# 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 $\beta=k_BT$. 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 :) – Ryan Thorngren Apr 11 '18 at 19:03