Dirac quantization of parametrized field theory

Abstract

Parametrized field theory (PFT) is free field theory on flat spacetime in a diffeomorphism invariant disguise. It describes field evolution on arbitrary (and in general, curved) foliations of the flat spacetime instead of only the usual flat foliations, by treating the ``embedding variables'' which describe the foliation as dynamical variables to be varied in the action in addition to the scalar field. A formal Dirac quantization turns the constraints of PFT into functional Schr\"odinger equations which describe evolution of quantum states from an arbitrary Cauchy slice to an infinitesimally nearby one. This formal Schr\"odinger picture-based quantization is unitarily equivalent to the standard Heisenberg picture-based Fock quantization of the free scalar field if scalar field evolution along arbitrary foliations is unitarily implemented on the Fock space. Torre and Varadarajan (TV) showed that for generic foliations emanating from a flat initial slice in spacetimes of dimension greater than 2, evolution is not unitarily implemented, thus implying an obstruction to Dirac quantization. We construct a Dirac quantization of PFT, unitarily equivalent to the standard Fock quantization, using techniques from loop quantum gravity (LQG) which are powerful enough to supercede the no-go implications of the TV results. The key features of our quantization include an LQG type representation for the embedding variables, embedding-dependent Fock spaces for the scalar field, an anomaly free representation of (a generalization of) the finite transformations generated by the constraints, and group averaging techniques. The difference between the $1+1$-dimensional case and the case of higher spacetime dimensions is that for the latter, only finite gauge transformations are defined in quantum theory, not the infinitesimal ones.