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 Schr\"odinger picture exists for arbitrary foliations as a unitary image of the Heisenberg picture for the theory. An explicit construction of the Schr\"odinger 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 Schr\"odinger picture physical states satisfy a functional Schr\"odinger equation which includes a slice-dependent $c$-number quantum correction, in accord with a proposal of Kucha\ifmmode \check{r}\else \v{r}\fi{}. The spatial diffeomorphism invariance of the Schr\"odinger picture physical states is established. Fundamental difficulties arise when trying to generalize these results to higher-dimensional spacetimes.
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