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This is a quite heavy approach, I remember my studying time, I couldn't really help now, but found these articles may help https://iopscience.iop.org/article/10.1088/2399-6528/aaa70a/pdf https://cds.cern.ch/record/553842/files/tha62.pdf https://www.youtube.com/watch?v=3CHjvjTPpEA Book references: Alberto Cabada auth. Green’s Functions in the Theory of ...

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the integral $$\int_0^{\infty} e^{i\epsilon t}dt$$ is not-defined for $\epsilon \in R$. For your integral, Mathematica probably also gave you the condition that ${\rm Im}\{\epsilon_\lambda - \epsilon\} < 0$. For real frequencies, we have to add a small infinitesimal in order to assure convergence. Thus $$\int_0^{\infty} dt e^{i\epsilon t} \to \int_0^{\... 0 Since the number of q-points with q<2k_F is relatively small, I was able to evaluate most points (q>2k_F) with a rough k-point grid, and then the problematic few (q<2k_F) with a higher-resolution k-point grid. Although still a somewhat costly brute force approach, it works and is much more computationally feasible than evaluating all q-... 0 The numerator of the 2-pt function is$$\begin{align} \left. \frac{\delta^2 Z[J,\bar{J}]}{\delta J_k\delta \bar{J}_{\ell}} \right|_{J=0=\bar{J}} ~=~&\left. \frac{\delta^2 e^{\frac{i}{\hbar}W_c[J,\bar{J}]}}{\delta J_k\delta \bar{J}_{\ell}} \right|_{J=0=\bar{J}}\cr ~=~&\frac{i}{\hbar} \left.\frac{\delta^2 W_c[J,\bar{J}]}{\delta J_k\delta \bar{J}_{\...

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Yes, it looks like $\delta^{ab}$ is effectively the Kronecker delta. Your expression of $G^{ab}_p$ cannot be correct because it is of the form $[(A\delta^{ab}+B)^{-1}]^{ab}$. In fact the expression given for $G^{ab}_p$ is just $((p^2+r)\delta^{ab}-\Sigma_p \delta^{ab})^{-1}$.

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Here are the steps: \begin{align} \Delta_F(x) &= \int \frac{d^4 k}{(2\pi)^4} \frac{1}{k^2-m^2+i\epsilon}e^{-k x} \end{align} We firstly to a complex integral upon $k^0$ around the upper plane and the lower plane, the contours are $\gamma^+$ and $\gamma^-$ and the associated divergences are $-\omega$ and $\omega$ respectively, with \$\omega = \sqrt{|\vec{k}...

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