Momentum-winding duality of the quantum rotor
= Momentum-winding duality of the quantum rotor
{title2=$\sum_ne^{-an^2}=\sqrt{\pi/a}\sum_me^{-\pi^2m^2/a}$}
<Poisson resummation> relates the momentum sum $\sum_ne^{-\beta\hbar^2n^2/(2I)}$ to the winding sum $\sqrt{2\pi I/(\beta\hbar^2)}\sum_me^{-2\pi^2Im^2/(\beta\hbar^2)}$. Gaussian Fourier transformation proves equality, including the prefactor. The momentum form converges rapidly at low temperature and the winding form at high temperature.