= Real-time winding representation of a quantum rotor kernel
{title2=$K=\sqrt{I/(2\pi i\hbar T)}\sum_m e^{iI(\Delta+2\pi m)^2/(2\hbar T)}$}
The <quantum rotor> with angle period $2\pi$ has both the spectral kernel $(2\pi)^{-1}\sum_n e^{in\Delta-i\hbar n^2T/(2I)}$ and the displayed winding representation. The <Poisson summation formula> converts one into the other by a <Gaussian integral>. Each winding sector has a constant-velocity classical path and action $I(\Delta+2\pi m)^2/(2T)$. Quadratic fluctuations give the same prefactor in every sector, so this is exact. Real time is understood by $T\to T-i0$, which fixes the square-root branch and convergence prescription.
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