In quantum field theory, many models can be defined by treating spacetime as a very fine lattice, but operators that are well-defined on the lattice don't always remain well-defined in the continuous-spacetime limit. In some cases, we can fix this by smearing the operator over a region of spacetime that remains finite in the continuum limit, but in other cases the best we can do is construct operators whose n-point correlation functions remain well-defined as long as the points remain separated from each other in spacetime, even though the operators themselves do not remain well-defined as ordinary operators on the Hilbert space. The construction uses a prescription called normal ordering, which is a way of modifying the original operator to make its correlation functions well-defined in the continuum limit. This article introduces the concept and shows how to efficiently work out explicit expressions for the normal-ordered versions of arbitrary powers of a free scalar quantum field.
