Conservation Laws and Non-Lie Symmetries

TL;DR

This paper presents a novel method for deriving conservation laws for linear PDEs using symmetries beyond Lie symmetries, exemplified by the Dirac equation's discrete symmetries including CPT.

Contribution

It introduces a new approach to generate conservation laws from any symmetry of the operator, extending beyond classical Lie symmetries.

Findings

Constructed conservation laws for the Dirac equation using non-Lie symmetries.

Identified a 64-dimensional Lie algebra of discrete symmetries including CPT.

Demonstrated the method's applicability to a broad class of linear PDEs.

Abstract

We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the Dirac equation were we use our construction to find a class of conservation laws associated with a 64 dimensional Lie algebra of discrete symmetries that includes CPT.