Will the order of taking partial derivatives matter in a noncommutative spacetime?

If so, what implications will that have on the way we do gauge theory? For example, will our Lagrangian now contain new terms which would normally cancel out in a commutative spacetime? An answer in the context of QFT or GR would be much appreciated.

  • $\begingroup$ What is a (non)commutative spacetime? $\endgroup$ – Ryan Unger Nov 9 '16 at 22:10
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    $\begingroup$ "Noncommutative spacetime" is a C* algebra, there isn't much sense to "partial derivatives" there, or "terms" in a Lagrangian. There are "covariant derivative" operators on the algebra, and Connes defines "action" in terms of traces and spectral data involving them. In a toy example it reduces to something like Higgs mechanism arxiv.org/abs/0811.0268 "The programme then is not to predict a Lagrangian, which on the contrary is taken as input but to find a noncommutative geometry which describes the standard model." $\endgroup$ – Conifold Nov 9 '16 at 23:30
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    $\begingroup$ I believe that Connes uses invariant language for building noncommutative spacetimes. Instead of partial derivatives, he uses the Dirac operator on $C^{*}$-algebra. The trace of this operator, appropriately regularized, is the "action for gravity and matter". My point is - are you sure that $\partial_{\mu}$ are well-defined on the general noncommutative space? Anyway, I am not qualified to answer questions about noncommutative geometry, but I would very much like to see it answered by somebody else. $\endgroup$ – Prof. Legolasov Nov 10 '16 at 5:07

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