  Quantum Energy Inequalities (QEIs) are results which limit the extent to
which the smeared renormalised energy density of the quantum field can be
negative, when averaged along a timelike curve or over a more general timelike
submanifold in spacetime. On globally hyperbolic spacetimes the
minimally-coupled massive quantum Klein--Gordon field is known to obey a
`difference' QEI that depends on a reference state chosen arbitrarily from the
class of Hadamard states. In many spacetimes of interest this bound cannot be
evaluated explicitly. In this paper we obtain the first `absolute' QEI for the
minimally-coupled massive quantum Klein--Gordon field on four dimensional
globally hyperbolic spacetimes; that is, a bound which depends only on the
local geometry. The argument is an adaptation of that used to prove the
difference QEI and utilises the Sobolev wave-front set to give a complete
characterisation of the singularities of the Hadamard series. Moreover, the
bound is explicit and can be formulated covariantly under additional (general)
conditions. We also generalise our results to incorporate adiabatic states.
