= Retarded circle-compactified massless propagator
{title2=$G_{\rm ret}=R^{-1}\sum_ne^{2\pi in\Delta w/R}G_{\rm ret}^{(4,m_n)},\quad m_n=2\pi|n|/R$}
For a massless field on Minkowski space times a circle of circumference $R$, the compact delta distribution expands as $R^{-1}\sum_ne^{2\pi in\Delta w/R}$. 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.
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