Noncommutative quantum field theory

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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^[1] are noncommutative. One commonly studied version of such theories has the "canonical" commutation relation:

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

where ${\displaystyle x^{\mu }}$ and ${\displaystyle x^{\nu }}$ are the hermitian generators of a noncommutative ${\displaystyle C^{*}}$-algebra of "functions on spacetime". That 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. In fact, this leads to an uncertainty relation for the coordinates analogous to the Heisenberg uncertainty principle.

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

One of the novel features of noncommutative field theories is the UV/IR mixing^[2] phenomenon in which the physics at high energies affects the physics at low energies which does not occur in quantum field theories in which the coordinates commute.

Other features include violation of Lorentz invariance due to the preferred direction of noncommutativity. Relativistic invariance can however be retained in the sense of twisted Poincaré invariance of the theory.^[3] The causality condition is modified from that of the commutative theories.