A Quantum Weak Energy Inequality for Spin-One Fields in Curved Spacetime

TL;DR

This paper establishes quantum weak energy inequalities for electromagnetic and spin-one fields in certain curved spacetimes, providing lower bounds on energy density averages that are valid across all Hadamard states.

Contribution

It derives explicit QWEIs for spin-one fields in globally hyperbolic spacetimes with specific topological conditions, including examples like Minkowski space and the Einstein universe.

Findings

QWEIs hold for all Hadamard states of spin-one fields.

Simplified bounds for static trajectories in ultrastatic spacetimes.

Application to Minkowski space and Einstein universe.

Abstract

Quantum weak energy inequalities (QWEI) provide state-independent lower bounds on averages of the renormalised energy density of a quantum field. We derive QWEIs for the electromagnetic and massive spin-one fields in globally hyperbolic spacetimes whose Cauchy surfaces are compact and have trivial first homology group. These inequalities provide lower bounds on weighted averages of the renormalized energy density as ``measured'' along an arbitrary timelike trajectory, and are valid for arbitrary Hadamard states of the spin-one fields. The QWEI bound takes a particularly simple form for averaging along static trajectories in ultrastatic spacetimes; as specific examples we consider Minkowski space [in which case the topological restrictions may be dispensed with] and the static Einstein universe. A significant part of the paper is devoted to the definition and properties of Hadamard states of spin-one fields in curved spacetimes, particularly with regard to their microlocal behaviour.