On single-photon wave function

TL;DR

This paper rigorously formulates the single-photon wave function within the Gupta-Bleuler quantization of the electromagnetic field, reconciling different theoretical approaches and clarifying the physical state space.

Contribution

It provides a rigorous mathematical framework for the single-photon wave function using Krein space and Gupta-Bleuler quantization, unifying previous results and approaches.

Findings

Constructed a Krein space for single photons with positive definite subspace.

Reconciled Bia{ }ynicki-Birula's wave function with perturbative QED approaches.

Established the physical equivalence of states differing by Krein norm zero.

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 in the Krein space (i.e. in the ordinary Hilbert space endowed with the Gupta-Bleuler operator $\eta$). Rigorous Gupta-Bleuler quantization of the free electromagnetic field is based on a generalization of ours (published formerly) of the Mackey theory of induced representations which includes representations preserving the indefinite Krein inner-product given by the Gupta-Bleuler operator and acting in the Krein space. The free electromagnetic field is constructed by application of the Segal's second quantization functor to the specific Krein-isometric representation. A closed subspace $\mathcal{H}_{\textrm{tr}}$ of the single-photon Krein space on which the indefinite Krein-inner-product is strictly positive is constructed such that the Krein-isometric single-photon representation generates modulo unphysical states precisely the action of a representation which preserves the positive inner product on $\mathcal{H}_{\textrm{tr}}$ induced by the Krein inner product, and is equal to the direct sum of ordinary unitary representations of helicity 1 and -1 respectively. Two states of single photon Krein space are physically equivalent whenever differ by a state of Krein norm zero and whose projection on $\mathcal{H}_{\textrm{tr}}$, in the sense of the Krein-inner-product, vanishes. In particulart it follows that the results of Bia{\l}ynicki-Birula on the single-photon wave function may be reconciled with the micro-local perturbative approach to QED initiated by St\"uckelberg and Bogoliubov.