# Matsubara frequencies as poles of distribution function

Is there any deeper meaning to why the bosonic/fermionic Matsubara frequencies appear as poles of their corresponding distribution functions (with an additional $$i$$)?

For example in the bosonic case we have: $$\omega_n=\frac{2n\pi}{\beta}$$ which are the poles of $$n_B=\frac{1}{e^{\beta \hbar \omega}-1}$$ if we say that $$\omega = i\omega_n$$

I know this is sometimes exploited when evaluating sums of functions of $$\omega_n$$.

• Mathematical techniques that happen to be applied to Physics usually don't carry physical meaning, interpreted or otherwise. Oct 26, 2016 at 9:58
• Sometimes they do though Oct 26, 2016 at 16:48

The point is that to achieve the sum over Matsubara frequencies $$\sum_{n} g(i\omega_n)$$ we can use a contour integral $$\oint_C g(z) f(z)$$ with the contour described in fig 1 here, so long as we choose an $f(z)$ with simple poles exactly at the Matsubara frequencies $\omega_n$. This determines $f(z)$ to be proportional to the Bose-Einstein distribution. In fact, if we think about the sum as computed in a correlation function, we could have derived the integral above instead by considering computing the expectation value in the thermal density matrix $\int dE n(\beta,E) |E\rangle\langle E|$ determined by the Bose-Einstein distribution $n(\beta,E)$. I think this is an equally good starting point of the logic, then deriving the Matsubara frequencies as the poles of the distribution function.