Local and Covariant Quantization of Linearized Einstein's Equations

Abstract

An analysis is given of all the possible quantizations of the linearized Einstein equations in terms of a weakly local and/or covariant potential hμν(x). The discussion is done without making the apparently arbitrary choices which characterize the standard formulations, and special attention is paid in proving those general features which follow from basic principles and are therefore common to all local and/or covariant formulations. It is shown that the requirement of locality and/or covariance alone implies that the Einstein equations cannot hold as mean values on a dense set of states, and therefore unphysical states must be introduced in an essential way. Moreover, the requirement that the Einstein equations hold as mean values on the physical states forces the existence of states of negative norm in order to define hμν as a local and/or covariant operator. Thus the characteristic features of Gupta's formulation are shown to be shared by any local and/or covariant theory. The arbitrary choices which occur in the representation of the field operator hμν, in the definition of the metric operator and in the choice of the subsidiary condition which identifies the physical states, are shown to lead to only a one-parameter family of theories. They can be classified according to the subsidiary condition (∂μhμν+q∂νhμμ)+Ψ=0,each q ≠ −¼ identifying a possible theory. This arbitrariness, which makes the literature on the subject rather confusing, is resolved by proving that all the theories are (isometrically) equivalent. Such formulations are discussed in the framework of axiomatic quantum field theory, with particular emphasis on their group theoretical contents. Finally, an extensive treatment of Gupta's formulation is given along the lines discussed by Wightman and Gärding for quantum electrodynamics.