Relativistic probability amplitudes II. The photon

TL;DR

This paper defines and analyzes the relativistic momentum and helicity probability amplitudes for photons, exploring their transformation properties, symmetries, and implications for photon localization and gauge invariance.

Contribution

It introduces explicit forms of photon probability amplitudes, examines their transformation under Lorentz and discrete symmetries, and discusses photon localization limitations due to helicity spectrum.

Findings

Photon momentum/helicity amplitudes are explicitly constructed.

Transformation properties under Lorentz, space inversion, and time reversal are derived.

Position eigenvectors for photons are not possible due to helicity constraints.

Abstract

We identify momentum/helicity probability amplitudes for the photon and find their relativistic transformation properties. We also find their behaviour under space inversion and time reversal. The discussion begins with a review of the unitary, irreducible representations of the Poincare group for massless particles. The little group, the set of Lorentz transformations that leave the momentum unchanged, is distinctly different for massless particles compared to the little group of rest frame rotations for a massive particle. We give a physical interpretation of the little group for a massless particle. The explicit forms of the Wigner rotations for general rotations and boosts are given. In normalized superpositions of the basis vectors, we identify the momentum/helicity probability amplitudes, show their probability interpretation and find their transformation properties. We see that position eigenvectors for the photon are not possible, not because of their masslessness but because of their limited helicity spectrum. Instead, to have a measure of localization for the photon, we use the expectations of the electromagnetic field strength operators in a coherent state with a mean photon number of unity. In the construction of the polarization vectors appearing in the field strengths, we see the connection between gauge invariance and Lorentz covariance.