SO(2,4)-covariant quantization of the Maxwell field in a conformally flat space

TL;DR

This paper develops an SO(2,4)-covariant quantization method for the electromagnetic field in conformally flat spaces, extending Minkowski space results via a geometric approach and gauge fixing.

Contribution

It introduces a covariant quantization scheme for Maxwell fields in conformally flat spaces using a six-dimensional geometric framework and the Gupta-Bleuler method.

Findings

Wightman functions expressed in Minkowski space

Quantization scheme preserves SO(2,4) symmetry

Hilbert space structure remains unchanged under Weyl rescaling

Abstract

We present an SO(2,4)-covariant quantization of the free electromagnetic field in conformally flat spaces (CFS). A CFS is realized in a six-dimensional space as an intersection of the null cone with a given surface. The smooth move of the latter is equivalent to perform a Weyl rescaling. This allows to transport the SO(2,4)-invariant quantum structure of the Maxwell field from Minkowski space to any CFS. Calculations are simplified and the CFS Wightman two-point functions are given in terms of their Minkowskian counterparts. The difficulty due to gauge freedom is surpassed by introducing two auxiliary fields and using the Gupta-Bleuler quantization scheme. The quantum structure is given by a vacuum state and creators/annihilators acting on some Hilbert space. In practice, only the Hilbert space changes under Weyl rescalings. Also the quantum SO(2,4)-invariant free Maxwell field does not distinguish between two CFSs.