Conserved and non-conserved Noether currents from the quantum effective action

TL;DR

This paper explores how the quantum effective action encodes both conserved and non-conserved Noether currents, especially for extended symmetries related to space-time geometry, revealing new divergence equations for certain currents.

Contribution

It generalizes the concept of Noether currents to include non-conserved currents associated with extended gauge transformations in quantum field theories.

Findings

Derived divergence equations for non-conserved currents.

Identified non-conserved currents for Weyl, spin, and shear symmetries.

Extended the understanding of symmetry currents beyond standard invariance.

Abstract

The quantum effective action yields equations of motion and correlation functions including all quantum corrections. We discuss here how it encodes also Noether currents at the full quantum level. Interestingly, the construction can be generalized beyond the standard symmetry transformations that leave the action invariant. We also discuss an extended set of transformations, which change the action by a term that is locally known on the level of the quantum effective action. Associated to such extended gauge transformations are currents for which we obtain a divergence-type equation of motion, but they are not conserved. We call them non-conserved Noether currents. We discuss in particular symmetries and extended transformations associated to space-time geometry for relativistic quantum field theories. These encompass local dilatations or Weyl gauge transformation, local Lorentz transformations and local shear transformations. Together they constitute the symmetry group of the frame bundle GL$(d)$. The corresponding non-conserved Noether currents are the dilatation or Weyl current, the spin current and the shear current. In particular for the latter we obtain a new divergence-type equations of motion.