Momentum-winding duality of the quantum rotor (${\sum }_n{e}^{-a{n}^2}=\sqrt{\pi /a}{\sum }_m{e}^{-{\pi }^2{m}^2/a}$)

Poisson resummation relates the momentum sum ${\sum }_n{e}^{-\beta {\hslash }^2{n}^2/(2I)}$ to the winding sum $\sqrt{2\pi I/(\beta {\hslash }^2)}{\sum }_m{e}^{-2{\pi }^{2}I{m}^2/(\beta {\hslash }^2)}$. Gaussian Fourier transformation proves equality, including the prefactor. The momentum form converges rapidly at low temperature and the winding form at high temperature.