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Say the time evolution for Hamiltonian $H$ is given by $U(t) = \exp(-iHt)$ and the corresponding evolution on the support of $P$ is $$PU(t)P = P\exp(-iHt)P = \exp(-iH_\text{eff}t) \equiv U_\text{eff}(t)$$ assuming $H_\text{eff}$ exists. The desired identity follows from $$ \lim_{\eta \rightarrow 0} \int_0^\infty dt \; U(t) e^{i (\epsilon + i \eta) t} = ...


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I highly recommend Richard D. Mattuck A Guide to Feynman Diagrams in the Many-Body Problem. You can read some pages here. It's a very surface level introduction, but the first 3 or so chapters are presented at what he calls a "kindergarten" level so you shouldn't have any problems understanding it. However, the last part is most definitely not ...


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There are many resources on many-body Green's functions (propagators) both on-line and in print. You may want to search "quantum field methods in many-particle systems" or "quantum field methods for condensed matter systems" or variations thereof. In any case, I personally recommend the oldie-but-goodie book by Fetter and Walecka, Quantum Theory of ...


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Every phase transition has an order parameter: something that vanishes above the transition temperature and is finite below. In superconductors, the order parameter is a complex quantity related to the superconducting gap: $\Delta = |\Delta| e^{i \phi}$. In BCS theory, there is a self-consistent equation for the gap: $\Delta_k = -\sum_q V_{kq} ...



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