 Solution

Continue the free-particle propagator to $t=-i\mathcal{T}$ and replace its coordinate mass by the angular inertia $I$. The Euclidean free-particle bridge amplitude at coincident endpoints isThis normalization includes the time-sliced Gaussian measure, not just a formal determinant with an unspecified multiplicative constant. Set $A=2{\pi }^{2}I/({\hslash }^2\beta )>0$. The correct Poisson summation formula is ${\sum }_{m}F(m)={\sum }_n{\int }_{-\infty }^{\infty }dxF(x)e^{-2\pi inx}$; the source PDF omits the integral sign before its $d\theta$. For $F(x)=e^{-A{x}^2+i2\pi fx}$ the Gaussian Fourier integral givesThe overall prefactor cancels exactly: $2\pi Z_0\sqrt{\pi /A}=1$, and ${\pi }^2/A=\beta {\hslash }^2/(2I)$. Hencein agreement with the angular-momentum trace. Pairing $m$ and $-m$ makes the winding expression real, and the momentum expression shows positivity. Both are periodic in flux and invariant under $f\mapsto -f$. At high temperature the zero-winding term gives $Z\simeq \sqrt{2\pi I/(\beta {\hslash }^2)}$, with exponentially small flux dependence; at low temperature the nearest integer angular-momentum state dominates, with two equal lowest terms at half-integer flux. These independent limits also check the normalization and the flux phase.