Quantum Energy Inequalities and local covariance I: Globally hyperbolic spacetimes

TL;DR

This paper develops a framework for Quantum Energy Inequalities (QEIs) that are locally covariant, enabling the transfer of energy bounds between isometric regions in different spacetimes, with applications to static and symmetric spacetimes.

Contribution

It introduces notions of locally covariant QEIs, shows their consistency with existing inequalities, and demonstrates their utility in deriving energy bounds in various spacetimes.

Findings

QEIs satisfy local covariance conditions

Constraints on energy density depend only on geometric quantities

Concrete calculations confirm bounds align with known energy densities

Abstract

We begin a systematic study of Quantum Energy Inequalities (QEIs) in relation to local covariance. We define notions of locally covariant QEIs of both 'absolute' and 'difference' types and show that existing QEIs satisfy these conditions. Local covariance permits us to place constraints on the renormalised stress-energy tensor in one spacetime using QEIs derived in another, in subregions where the two spacetimes are isometric. This is of particular utility where one of the two spacetimes exhibits a high degree of symmetry and the QEIs are available in simple closed form. Various general applications are presented, including a priori constraints (depending only on geometric quantities) on the ground-state energy density in a static spacetime containing locally Minkowskian regions. In addition, we present a number of concrete calculations in both two and four dimensions which demonstrate the consistency of our bounds with various known ground- and thermal state energy densities. Examples considered include the Rindler and Misner spacetimes, and spacetimes with toroidal spatial sections. In this paper we confine the discussion to globally hyperbolic spacetimes; subsequent papers will also discuss spacetimes with boundary and other related issues.