  We generalize some results of Ford and Roman constraining the possible
behaviors of renormalized expected stress-energy tensors of a free massless
scalar field in two dimensional Minkowski spacetime. Ford and Roman showed that
the energy density measured by an inertial observer, when averaged with respect
to that observers proper time by integrating against some weighting function,
is bounded below by a negative lower bound proportional to the reciprocal of
the square of the averaging timescale. However, the proof required a particular
choice for the weighting function. We extend the Ford-Roman result in two ways:
(i) We calculate the optimum (maximum possible) lower bound and characterize
the state which achieves this lower bound; the optimum lower bound differs by a
factor of three from the bound derived by Ford and Roman for their choice of
smearing function. (ii) We calculate the lower bound for arbitrary, smooth
positive weighting functions. We also derive similar lower bounds on the
spatial average of energy density at a fixed moment of time.
