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Basically the Green Function can be put in terms of eigenfunctions (or eigenmodes) like so: $$ G(x,x')=\sum_{\text{relevant modes}}u^{*}(x')u(x) $$ in some cases the sum turns to integral. One of the basic premises of Sturm-Liouville theorem (I hope I spelled it correctly), is that given a Linear operator $\hat{L}$, and an equation: $$ \hat{L}y(x)=f(x) $$ ...


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I think some of your confusion stems from the fact that there are two different kinds of vacuua in QFT. First there is the vacuum of the free theory, usually denoted $|0\rangle $, second there is the full (interacting) vacuum, usually denoted $|\Omega \rangle$. What we want to calculate are the different quantities in the full theory like: \begin{equation} ...


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Lets start with the second question: Also, should I worry about taking $T\to\infty(1-i\epsilon)$ instead of $T\to \infty$ (which would the natural thing to do in (1))? And there lets start with asking ourselves "why" do we do the limit in the first place? The answer is -- we don't want the expectation for the $|\phi_a\rangle$ and $|\phi_b\rangle$ ...



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