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Noncommutative quantum field theory is a branch of quantum field theory in which the spatial co-ordinates do not commute:

${\displaystyle =i\theta ^{\mu \nu }\,\!}$

which means that (with any given set of axes), it is impossible to accurately measure the position of a particle with respect to more than one axis.

Various lower limits have been claimed for the noncommutative scale, (i.e. how accurately positions can be measured) but there is currently no evidence in favour of such theory (or grounds for completely ruling them out).

Based on the the noncommutative geometry of Alain Connes, the recent interest by the particle physics community was driven by a paper by Nathan Seiberg and Edward Witten. Their most novel feature is the UV/IR mixing in which the physics at high energies has (some limited) effects on physics at low energies which does not occur in quantum field theories in which the co-ordinates commute.

There is a good review of noncommutative quantum field theories freely available on the web.

It is possible to have a noncommuting time co-ordinate but this causes many problems and most research is restricted to so-called "space-space" noncommutativity.

See for example this paper and this paper.