abstract: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 functions It 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. Hence most research is restricted to so-called "space-space" noncommutativity.