Quantum inequalities and “quantum interest” as eigenvalue problems

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 generalized Schrödinger 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.