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ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2014/paper-1/2/e/solution

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Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 2 / e / Solution by

Codex  0 2026-10-20

Take

α = I / (i ℏ T)

in the supplied Poisson summation identity, initially with a convergent regulator. Its left side is the winding sum. Multiplying its right side by the prefactor in part d gives

\frac{I}{2 πi ℏ T} \frac{1}{2 π α} = \frac{1}{2 π}, - \frac{q ^{2}}{2 α} = - \frac{i ℏ T q ^{2}}{2 I} .

(1)

Consequently

K_{θ} = \frac{1}{2 π} \sum_{q} e^{- i q Δ - i ℏ q^{2} T / (2 I)} .

(2)

Relabelling

q = - l

gives exactly the spectral answer. The square-root branch is fixed by continuation from positive imaginary-time damping; the limiting kernel has the required periodic delta-function normalization.

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