Canonical SO(2,4)-invariant quantization in conformally flat spaces

TL;DR

This paper presents a method for quantizing SO(2,d)-invariant fields in conformally flat spaces by leveraging Weyl equivalence and canonical quantization, ensuring the invariance structure is preserved across curved spaces.

Contribution

It introduces a systematic approach to quantize invariant fields in conformally flat spaces using Minkowski space as a reference, maintaining the symmetry structure.

Findings

Quantum fields are constructed with two-point functions related to Minkowski space.

SO(2,d)-invariant quantum fields do not distinguish between different conformally flat spaces.

The method preserves the invariance structure across curved and flat conformally flat spaces.

Abstract

We show how to quantize SO(2,d)-invariant fields in d > 2 dimensional conformally flat spaces (CFS). The Weyl equivalence between CFSs is exploited to perform the quantization process in Minkowski space then transport the entire SO(2,d)-invariant structure to curved CFSs. We make use of the canonical quantization scheme and a special careful is made to specify a scalar product, technically related to a Cauchy surface. The latter is chosen to be common to all globally hyperbolic CFSs in order to relate the different associated Hilbert spaces. The quantum fields are constructed and the two-point functions are given in terms of their minkowskian counterparts. It appears that an SO(2,d)-invariant quantum field does not locally distinguish between two different CFSs.