A note on conserved quantities for electromagnetic waves

TL;DR

This paper explains the infinite conserved quantities in electromagnetic waves, their symmetries, and their relation to optical angular momentum, magnetic helicity, and conformal field theory, including quantum anomalies.

Contribution

It provides a simple framework to identify and compute conserved quantities and symmetries in electromagnetic waves, linking classical and quantum aspects.

Findings

Magnetic helicity is conserved for beams and pulses.

An infinite set of conserved quantities relates to Virasoro generators.

Quantum case shows Virasoro algebra with a central charge.

Abstract

Electromagnetic waves carry an infinite number of conserved quantities. We give a simple explanation of this fact, which also shows how to write down conserved quantities at will and calculate their associated symmetry transformations. This framework is then used to discuss decompositions of optical angular momentum, and to prove that magnetic helicity is conserved for beams and pulses. Finally we describe an infinite set of electromagnetic conserved quantities that corresponds to the Virasoro generators of conformal field theories. In the quantum case the Virasoro generators acquire a central charge in their algebra, an example of a quantum anomaly.