  Motivated by traversable wormhole constructions that require large amounts of
negative energy, we explore constraints on the amount of negative energy that
can be carried by a free Dirac field in a slab-shaped region between two
parallel spatial planes. Specifically, we ask what is the minimum possible
uniform energy density that can exist at some time, considering all possible
states and all possibilities for the physics outside the slab. The vacuum state
where we identify the two sides of the slab with antiperiodic boundary
conditions gives one possible state with uniform negative energy, but we argue
that states with more negative energy exist above 1+1 dimensions. Technically,
we reduce the problem to studying a massive Dirac field on an interval in 1+1
dimensions and numerically search for states with uniform energy density in a
lattice regulated model. We succeed in finding states with enhanced negative
energy (relative to the antiperiodic vacuum) which also appear to have a
sensible continuum limit. Our results for the mass-dependence of the minimum
uniform energy density in 1+1 dimensions suggest that for a 3+1 dimensional
massless Dirac fermion, it is possible to have states with arbitrarily large
uniform negative energy density in an arbitrarily wide slab.
