Local quantum field operator ($\mathcal{O}(x)$)

A local quantum field operator is supported at a spacetime point, with operator-valued distributions understood by smearing. Products at the same point generally require renormalization; these give composite operators. This is distinct from the topos-theoretic meaning of local operator.

A composite operator is a renormalized local product of quantum fields and their derivatives. Operators with compatible quantum numbers can undergo composite operator mixing, and products at coincident insertion points can need contact terms in a correlation function.

A contact term is a distribution supported where insertion points coincide. Its Fourier transform is local in momentum, typically polynomial. Renormalizing an operator product can require additive contact counterterms even if each operator has zero anomalous dimension.

Renormalization can relate bare operators to a matrix of renormalized operators, ${\mathcal{O}}_{i,B}=Z_{ij}{\mathcal{O}}_{j,R}$. The corresponding anomalous dimension matrix acts on each insertion slot in the Callan-Symanzik equation.