Quantum corrections to the Lorentz algebra due to mixed gravitational-$U(1)$-chiral anomalies

TL;DR

This paper investigates how quantum anomalies in chiral fermions interacting with gauge fields in curved spacetime modify the Lorentz algebra, revealing that quantum effects lead to a non-trivial, order-sensitive correction to the algebra.

Contribution

It provides the first calculation of quantum corrections to the Lorentz algebra caused by mixed gravitational-$U(1)$-chiral anomalies in a Riemann-Cartan background.

Findings

Quantum corrections alter the Lorentz algebra structure.

Swapping Lorentz generators results in different correction terms.

Anomalies induce order-sensitive modifications to the algebra.

Abstract

We calculate the quantum corrections to the Lorentz algebra for chiral Weyl fermions interacting with an external $U(1)$ gauge field in a background Riemann-Cartan (RC) spacetime. This was achieved by setting up the equal-time commutation relations (ETCR) for the canonical spin current of chiral Weyl fermions. Furthermore, these quantum corrections lead to an order sensitive commutator, i.e., swapping the Lorentz generators in the commutator doesn't merely lead to a sign change, but rather a completely different correction term to the Lorentz algebra. Thus, the algebra of Lorentz is altered due to anomalies associated with chiral particles.