Quantum inequalities and `quantum interest' as eigenvalue problems

TL;DR

This paper reformulates quantum inequalities as eigenvalue problems involving a generalized Schrödinger operator, providing new bounds on energy densities and insights into the quantum interest conjecture in quantum field theory.

Contribution

It introduces a novel eigenvalue problem approach to quantum inequalities and applies it to analyze the quantum interest conjecture and energy compensation in quantum fields.

Findings

Energy density bounds verified for moving mirrors in 2D.

Derived optimal bounds on quantum interest rate.

Showed positive delta pulses cannot fully compensate negative ones in 4D.

Abstract

Quantum inequalities (QI's) provide lower bounds on the averaged energy density of a quantum field. We show how the QI's for massless scalar fields in even dimensional Minkowski space may be reformulated in terms of the positivity of a certain self-adjoint operator - a generalised Schroedinger operator with the energy density as the potential - and hence as an eigenvalue problem. We use this idea to verify that the energy density produced by a moving mirror in two dimensions is compatible with the QI's for a large class of mirror trajectories. In addition, we apply this viewpoint to the `quantum interest conjecture' of Ford and Roman, which asserts that the positive part of an energy density always overcompensates for any negative components. For various simple models in two and four dimensions we obtain the best possible bounds on the `quantum interest rate' and on the maximum delay between a negative pulse and a compensating positive pulse. Perhaps surprisingly, we find that - in four dimensions - it is impossible for a positive delta-function pulse of any magnitude to compensate for a negative delta-function pulse, no matter how close together they occur.