A Quantum Weak Energy Inequality for Dirac fields in curved spacetime

TL;DR

This paper proves quantum weak energy inequalities for Dirac and Majorana fields of any mass in four-dimensional curved spacetimes, establishing lower bounds on averaged energy densities along timelike curves.

Contribution

It extends QWEIs to massive Dirac fields in four-dimensional curved spacetimes, using microlocal analysis and Hadamard states.

Findings

Established QWEIs for massive Dirac fields in 4D spacetimes

Applied microlocal analysis to define energy density in curved spacetime

Provided lower bounds for energy density averaged along timelike curves

Abstract

Quantum fields are well known to violate the weak energy condition of general relativity: the renormalised energy density at any given point is unbounded from below as a function of the quantum state. By contrast, for the scalar and electromagnetic fields it has been shown that weighted averages of the energy density along timelike curves satisfy `quantum weak energy inequalities' (QWEIs) which constitute lower bounds on these quantities. Previously, Dirac QWEIs have been obtained only for massless fields in two-dimensional spacetimes. In this paper we establish QWEIs for the Dirac and Majorana fields of mass $m\ge 0$ on general four-dimensional globally hyperbolic spacetimes, averaging along arbitrary smooth timelike curves with respect to any of a large class of smooth compactly supported positive weights. Our proof makes essential use of the microlocal characterisation of the class of Hadamard states, for which the energy density may be defined by point-splitting.