  For the quantised, massless, minimally coupled real scalar field in
four-dimensional Minkowski space, we show (by an explicit construction) that
weighted averages of the null-contracted stress-energy tensor along null
geodesics are unbounded from below on the class of Hadamard states. Thus there
are no quantum inequalities along null geodesics in four-dimensional Minkowski
spacetime. This is in contrast to the case for two-dimensional flat spacetime,
where such inequalities do exist. We discuss in detail the properties of the
quantum states used in our analysis, and also show that the renormalized
expectation value of the stress energy tensor evaluated in these states
satisfies the averaged null energy condition (as expected), despite the
nonexistence of a null-averaged quantum inequality. However, we also show that
in any globally hyperbolic spacetime the null-contracted stress energy averaged
over a timelike worldline does satisfy a quantum inequality bound (for both
massive and massless fields). We comment briefly on the implications of our
results for singularity theorems.
