from non-invariant domains of symmetry generators.

Extra terms in the Heisenberg equation account for non-conservation of charges.

Application to conformal symmetry breaking in two-dimensional quantum mechanics.

Abstract

The origin of the anomalies is analyzed. It is shown that they are due to the fact that the generators of the symmetry do not leave invariant the domain of definition of the Hamiltonian and then a term, normally forgotten in the Heisenberg equation, gives an extra contribution responsible for the non conservation of the charges. This explanation is equivalent to that of the Fujikawa in the path integral formalism. Finally, this approach is applied to the conformal symmetry breaking in two-dimensional quantum mechanics.