Imaginary-time-ordered correlation function

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.