Quantum Energy Inequalities along stationary worldlines

TL;DR

This paper derives explicit quantum energy inequality bounds for a massless scalar field along stationary worldlines in Minkowski space, linking these bounds to curvature invariants and illustrating them with key examples.

Contribution

It provides the first closed-form expressions for QEIs along stationary worldlines, expanding understanding of quantum energy constraints in flat spacetime.

Findings

Explicit QEI bounds are derived for stationary worldlines.

Bounds for inertial and accelerated worldlines match known vacua.

The bounds are consistent with constant energy densities in the limit.

Abstract

Quantum energy inequalities (QEIs) are lower bounds on the averaged energy density of a quantum field. They have been proved for various field theories in general curved spacetimes but the explicit lower bound is not easily calculated in closed form. In this paper we study QEIs for the massless minimally coupled scalar field in four-dimensional Minkowski spacetime along stationary worldlines - curves whose velocity evolves under a 1-parameter Lorentz subgroup -- and find closed expressions for the QEI bound, in terms of curvature invariants of the worldline. Our general results are illustrated by specific computations for the six protoypical stationary worldlines. When the averaging period is taken to infinity, the QEI bound is consistent with a constant energy density along the worldline. For inertial and uniformly linearly accelerated worldlines, this constant value is attained by the Minkowski and Rindler vacuums respectively. It is an open question as to whether the bounds for other stationary worldlines are attained by other states of interest.