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

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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.