Bounds on negative energy densities in flat spacetime

Abstract

We generalise results of Ford and Roman which place lower bounds -known as quantum inequalities -on the renormalised energy density of a quantum field averaged against a choice of sampling function. Ford and Roman derived their results for a specific non-compactly supported sampling function; here we use a different argument to obtain quantum inequalities for a class of smooth, even and non-negative sampling functions which are either compactly supported or decay rapidly at infinity. Our results hold in d-dimensional Minkowski space (d ≥ 2) for the free real scalar field of mass m ≥ 0. We discuss various features of our bounds in 2 and 4 dimensions. In particular, for massless field theory in 2-dimensional Minkowski space, we show that our quantum inequality is weaker than Flanagan's optimal bound by a factor of 3 2 .

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

“…The bound is finite, owing to the rapid decay of g. In fact the bound given in [10] is slightly tighter than this, but (1) will suffice for our present purposes. For later reference, let us note the scaling behaviour of the bound (1).…”

“…They have since been established for the free Klein-Gordon [22,24,26,38,10,16,8,19,47,20], Dirac [47,17,12], Maxwell [26,37,14] and Proca [14] quantum fields in both flat and curved spacetimes, the RaritaSchwinger field in Minkowski space [49], and also for general unitary positiveenergy conformal field theories in two-dimensional Minkowski space [11]. We will not give a full history of the development of the subject, referring the reader to the recent reviews [9,42].…”

“…To give a flavour of the sort of results obtained, we give an example in which the energy density of a scalar field of mass m is averaged along the inertial trajectory (t, 0) in Minkowski space. It can be obtained by elementary means [10] or as a special case of the rigorous result [8].…”

Stability of Quantum Systems at Three Scales: Passivity, Quantum Weak Energy Inequalities and the Microlocal Spectrum Condition

“…However, as pointed out by Ford [17], one may argue on heuristic grounds that there must be constraints on the intensity and spatio-temporal extension of negative values of the energy density, as otherwise one could produce situations in which macroscopic violations of the second law of thermodynamics occur. This idea was further developed by Ford and several other authors (see, e.g., [18,19,32,12,11,13,14,16,47,15,30]) and led to a form of such constraints which are now called quantum weak energy inequalities (abbreviated, QWEIs) in the terminology of [15], or often simply quantum inequalities (QIs). To explain their nature, consider some quantum field propagating on a Lorentzian spacetime (M, g), and let T µν (x) ω be the expectation value of the energy-momentum tensor in a state (expectation functional) ω at some spacetime point x.…”

“…for all states ω of finite particle number and energy, where Γ d is a state-independent universal constant depending only on the spacetime dimension [19,16,11]. Thus the intensity of the weighted negative energy is at most proportional to an inverse power of its mean duration, i.e.…”