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

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.