Matsubara frequency

Summary

Matsubara frequencies are the discrete frequencies that appear when quantum statistical mechanics at temperature $T$ is written in imaginary time. In this formalism the time variable runs over an interval of length $\beta = 1/(k_B T)$, and bosonic quantities are periodic in it while fermionic quantities are antiperiodic. Their Fourier series therefore use the frequencies $2\pi n / \beta$ for bosons and $(2n+1)\pi/\beta$ for fermions, $n \in \Z$. Sums over these frequencies, computed by the residue theorem, produce the Bose–Einstein and Fermi–Dirac occupation factors.