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

The defining integral for the retarded Green function has support only at $t\ge 0$. If $\omega ={\omega }_R+i{\omega }_I$ with ${\omega }_I>0$, then $e^{i\omega t}=e^{i{\omega }_{R}t}{e}^{-{\omega }_{I}t}$ supplies exponential damping. Under the usual tempered-growth condition on the thermal commutator, the integral and all its $\omega$ derivatives converge locally uniformly. ThereforeThis is the frequency-space expression of causality.

Let $H|a⟩=E_a|a⟩$, $p_a=e^{-E_a/T}$, and take $A$ Hermitian. Inserting energy eigenstates givesOn the support of the delta function, $E_b-E_a=\omega$ andThis has the same sign as $\omega$. Every remaining factor is nonnegative, henceThis thermal spectral-positivity statement also shows that $\mathrm{Im}{G}_R$ is odd after pairing $a,b$.

A path integral is obtained by slicing an evolution operator into short time intervals and inserting complete sets of field eigenstates between successive factors. An operator inserted at a later slice therefore appears to the left of one inserted at an earlier slice. Summing over the intermediate fields preserves this ordering, so real-time path integrals generate time-ordered correlators and Euclidean thermal path integrals generate imaginary-time-ordered correlation functions.

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

Periodicity on the interval $0\le \tau <\beta$ gives the bosonic Matsubara frequenciesThe Fourier coefficients areThe notation emphasizes that these data lie at imaginary real-time frequencies $i{\omega }_n$.

The energy-eigenstate expansion of $C(\tau )$ for $0<\tau <\beta$ isFourier integration and $e^{i{\omega }_n\beta }=1$ giveComparing this with the spectral expression in part ii yields the spectral representation of a thermal correlation function

The Drude responsehas a pole at $\omega =-i/t_0$. Analyticity in the upper half plane therefore requiresFor real frequency,The sign condition from part ii then requiresso with a nonzero causal relaxation time the convention used in this question has ${\sigma }_0<0$. In conventions where the physical conductivity is defined with an additional minus sign, its static value is positive. The parameter $t_0$ is the relaxation time: after forcing is removed, the corresponding current or response decays as $e^{-t/t_0}$.

The spectral density isSubstitution into the spectral representation, closing the contour in the half plane selected by the sign of ${\omega }_n$, givesIn particular $G(0)=-{\sigma }_0\ge 0$ under the sign convention established in part vii. The absolute value is required because a bosonic Hermitian-operator Matsubara correlator is even in ${\omega }_n$.