Quantum Fields at Any Time

TL;DR

This paper demonstrates that the quantum evolution of a free field in flat 2D spacetime can be consistently described along arbitrary spacelike slices, establishing the existence of a unitary Schrödinger picture compatible with the Heisenberg picture.

Contribution

It provides an explicit construction of the Schrödinger picture for arbitrary foliations and shows its unitarity, extending the understanding of quantum field theory in non-inertial frames.

Findings

Unitary evolution along arbitrary spacelike foliations is established.

The Schrödinger picture exists as a unitary image of the Heisenberg picture.

Physical states satisfy a slice-dependent Schrödinger equation with quantum corrections.

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.