# Interaction vertex in non-commutative QFT

If $$\hat{S}_{1}=i \int d^{d} x \mathcal{L}_{I}$$ and \begin{aligned} V\left(x_{1}, x_{2}, \ldots, x_{n}\right) & \equiv \int\left[\prod_{j=1}^{n} \frac{d k_{j}}{(2 \pi)^{d}}\right] e^{i k_{\mu}^{j} x_{j}^{\mu}} \exp \left(-\frac{i}{2} \sum_{i with interaction Lagrangian taken to be $$\mathcal{L}_{I}=\frac{g}{4 !} \phi(x) * \phi(x) * \phi(x) * \phi(x)$$ applying Wick's theorem leads to \begin{aligned} & \int\left(\prod_{j=1}^{4} d x_{j}\right) e^{i p_{\mu}^{1} x_{1}^{\mu}+i p_{\mu}^{2} x_{2}^{\mu}+i p_{\mu}^{3} x_{3}^{\mu}+i p_{\mu}^{4} x_{4}^{\mu}} \left\langle 0\left|T\left\{\phi(x_{1}) \phi(x_{2}) \phi(x_{3}) \phi(x_{4}) \hat{S}_{1}\right\}\right| 0 \right\rangle \\ =& i \frac{g}{4 !} \tilde{G}\left(p_{1}\right) \tilde{G}\left(p_{2}\right) \tilde{G}\left(p_{3}\right) \tilde{G}\left(p_{4}\right) \\ \times & \int\left(\prod_{j=1}^{4} d x_{j}\right) V\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \sum_{P} e^{\left(i p_{\mu}^{1} x_{\alpha_{1}}^{\mu}+i p_{\mu}^{2} x_{\alpha_{2}}^{\mu}+i p_{\mu}^{3} x_{\alpha_{3}}^{\mu}+i p_{\mu}^{4} x_{\alpha_{4}}^{\mu}\right)} \end{aligned} where $$\sum_P$$ extends over all permutations of $$\alpha_{1}, \alpha_{2}, \alpha_{3}$$ and $$\alpha_{4}$$. I understand that from normal ordering we get the four 2-point Green's functions but how does one get the term $$\sum_{P} e^{\left(i p_{\mu}^{1} x_{\alpha_{1}}^{\mu}+i p_{\mu}^{2} x_{\alpha_{2}}^{\mu}+i p_{\mu}^{3} x_{\alpha_{3}}^{\mu}+i p_{\mu}^{4} x_{\alpha_{4}}^{\mu}\right)}$$?

• It's from arxiv.org/abs/hep-th/0301237 eq. (102) Nov 21 '20 at 17:03
• Looks like they're just Fourier transforming. In case you're asking about the sum over permutations, that just comes from Wick's theorem Nov 22 '20 at 16:19