Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 337 / 1 / iv / Solution

Let $\beta =1/T$ and $A(\tau )=e^{H\tau }A{e}^{-H\tau }$. For $0<\tau <\beta$, cyclicity of the trace givesThis is the bosonic Kubo--Martin--Schwinger condition. It identifies the two time orderings across the end of the thermal interval, so