Finite-temperature quantum field theory

Finite-temperature quantum field theory represents thermal expectation values by evolution through an imaginary-time interval of length $\beta =1/T$.

For a bosonic operator $A$, the retarded Green function is $G_R(t)=-i\theta (t)⟨[A(t),A(0)]⟩$. Causality makes its Fourier transform analytic in the upper half frequency plane.

The Lehmann representation expands a thermal correlation function in energy eigenstates. Its poles occur at transition frequencies, and its residues are thermal occupation differences times operator matrix elements.

The spectral function records the weighted energy differences connected by an operator. With the convention $G_R=-i\theta ⟨[A,A]⟩$, it is proportional to $-\mathrm{Im}{G}_R$.

The fluctuation-dissipation theorem relates an equilibrium correlation function to the dissipative part of the corresponding linear response. If $\rho =-2\mathrm{Im}{G}_R$, one common convention is $S(\omega )=\rho (\omega )/(1-e^{-\omega /T})$, which becomes $S(\omega )\simeq T\rho (\omega )/\omega$ in the classical limit $|\omega |\ll T$.

A Matsubara correlator is obtained from the retarded spectral density by a Cauchy transform evaluated at an imaginary frequency.

A Drude response has a single relaxation pole, $G_R(\omega )={\sigma }_0/(1-i\omega {\tau }_0)$. Causality places the pole in the lower half plane when ${\tau }_0>0$.

An imaginary-time-ordered correlation function orders operators along the Euclidean thermal circle. Bosonic correlators are periodic with period $\beta$.

The Kubo--Martin--Schwinger condition uses cyclicity of the thermal trace to relate correlation functions separated by imaginary time $\beta$.

Bosonic Matsubara frequencies are ${\omega }_n=2\pi nT$. They are the Fourier modes on the periodic imaginary-time circle.

For $E>0$, contour summation over bosonic Matsubara frequencies givesIt separates into the zero-point term $1/(2E)$ and a thermal Bose--Einstein occupation term.