In thermal quantum field theory, bosonic and fermionic quantum fields are respectively periodic or antiperiodic in imaginary time , with periodicity . Matsubara summation refers to the technique of expanding these fields in Fourier series
The frequencies are called the Matsubara frequencies, taking values from either of the following sets (with ):
bosonic frequencies:
fermionic frequencies:
which respectively enforce periodic and antiperiodic boundary conditions on the field .
Once such substitutions have been made, certain diagrams contributing to the action take the form of a so-called Mastubara summation
The summation will converge if tends to 0 in limit in a manner faster than . The summation over bosonic frequencies is denoted as (with ), while that over fermionic frequencies is denoted as (with ). 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.[2][3][4]
Generally speaking, if at , a certain Feynman diagram is represented by an integral , at finite temperature it is given by the sum .