  This paper initiates a program which seeks to study the allowed spatial
distributions of negative energy density in quantum field theory. Here we deal
with free fields in Minkowski spacetime. Known restrictions on time integrals
of the energy density along geodesics, the averaged weak energy condition and
quantum inequalities are reviewed. These restrictions are then used to discuss
some possible constraints on the allowable spatial distributions of negative
energy. We show how some geometric configurations can either be ruled out or
else constrained. We also construct some explicit examples of allowed
distributions. Several issues related to the allowable spatial distributions
are also discussed. These include spacetime averaged quantum inequalities in
two-dimensional spacetime, the failure of generalizations of the averaged weak
energy condition to piecewise geodesics, and the issue of when the local energy
density is negative in the frame of all observers.
