Mass Dependence of Quantum Energy Inequality Bounds

TL;DR

This paper refines quantum energy inequality bounds for massive quantum fields in Minkowski spacetime, showing they decay exponentially with mass and providing new mathematical insights into eigenvalue asymptotics.

Contribution

It introduces more stringent, exponentially decaying bounds for massive fields and analyzes eigenvalue asymptotics of polyharmonic operators, extending previous quantum energy inequality results.

Findings

Bounds decay exponentially with mass

Eigenvalue asymptotics of polyharmonic equations determined

Comparison with ground state energy on a cylinder

Abstract

In a recent paper [J. Math. Phys. 47 082303 (2006)], Quantum Energy Inequalities were used to place simple geometrical bounds on the energy densities of quantum fields in Minkowskian spacetime regions. Here, we refine this analysis for massive fields, obtaining more stringent bounds which decay exponentially in the mass. At the technical level this involves the determination of the asymptotic behaviour of the lowest eigenvalue of a family of polyharmonic differential equations, a result which may be of independent interest. We compare our resulting bounds with the known energy density of the ground state on a cylinder spacetime. In addition, we generalise some of our technical results to general $L^p$-spaces and draw comparisons with a similar result in the literature.