The Green's function of an inhomogeneous linear differential operator, defined on a domain with specified initial conditions or boundary conditions, is its impulse response, and mathematically we define the point particles of statistical mechanics as Dirac delta functions, which is to say impulses. At a finite temperature T, the Green's functions are periodic in imaginary time with a period of 2\beta = 2/T. Therefore, their Fourier transforms contain only a discrete set of frequencies called Matsubara frequencies.
