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

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.