In one minute

Christopher Fewster and Thomas Roman show, by explicit construction for the massless scalar field in four-dimensional flat spacetime, that weighted null averages of the stress-energy tensor have no lower bound over Hadamard states. So no quantum inequality holds along null geodesics in four dimensions, unlike two-dimensional flat spacetime where one does. The difference arises because in two dimensions any two field modes move either parallel or antiparallel, while the four-dimensional proof uses modes almost parallel to the geodesic. These states still satisfy the averaged null energy condition. In any globally hyperbolic spacetime, null-contracted stress-energy averaged along a timelike worldline does obey a quantum inequality, for massive and massless fields alike.

Null energy conditions in quantum field theory