# What is the interpretation of Matsubara frequencies?

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?

• 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 :) Apr 11, 2018 at 19:03

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.