Systematic Derivation of Noether Point Symmetries in Special Relativistic Field Theories

TL;DR

This paper systematically derives Noether point symmetries and conserved currents in special relativistic field theories without assuming transformation laws beforehand, analyzing various fields and the role of gauge invariance.

Contribution

It provides a didactic, systematic method to derive symmetries directly from the Lagrangian, including less common symmetries like superposition and gauge invariance.

Findings

Derived symmetry transformations for real and complex scalar fields.

Identified conserved currents associated with these symmetries.

Analyzed symmetries in charged scalar particles under external fields.

Abstract

A didactic and systematic derivation of Noether point symmetries and conserved currents is put forward in special relativistic field theories, without a priori assumptions about the transformation laws. Given the Lagrangian density, the invariance condition develops as a set of partial differential equations determining the symmetry transformation. The solution is provided in the case of real scalar, complex scalar, free electromagnetic, and charged electromagnetic fields. Besides the usual conservation laws, a less popular symmetry is analyzed: the symmetry associated with the linear superposition of solutions, whenever applicable. The role of gauge invariance is emphasized. The case of the charged scalar particle under external electromagnetic fields is considered, and the accompanying Noether point symmetries determined.