A Quantum Weak Energy Inequality¶for Dirac Fields in Curved Spacetime

Abstract

Quantum fields are well known to violate the weak energy condition of general relativity: the renormalised energy density at any given point is unbounded from below as a function of the quantum state. By contrast, for the scalar and electromagnetic fields it has been shown that weighted averages of the energy density along timelike curves satisfy 'quantum weak energy inequalities' (QWEIs) which constitute lower bounds on these quantities. Previously, Dirac QWEIs have been obtained only for massless fields in two-dimensional spacetimes. In this paper we establish QWEIs for the Dirac and Majorana fields of mass m ≥ 0 on general four-dimensional globally hyperbolic spacetimes, averaging along arbitrary smooth timelike curves with respect to any of a large class of smooth compactly supported positive weights. Our proof makes essential use of the microlocal characterisation of the class of Hadamard states, for which the energy density may be defined by point-splitting.

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

“…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].…”

“…Moreover, the same is true if f is replaced by a partial differential operator with smooth coefficients compactly supported in the coordinate patch. Microlocal formulations of the Hadamard condition are also known for the Dirac [34,30,44], Maxwell and Proca fields [14]. They may be regarded as local remnants of the spectrum condition, i.e., the Minkowski space requirement that the joint spectrum of the generators P µ of spacetime translations should lie in the future causal cone.…”

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

“…In the particular case A = W (h) for h ∈ D(M; R), this gives 15) and, noting that each term in (A.15) is a product of smooth Klein-Gordon solutions, we use the following lemma.…”

“…The important point is that the constant c g,γ bounding the weighted integral of the energy density along γ from below may depend on the weight-function g and the curve γ, but is state-independent. To comment on this constraint, we note first that an inequality of the form (1.1) has been shown to hold for the class of Hadamard states ω of the free Klein-Gordon field and the Dirac field on arbitrary, globally hyperbolic spacetimes [12,15]. Moreover, in more special situations, the inequality (1.1) was obtained in a more specific form.…”

“…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.…”