Physical Theories with Average Symmetry

TL;DR

This paper introduces the concept of average symmetry in physical laws, where invariance holds only on average under transformations with noise, leading to a modified Noether's theorem with dissipative currents.

Contribution

It extends symmetry concepts to include average invariance under noisy transformations and derives a modified Noether's theorem applicable in such contexts.

Findings

Actions with average symmetry obey a modified Noether's theorem.

Average symmetry involves invariance in the presence of random noise.

Potential implications for symmetry violations in quantum gravity theories.

Abstract

This Letter probes the existence of physical laws invariant only in average when subjected to some transformation. The concept of a symmetry transformation is broadened to include corruption by random noise and average symmetry is introduced by considering functions which are invariant only in average under these transformations. It is then shown that actions with average symmetry obey a modified version of Noether's Theorem with dissipative currents. The relation of this with possible violations of physical symmetries, as for instance Lorentz invariance in some quantum gravity theories, is briefly commented.