In quantum field theory, each region of spacetime has an associated set of observables. Observables associated with individual points in smooth spacetime – or more generally with lower-dimensional submanifolds of smooth spacetime – are typically undefined as operators on a Hilbert space. Starting with a formulation that treats time as continuous but space as discrete, this article explains how to use smearing in time to construct observables whose resolution is much coarser than the discretization scale. The construction works for any renormalized operator. (This article reviews what that means.) This article also explains why smearing only in space and smearing in "euclidean spacetime" are not as effective. The general concepts are illustrated using operators constructed from a free scalar field.
