Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2014 / Paper 1 / 2

A Wick rotation $t=-i{\tau }_E$ changes real-time evolution into $e^{-H{\tau }_E/\hslash }$. At ${\tau }_E=\beta \hslash$, taking the trace integrates equal endpoints:Thus temperature sets the imaginary-time circumference, and bosonic coordinate paths are periodic. The measure is fixed by the short-time propagator. The trace sums paths, not just a single stationary trajectory; a quadratic action is special because the saddle plus Gaussian fluctuations is exact.

Put $\mathcal{T}=\beta \hslash$ and $x=\omega \mathcal{T}$. With equal endpoints $q$, the Euclidean classical path and its action areWrite $q({\tau }_E)=q_{\mathrm{cl}}+\eta$, where $\eta (0)=\eta (\mathcal{T})=0$. Sine modes have fluctuation eigenvalues $(n\pi /\mathcal{T}{)}^2+{\omega }^2$. Relative to the free-particle determinant, their Gaussian integral contributesMultiplying the free prefactor $\sqrt{m/(2\pi \hslash \mathcal{T})}$ gives the Euclidean oscillator diagonal kernelIntegrating this over the common endpoint is precisely the trace. All fluctuations here are Dirichlet fluctuations about a fixed-endpoint path; periodicity is imposed on the full trace after endpoint integration.

The endpoint integral is a Gaussian with coefficient $A=(m\omega /\hslash )\tanh (x/2)$. ThereforeIt agrees with ${\sum }_{n\ge 0}{e}^{-x(n+1/2)}$, tends to $1/x$ at high temperature, and to $e^{-x/2}$ at low temperature. The last limit retains the oscillator zero-point energy.

For unnormalized coherent states, $|z⟩=e^{z{a}^{†}}|0⟩$, $⟨w|z⟩=e^{\bar{w}z}$ and $1=\int d^{2}z{e}^{-|z{|}^2}|z⟩⟨z|/\pi$. Insert this identity between short imaginary-time evolutions. The overlaps generate $\bar{\varphi }{\partial }_{\tau }\varphi$, while the normal-ordered symbol of $H=\hslash \omega (a^{†}a+1/2)$ givesHere $\tau ={\tau }_E/\hslash$ runs from zero to $\beta$; the exponent is dimensionless. The bosonic trace makes the complex fields periodic, not antiperiodic.

The printed expression omits the explicit vacuum factor. It represents the full oscillator only if a symmetric/Weyl determinant prescription implicitly includes that factor. With the ordinary normal-ordered time-sliced coherent-state measure, it instead represents $H=\hslash \omega a^{†}a$. The continuum formula alone does not specify the energy-zero convention.

Let $\epsilon =\hslash \omega$, use $M$ time slices and $q=e^{-\beta \epsilon /M}$. The exact normal-ordered coherent Gaussian has cyclic matrix $A=1-qP$, where $P$ shifts one slice. Its eigenvalues are $1-q{e}^{2\pi in/M}$ andRestoring the constant energy givesconsistent with the coordinate path integral. Alternatively, pairing positive and negative Matsubara frequencies symmetrically yields a determinant proportional to $x{\prod }_{n\ge 1}[1+(x/2\pi n{)}^2]=2\sinh (x/2)$. This symmetric prescription already incorporates the zero-point convention and must not receive the vacuum factor twice. A naive unregulated frequency product can hide this difference.