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In QFT, the Matsubara frequencies are defined as $$\omega_n=\dfrac{2n\pi}{\hbar\beta}\quad\text{(bosons)}\quad\text{or}\quad\omega_n=\dfrac{(2n+1)\pi}{\hbar\beta}\quad\text{(fermions)},$$ where $\beta=1/k_BT$. In the literature you often encounter them in Green functions: $$G({\bf{k}},\omega_n)=\dfrac{1}{-i\omega_n+\xi_{{\bf{k}}}}.$$

What are these $\omega_n$ the frequencies of? What do they relate to physically?

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  • $\begingroup$ These frequencies are the poles of the Bose-Einstein and Fermi-Dirac distributions, respectively. See my answer here physics.stackexchange.com/questions/288944/… . One can probably think of them in terms of some kind of thermal peak-broadening, but I'd have to think about it. Anyway, they cannot be that physical because they are so universal :) $\endgroup$ Apr 11, 2018 at 19:03

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The imaginary-time Green-function is a periodic function on the interval $(- \hbar \beta , + \hbar \beta)$. Expanding this function w.r.t. its Fourier-series and using the periodicity-properties of the Green-function (i.e. $\mathscr{G} ( \tau + \hbar \beta) = \pm \mathscr{G} ( \tau)$), one sees that the Fourier-series becomes a series in the Matsubara-frequencies.

In this sense, these frequencies have the same meaning as any other set of frequencies that you obtain from the expansion of a periodic function in time.

Notice however, that this Green-function is usually analytically continued to the retarded Green-function from which almost every physical observable can be inferred, either by direct computation, or by computation / transitioning to other Green-functions.

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