In thermal quantum field theory, the Matsubara summation (named after Takeo Matsubara) is a technique used to simplify calculations involving Euclidean (imaginary time) path integrals. In thermal quantum field theory, bosonic and fermionic quantum fields ϕ ( τ ) {\displaystyle \phi (\tau )} are respectively periodic or antiperiodic in imaginary time τ {\displaystyle \tau } , with periodicity β = ℏ / k B T {\displaystyle \beta =\hbar /k_{\rm {B}}T} . Matsubara summation refers to the technique of expanding these fields in Fourier series
ϕ ( τ ) = 1 β ∑ n e − i ω n τ ϕ ( i ω n ) ⟺ ϕ ( i ω n ) = 1 β ∫ 0 β d τ e i ω n τ ϕ ( τ ) . {\displaystyle \phi (\tau )={\frac {1}{\sqrt {\beta }}}\sum _{n}e^{-i\omega _{n}\tau }\phi (i\omega _{n})\iff \phi (i\omega _{n})={\frac {1}{\sqrt {\beta }}}\int _{0}^{\beta }d\tau \ e^{i\omega _{n}\tau }\phi (\tau ).}
The frequencies ω n {\displaystyle \omega _{n}} are called the Matsubara frequencies, taking values from either of the following sets (with n ∈ Z {\displaystyle n\in \mathbb {Z} } ):
bosonic frequencies: ω n = 2 n π β , {\displaystyle \omega _{n}={\frac {2n\pi }{\beta }},}
fermionic frequencies: ω n = ( 2 n + 1 ) π β , {\displaystyle \omega _{n}={\frac {(2n+1)\pi }{\beta }},}
which respectively enforce periodic and antiperiodic boundary conditions on the field ϕ ( τ ) {\displaystyle \phi (\tau )} . Once such substitutions have been made, certain diagrams contributing to the action take the form of a so-called Matsubara summation
S η = 1 β ∑ i ω n g ( i ω n ) . {\displaystyle S_{\eta }={\frac {1}{\beta }}\sum _{i\omega _{n}}g(i\omega _{n}).}
The summation will converge if g ( z = i ω ) {\displaystyle g(z=i\omega )} tends to 0 in z → ∞ {\displaystyle z\to \infty } limit in a manner faster than z − 1 {\displaystyle z^{-1}} . The summation over bosonic frequencies is denoted as S B {\displaystyle S_{\rm {B}}} (with η = + 1 {\displaystyle \eta =+1} ), while that over fermionic frequencies is denoted as S F {\displaystyle S_{\rm {F}}} (with η = − 1 {\displaystyle \eta =-1} ). η {\displaystyle \eta } is the statistical sign. In addition to thermal quantum field theory, the Matsubara frequency summation method also plays an essential role in the diagrammatic approach to solid-state physics, namely, if one considers the diagrams at finite temperature.
Generally speaking, if at T = 0 K {\displaystyle T=0\,{\text{K}}} , a certain Feynman diagram is represented by an integral ∫ T = 0 d ω g ( ω ) {\textstyle \int _{T=0}\mathrm {d} \omega \ g(\omega )} , at finite temperature it is given by the sum S η {\displaystyle S_{\eta }} .
Summation formalism
General formalism
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