 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-337/1/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

β = 1/ T

and

A (τ) = e^{H τ} A e^{- H τ}

. For

0 < τ < β

, cyclicity of the trace gives

Tr (e^{- β H} A (τ) A (0)) = Tr (e^{- β H} A (0) A (τ - β)) .

This is the bosonic Kubo--Martin--Schwinger condition. It identifies the two time orderings across the end of the thermal interval, so

C (τ + β) = C (τ) .

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