Matsubara formalism

In thermal quantum field theory, the Matsubara frequency summation (named after Takeo Matsubara) is the summation over discrete imaginary frequencies. It takes the following form

where ${\displaystyle \beta =\hbar /k_{B}T}$ is the inverse temperature and the frequencies ${\displaystyle \omega _{n}}$ are usually taken from either of the following two sets (with ${\displaystyle n\in \mathbb {Z} }$):

The summation will converge if ${\displaystyle g(z=i\omega )}$ tends to 0 in ${\displaystyle z\to \infty }$ limit in a manner faster than ${\displaystyle z^{-1}}$. The summation over bosonic frequencies is denoted as ${\displaystyle S_{B}}$ (with ${\displaystyle \eta =+1}$), while that over fermionic frequencies is denoted as ${\displaystyle S_{F}}$ (with ${\displaystyle \eta =-1}$). ${\displaystyle \eta }$ is the statistical sign.

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