Quantum Energy Inequalities for the Non-Minimally Coupled Scalar Field

TL;DR

This paper extends quantum energy inequalities to non-minimally coupled scalar fields, showing they are unbounded below but can satisfy a state-dependent lower bound in curved spacetimes.

Contribution

It derives a generalized quantum energy inequality for non-minimally coupled scalar fields, allowing for state-dependent lower bounds in curved spacetimes.

Findings

Averaged energy densities are unbounded from below for non-minimally coupled fields.

A generalized QEI with state-dependent bounds is established.

The state-dependence imposes a nontrivial restriction on energy density in Minkowski space.

Abstract

In this paper we discuss local averages of the energy density for the non-minimally coupled scalar quantum field, extending a previous investigation of the classical field. By an explicit example, we show that such averages are unbounded from below on the class of Hadamard states. This contrasts with the minimally coupled field, which obeys a state-independent lower bound known as a Quantum Energy Inequality (QEI). Nonetheless, we derive a generalised QEI for the non-minimally coupled scalar field, in which the lower bound is permitted to be state-dependent. This result applies to general globally hyperbolic curved spacetimes for coupling constants in the range $0<\xi\leq 1/4$. We analyse the state-dependence of our QEI in four-dimensional Minkowski space and show that it is a nontrivial restriction on the averaged energy density in the sense that the lower bound is of lower order, in energetic terms, than the averaged energy density itself.