Manifestly covariant canonical quantization of gravity and diffeomorphism anomalies in four dimensions

Canonical quantization of gravity requires knowledge about the representation theory of its constraint algebra, which is physically equivalent to the algebra of arbitrary 4-diffeomorphisms. All interesting lowest-energy representations are projective, making the relevant algebra into a four-dimensional generalization of the Virasoro algebra. Such diffeomorphism anomalies are invisible in field theory, because the relevant cocycles are functionals of the observer's trajectory in spacetime. The multi-dimensional Virasoro algebra acts naturally in the phase space of arbitrary histories, with dynamics playing the role of first-class constraints. General relativity is regularized by expanding all fields in Taylor series around the observer's trajectory, and truncating at some fixed order. This regularized but manifestly general-covariant theory is quantized in the history phase space, and dynamics is imposed afterwards, in analogy with BRST quantization. Infinities arise when the regularization is removed; it is presently unclear how these should be dealt with.