Stress tensors for instantaneous vacua in dimensions

Abstract

The regularized expectation value of the stress - energy tensor for a massless bosonic or fermionic field in 1 + 1 dimensions is calculated explicitly for the instantaneous vacuum relative to any Cauchy curve, in terms of the length L of the curve, the local extrinsic curvature K of the curve, its derivative with respect to proper distance x along the curve and the scalar curvature R of the spacetime. For example, for an untwisted one-component massless bosonic field, , in an orthonormal frame with the spatial vector parallel to the curve. The calculation uses merely the energy - momentum conservation law and the trace anomaly, for which a very simple derivation is also given herein, as well as the expression for the Casimir energy of bosonic and fermionic fields twisted by an arbitrary amount in . The two coordinate and conformal invariants of a quantum state that are nonlocally determined by the stress - energy tensor are given, as well as various applications.