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I learn noncommutative quantum field theory now. Here, this topic is treated: arXiv:hep-th/0109162

I understood basic equations, but I don't really understand Feynman rules for noncommutative case. I have the following questions:

  1. Considering the action $S$ of a nonc. model (e.g. Moyal product model) and now I split the action like this: $$S = \int_{0}^{T_1} dt \int d^3x L + \int_{T_1}^{T_2} dt \int d^3x L.$$ If I compute $e^{iS}$ can I write $$e^{iS}=e^{i\int_{0}^{T_1} dt \int d^3x L }e^{i\int_{T_1}^{T_2} dt \int d^3x L}$$ or is the Baker-Campbell-Hausdorff formula required? The noncommutativity in Moyal product theory concerns only the change of multiplication order of fields, but is every multiplication in such a theory dependent on order? Is in a Moyal product theory $$e^{iS} = e^{i\int_{0}^{T_1} dt \int d^3x L } \star e^{i\int_{T_1}^{T_2} dt \int d^3x L}$$ Or can I treat an action like it is commutative?

  2. Planar graphs and nonplanar graphs: Why in noncommutative graphs one has two lines parallel to each other but one with opposite direction of the other? How such graphs can be obtained?

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kryomaxim
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1 Answers1

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  1. Often when physicists refer to non-commutative field theory, they are talking about a star product $\star$ within the Lagrangian density, e.g. non-commutative $\phi^4$-theory is $$ {\cal L}~=~-\frac{1}{2}\partial_{\mu} \phi ~\partial^{\mu} \phi -\frac{1}{2} m^2\phi^2 - \frac{\lambda}{4!} \phi\star\phi\star\phi\star\phi, $$ i.e. upstairs$^1$ in the path integral.

    The star product $\star$ is often assumed to be non-commutative only in spatial directions. Then the star product $\star$ does not interfere with the time-slicing/ordering prescription in the path integral, and there will be no $\star$-differentiations downstairs in the path integral. See also e.g. my Phys.SE answer here and links therein.

  2. Planar non-commutative Feynman diagrams use 't Hooft's double index/line notation. This makes sense because the star product $\star$ may be viewed as multiplication of (possibly infinite-dimensional) matrices. The momentum $p=\ell_a-\ell_b$ in a double-line propagator is the difference between the momenta of the two single-lines. In this way the overall momentum conservation is automatically implemented, and vertices take a simpler form.

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$^1$ A correlator function $\langle F \rangle$ in the path integral formulation is schematically of the form $\langle F \rangle=\frac{1}{Z} \int F e^{\frac{i}{\hbar}S}$. The words downstairs and upstairs refer to $F$ and $S$, respectively, for hopefully obvious reasons.

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