Quantum Fields at Any Time

Abstract

The canonical quantum theory of a free field using arbitrary foliations of a flat two-dimensional spacetime is investigated. It is shown that dynamical evolution along arbitrary spacelike foliations is unitarily implemented on the same Fock space as that associated with inertial foliations. It follows that the Schrodinger picture exists for arbitrary foliations as a unitary image of the Heisenberg picture for the theory. An explicit construction of the Schrodinger picture image of the Heisenberg Fock space states is provided. The results presented here can be interpreted in terms of a Dirac constraint quantization of parametrized field theory. In particular, it is shown that the Schrodinger picture physical states satisfy a functional Schrodinger equation which includes a slice-dependent c-number quantum correction, in accord with a proposal of Kuchar. The spatial diffeomorphism invariance of the Schrodinger picture physical states is established. Fundamental difficulties arise when trying to generalize these results to higher-dimensional spacetimes.