Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 337 (Applications of Quantum Field Theory)

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.

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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 $a,b$.

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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.

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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

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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 $i{\omega }_n$.

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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

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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 $e^{-t/t_0}$.

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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 ${\omega }_n$.

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In the long-wavelength $O(N)$ description of an antiferromagnet, $v$ is the antiferromagnetic spin wave velocity and $g$ controls the stiffness or strength of quantum fluctuations. The field $\lambda$ is an auxiliary Lagrange multiplier enforcing the fixed-length constraint on the order parameter. At the translation-invariant saddle, $i\lambda =m^2$ shifts every propagator denominator and $m$ is the excitation gap, which explains the name gap equation.

The Large-N expansion makes fluctuations of the auxiliary field relatively small. Its leading saddle-point condition is then self-consistent and becomes the displayed gap equation.

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As $T\to 0$, the Matsubara sum becomes $\int d\omega /(2\pi )$. Put $p_0=\omega /v$ and $M=m/v$. The factors of $v$ cancel from both sides, leavingUsing four-dimensional spherical coordinates,At the onset of spontaneous symmetry breaking, the symmetric-phase gap closes. ThusFor $g<g_c$ the constraint is instead satisfied by an ordered condensate; a positive symmetric gap exists for $g>g_c$.

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Subtracting the critical equation from the massive equation givesAs $g\downarrow g_c$ from above,Therefore, up to constants inside the slowly varying logarithm,The square-root mean-field power is modified by a logarithm. It is therefore not a pure quantum-critical power law; this is an upper-critical-dimension logarithmic correction.

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The first term is a gauge choice for the Spin coherent-state Berry phase. For a closed spin trajectory, changing the surface used to evaluate the solid angle changes the action byThe path-integral phase $e^{i{S}_B}$ must be independent of that choice, soHenceand the spin is integer or half-integer.

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Write $\theta =\pi /2-\vartheta$ and $\varphi =\phi$. To quadratic order,After dropping the total derivative $S{\phi }_t$, the quadratic Lagrangian density isIts Euler--Lagrange equations areFor modes proportional to $e^{i(kx-\omega t)}$, nontrivial amplitudes requireThusAt short wavelength the anisotropy is negligible and the quadratic dispersion is that of a conventional ferromagnetic magnon. At long wavelength, easy-plane anisotropy makes the out-of-plane fluctuation the conjugate density of the in-plane phase; eliminating it produces the linear Goldstone sound mode characteristic of a superfluid.

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