  The expected stress-energy tensor <T_{ab}> of quantum fields generically
violates the local positive energy conditions of general relativity. However,
<T_{ab}> may satisfy some nonlocal conditions such as the averaged null energy
condition (ANEC), which would rule out traversable wormholes. Although ANEC
holds in Minkowski spacetime, it can be violated in curved spacetimes if one is
allowed to choose the spacetime and quantum state arbitrarily, without
imposition of the semiclassical Einstein equation G_{ab} = 8 \pi <T_{ab}>. In
this paper we investigate whether ANEC holds for solutions to this equation, by
studying a free, massless scalar field with arbitrary curvature coupling in
perturbation theory to second order about the flat spacetime/vacuum solution.
We "reduce the order" of the perturbation equations to eliminate spurious
solutions, and consider the limit in which the lengthscales determined by the
incoming state are much larger than the Planck length. We also need to assume
that incoming classical gravitational radiation does not dominate the first
order metric perturbation. We find that although the ANEC integral can be
negative, if we average the ANEC integral transverse to the geodesic with a
suitable Planck scale smearing function, then a strictly positive result is
obtained in all cases except for the flat spacetime/vacuum solution. This
result suggests --- in agreement with conclusions drawn by Ford and Roman from
entirely independent arguments --- that if traversable wormholes do exist as
solutions to the semiclassical equations, they cannot be macroscopic but must
be ``Planck scale''. A large portion of our paper is devoted to the analysis of
general issues concerning the nature of the semiclassical Einstein equation and
of prescriptions for extracting physically relevant solutions.
