  Quantum field theory violates all the classical energy conditions of general
relativity. Nonetheless, it turns out that quantum field theories satisfy
remnants of the classical energy conditions, known as Quantum Energy
Inequalities (QEIs), that have been developed by various authors since the
original pioneering work of Ford in 1978. These notes provide an introduction
to QEIs and also to some of the techniques of quantum field theory in curved
spacetime (particularly, the use of microlocal analysis together with the
algebraic formulation of QFT) that enable rigorous and general QEIs to be
derived. Specific examples are computed for the free scalar field and their
consequences are discussed. QEIs are also derived for the class of unitary,
positive energy conformal field theories in two spacetime dimensions. In that
setting it is also possible to determine the probability distribution for
individual measurements of certain smearings of the stress-energy tensor in the
vacuum state.
