In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.

all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions.

In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s.

In theoretical physics, quantum field theory in curved spacetime is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime.

In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functionsIt is possible to have a noncommuting time coordinate as in the paper by Doplicher, Fredenhagen and Roberts mentioned below, but this causes many problems such as the violation of unitarity of the S-matrix.