  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.
