Photons in the Quantum World

TL;DR

This paper explores the internal symmetries of photons using group theory, connecting their helicity to electromagnetic fields and gauge invariance through Lorentz transformations and spinor representations.

Contribution

It provides a detailed group-theoretic analysis of photon helicity and gauge invariance within the framework of Wigner's little group and SL(2,c) spinor formalism.

Findings

Photon helicity relates to electromagnetic field orientation.

Gauge degree of freedom corresponds to Lorentz-boosted rotations.

Polarization of neutrinos arises from gauge invariance.

Abstract

Einstein's photo-electric effect allows us to regard electromagnetic waves as massless particles. Then, how is the photon helicity translated into the electric and magnetic fields perpendicular to the direction of propagation? This is an issue of the internal space-time symmetries defined by Wigner's little group for massless particles. It is noted that there are three generators for the rotation group defining the spin of a particle at rest. The closed set of commutation relations is a direct consequence of Heisenberg's uncertainty relations. The rotation group can be generated by three two-by-two Pauli matrices for spin-half particles. This group of two-by-two matrices is called SU(2), with two-component spinors. The direct product of two spinors leads to four states leading to one spin-0 state and one spin-1 state with three sub-states. The SU(2) group can be expanded to another group of two-by-two matrices called SL(2,c), which serves as the covering group for the group of Lorentz transformations. In this Lorentz-covariant regime, it is possible to Lorentz-boost the particle at rest to its infinite-momentum or massless state. Also in this SL(2,c) regime, there are four spin states for each particle, as in the case of the Dirac equation. The direct product of two SL(2,c) spinors thus leads sixteen states. Among them, four of them can be used for the electromagnetic four-potential, and six for the Maxwell tensor. The gauge degree of freedom is shown to be a Lorentz-boosted rotation. The polarization of massless neutrinos is interpreted as a consequence of the gauge invariance.