Stress Tensors for Instantaneous Vacua in 1+1 Dimensions

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 surface (here a spacelike curve) in terms of the length L of the curve (if closed), the local extrinsic curvature K of the curve, its derivative K ′ with respect to proper distance x along the curve, and the scalar curvature R of the spacetime: T00 = − π/(6L2) − K2/(24π), T01 = −K ′/(12π), T11 = − π/(6L2)−K2/(24π) +R/(24π), in an orthonormal frame with the spatial vector parallel to the curve. Here = 1 for an untwisted (i.e., periodic in x) one-component massless bosonic field or for a twisted (i.e., antiperiodic in x) two-component massless fermionic field, = −12 for a twisted one-component massless bosonic field, and = −2 for an untwisted two-component massless fermionic field. 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 RD−1 × S1). The two coordinate and conformal invariants of a quantum state that are (nonlocally) determined by the stress-energy tensor are given. Applications to topologically modified deSitter spacetimes, to a flat cylinder, and to Minkowski spacetime are discussed. ∗Alberta-Thy-02-96, gr-qc/9603005 †Internet address: [email protected]