A commentary on single-photon wave function advocated by Bia{\l}ynicki-Birula

TL;DR

This paper integrates the concept of the single-photon wave function within the Gupta-Bleuler formalism for quantized electromagnetic fields, reconciling different theoretical approaches to photon wave functions.

Contribution

It generalizes Mackey's theory of induced representations to include indefinite inner products, enabling rigorous Gupta-Bleuler quantization of the electromagnetic field.

Findings

Reconciles Bia{ }ynicki-Birula's photon wave function with causal QED

Provides a rigorous mathematical framework for photon wave functions

Extends representation theory to indefinite inner-product spaces

Abstract

We present in this paper how the single-photon wave function for transversal photons (with the direct sum of ordinary unitary representations of helicity 1 and -1 acting on it) is subsumed within the formalism of Gupta-Bleuler for the quantized free electromagnetic field. Rigorous Gupta-Bleuler quantization of the free electromagnetic field is based on our generalization (published formerly) of the Mackey theory of induced representations which includes representations preserving the indefinite Krein inner-product given by the Gupta-Bleuler operator. In particular it follows that the results of Bia{\l}ynicki-Birula on the single-photon wave function may be reconciled with the causal perturbative approach to QED.