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?

  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.


$^1$ 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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