An improved Noether's theorem for spacetime symmetries

TL;DR

This paper presents a unified, systematic approach to deriving improved stress-energy tensors in relativistic field theories, encompassing Lorentz, scale, and conformal invariance, without guesswork and with algebraic guarantees.

Contribution

It introduces a novel method that unifies the derivation of improvement terms for the stress-energy tensor across various spacetime symmetries in a single framework.

Findings

Unified derivation of stress-energy tensor improvements

No guesswork involved in the derivation process

Ensures algebraic properties of the tensor off-shell

Abstract

We exploit an ambiguity somewhat hidden in Noether's theorem to derive systematically, for relativistic field theories, the stress-energy tensor's improvement terms that are associated with additional spacetime symmetries beyond translations. We work out explicitly the cases of Lorentz invariance, scale invariance, and full conformal invariance. The main idea is to use, directly in the translation Noether theorem, the fact that these additional symmetries can be thought of as suitably modulated translations. Compared to more standard derivations of the improvement terms, ours (1) unifies all different cases in a single framework, (2) involves no guesswork, (3) yields the desired algebraic properties (symmetry and/or tracelessness) of the stress-energy tensor off-shell, and (4) unifies the translation Noether theorem with those of the additional spacetime symmetries, yielding at the same time both the improved stress-energy tensor and the additional Noether currents.