Quantum Energy Inequalities in two-dimensional conformal field theory
Christopher J. Fewster, Stefan Hollands
TL;DR
The work establishes sharp quantum energy inequalities for unitary, two-dimensional conformal field theories with a stress-energy tensor, showing that weighted integrals of the chiral stress-energy density are bounded below by a state-independent expression that scales with the central charge. The authors develop an axiomatic framework based on unitary multiplier representations of the universal cover of ${ m Diff}_+(S^1)$, derive the transformation laws (including the Schwarzian anomaly), and prove the bound for both single- and two-component stress-energy Particles, including boundary CFTs and moving mirrors. They then demonstrate how these QEIs apply to worldline and worldvolume averages and discuss moving-mirror settings, showing the bounds remain sharp and physically meaningful. Finally, they show how to realize the framework in concrete CFTs via unitary highest-weight Virasoro representations, including minimal and rational models, by exponentiating to unitary multiplier representations with a Bott cocycle, thereby confirming the broad applicability of QEIs to interacting 2D CFTs.
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 two-dimensional 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.