On the canonical quantization of the electromagnetic field and the emergence of gauge invariance

TL;DR

This paper explores the canonical quantization of the electromagnetic field, demonstrating how gauge invariance naturally arises from the positive energy spectrum condition and the structure of the physical state space.

Contribution

It shows that the Gupta-Bleuler condition emerges from the positive definiteness of the energy spectrum within canonical quantization.

Findings

Gauge invariance emerges as a property of physical states.

The group structure of the theory is identified as SU(2) ⊗ SU(1,1).

Coherent states are necessary for longitudinal and scalar photon fields.

Abstract

In the framework of the canonical quantization of the electromagnetic field, we impose as primary condition on the dynamics the positive definiteness of the energy spectrum. This implies that (Glauber) coherent states have to be considered for the longitudinal and the scalar photon fields. As a result we obtain that the relation holds which in the traditional approach is called the Gupta-Bleuler condition. Gauge invariance emerges as a property of the physical states. The group structure of the theory is recognized to be the one of $SU(2) \otimes SU(1,1)$.