In one minute

L. H. Ford, Adam Helfer and Thomas Roman start from the fact that quantum fields can produce local negative energy densities after renormalisation. This is possible because renormalisation involves an infinite subtraction, even though the classical energy density looks positive definite. For a massless free scalar field they construct states that superpose the vacuum with multi-mode two-particle states. These states can place an arbitrarily large amount of negative energy in a given region of space at a fixed time. Each state in the sequence used is of Hadamard form, so the energy density can be made as negative as desired with physically acceptable states. This gives an explicit counterexample: no spatially averaged quantum inequality exists in four-dimensional spacetime.

Spatially Averaged Quantum Inequalities Do Not Exist in Four-Dimensional Spacetime