Generalized symmetries and conservation laws of (1+1)-dimensional Klein-Gordon equation

TL;DR

This paper explicitly derives the algebra of generalized symmetries and conservation laws of the (1+1)-dimensional Klein-Gordon equation using nonstandard computational techniques, revealing a structured symmetry and conservation law framework.

Contribution

It introduces a novel explicit computation of the symmetry algebra and conservation laws for the Klein-Gordon equation using light-cone variables and universal enveloping algebra methods.

Findings

The symmetry algebra is described in terms of the universal enveloping algebra.

The conservation laws are generated by a single first-order law.

Minimal order conserved currents are identified for each conservation law.

Abstract

Using advantages of nonstandard computational techniques based on the light-cone variables, we explicitly find the algebra of generalized symmetries of the (1+1)-dimensional Klein-Gordon equation. This allows us to describe this algebra in terms of the universal enveloping algebra of the essential Lie invariance algebra of the Klein-Gordon equation. Then we single out variational symmetries of the corresponding Lagrangian and compute the space of local conservation laws of this equation, which turns out to be generated, up to the action of generalized symmetries, by a single first-order conservation law. Moreover, for every conservation law we find a conserved current of minimal order that is contained in this conservation law.