Quantum Energy Inequalities in Two-Dimensional Conformal Field Theory

Abstract

Quantum energy inequalities (QEIs) are state-independent lower bounds on weighted averages of the stress-energy tensor, and have been established for several free quantum field models. We present rigorous QEI bounds for a class of interacting quantum fields, namely the unitary, positive energy conformal field theories (with stress-energy tensor) on twodimensional Minkowski space. The QEI bound depends on the weight used to average the stress-energy tensor and the central charge(s) of the theory, but not on the quantum state. We give bounds for various situations: averaging along timelike, null and spacelike curves, as well as over a spacetime volume. In addition, we consider boundary conformal field theories and more general 'moving mirror' models.Our results hold for all theories obeying a minimal set of axioms which-as we show-are satisfied by all models built from unitary highest-weight representations of the Virasoro algebra. In particular, this includes all (unitary, positive energy) minimal models and rational conformal field theories. Our discussion of this issue collects together (and, in places, corrects) various results from the literature which do not appear to have been assembled in this form elsewhere.

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

“…By threading a spacetime volume by worldlines, these bounds imply the existence of spacetimeaveraged QEIs, which may also be obtained directly, as sketched below. It is known that compactly supported weighted averages over spacelike hypersurfaces [23] or null lines [15] are not generally bounded from below, except for two-dimensional conformal fields [19,11].…”

“…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].…”

Conformal Covariance and Positivity of Energy in Charged Sectors

“…3.2). As they are nonnegative but not smooth, to conclude that T (f 1 ) and T (f 2 ) are bounded from below we cannot use the result stated in [11]. However, it turns out to be (Prop.…”

“…Thus each term in itself (although not bounded) can be considered in a given charged sector. Moreover, it has been recently proved by Fewster and Holland [11] that the stress-energy tensor evaluated on a nonnegative function is bounded from below. These operators then, being local elements, remain bounded from below also in the charged sector.…”

Averaged null energy condition in spacetimes with boundaries

“…In two dimensions more can be said: in Minkowski space one may prove the ANEC for general (interacting) quantum fields with mass [14], or for unitary, positive energy conformal fields [15]; precise classes of states for which these results hold are delineated in each case. In general globally hyperbolic two-dimensional spacetimes, [13] established the ANEC on complete achronal null geodesics for the minimally coupled free scalar field.…”