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 Schrödinger picture exists for arbitrary foliations as a unitary image of the Heisenberg picture for the theory. An explicit construction of the Schrödinger 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 Schrödinger picture physical states satisfy a functional Schrödinger equation which includes a slice-dependent c-number quantum correction, in accord with a proposal of Kuchař. The spatial diffeomorphism invariance of the Schrödinger picture physical states is established. Fundamental difficulties arise when trying to generalize these results to higher-dimensional spacetimes.