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My favorite reference for these sorts of things that straddle physics and geometry is Frankel's "The geometry of physics". In the chapter on harmonic forms, you will find what he refers to simply as "Hodge's Theorem". It's a little more general than you need, because it applies to general $p$-forms, and you only need functions (0-forms). So I'll ...

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The Green function you gave above is not the Green function for an insulator, which should generically has no pole if there is no Fermi surface. The Volovik argument is a bit circular, since you know from the beginning that $p=p_{F}$ is the Fermi momentum, defining the Fermi surface. When $p\approx p_{F}$ you can infer the form of the Green function as you ...

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All timelike geodesics in Minkowski spacetime start at past timelike infinity and end at future timelike infinity. The worldlines of Rindler observers are not geodesics, whereas the worldlines of Minkowski observers are. Heuristically think of a flat Euclidean plane. There are plenty of inextendible curves that don't go to infinity, but all geodesics start ...

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