Duality and helicity: the photon wave function approach

TL;DR

This paper develops a photon wave function framework based on the Riemann-Silberstein vector, exploring duality symmetry, conserved helicity, and related conserved quantities through different Lagrangian formulations.

Contribution

It introduces a photon wave equation derived from a Dirac/Weyl-type action and connects it to Klein-Gordon and Chern-Simons frameworks for helicity conservation.

Findings

The Dirac/Weyl-type approach yields zero conserved charge.

The Klein-Gordon-type approach recovers Lipkin's 'zilch'.

The framework unifies different descriptions of photon helicity.

Abstract

The photon wave equation proposed in terms of the Riemann-Silberstein vector is derived from a first-order Dirac/Weyl-type action principle. It is symmetric w.r.t. duality transformations, but the associated Noether quantity vanishes. Replacing the fields by potentials and using instead a quadratic Klein-Gordon-type Lagrangian allows us to recover the double-Chern-Simons expression of conserved helicity and is shown to be equivalent to recently proposed alternative frameworks. Applied to the potential-modified theory the Dirac/Weyl-type approach yields again zero conserved charge, whereas the Klein-Gordon-type approach applied to the original setting yields Lipkin's "zilch".