Quantum Inequalities in Quantum Mechanics

Abstract

Abstract. We study a phenomenon occuring in various areas of quantum physics, in which an observable density (such as an energy density) which is classically pointwise nonnegative may assume arbitrarily negative expectation values after quantisation, even though the spatially integrated density remains nonnegative. Two prominent examples which have previously been studied are the energy density (in quantum field theory) and the probability flux of rightwardsmoving particles (in quantum mechanics). However, in the quantum field context, it has been shown that the magnitude and space-time extension of negative energy densities are not arbitrary, but restricted by relations which have come to be known as 'quantum inequalities'. In the present work, we explore the extent to which such quantum inequalities hold for typical quantum mechanical systems. We derive quantum inequalities of two types. The first are 'kinematical' quantum inequalities where spatially averaged densities are shown to be bounded below. Specifically, we obtain such kinematical quantum inequalities for the current density in one spatial dimension (imposing constraints on the backflow phenomenon) and for the densities arising in Weyl-Wigner quantization. The latter quantum inequalities are direct consequences of sharp Gårding inequalities. The second type are 'dynamical' quantum inequalities where one obtains bounds from below on temporally averaged densities. We derive such quantum inequalities in the case of the energy density in general quantum mechanical systems having suitable decay properties on the negative spectral axis of the total energy.Furthermore, we obtain explicit numerical values for the quantum inequalities on the onedimensional current density, using various spatial averaging weight functions. We also improve the numerical value of the related 'backflow constant' previously investigated by Bracken and Melloy. In many cases our numerical results are controlled by rigorous error estimates.

Averaged energy inequalities for the nonminimally coupled classical scalar field

“…But it is possible to write this term as a difference of positive terms. To do so, we use a simple identity, which was used in [6] to derive energy inequalities in quantum mechanics. In a slightly different form, it is given by…”

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

“…Note that this QEI applies to the normal ordered stress-energy tensor, rather than the renormalised tensor. 7 By adding a term to the both sides which depends on the renormalised stress-energy tensor in state ψ 0 and certain other smooth local geometric terms, this defect can be remedied. (The bound is then typically not a 'closed form' expression.…”

Quantum backflow and scattering

“…This effect was first described by Allcock in the context of the arrival time problem in quantum mechanics [1], and then discussed in detail by Bracken and Melloy [2]. More recently, the backflow effect has attracted renewed interest [3][4][5][6][7], partially related to a proposed experiment to measure it [8], and partially because of its connection to other "quantum inequalities" appearing in quantum field theory [3,9] .…”