 Retarded circle-compactified massless propagator

For a massless field on Minkowski space times a circle of circumference $R$, the compact delta distribution expands as $R^{-1}{\sum }_n{e}^{2\pi in\Delta w/R}$. The noncompact retarded Green function in each Fourier sector is that of a four-dimensional field with mass $m_n$. Equivalently, the time-frequency denominator is ${\mathbf{k}}^2+m_n^2-(\omega +i0{)}^2$, putting both poles below the real axis. At equal compact positions the sum has one zero mode and a doubly counted nonzero mass tower.