Maxwell reduction on a circle ($m_n=|n|/r$)

A real five-dimensional Maxwell field on a circle of radius $r$ has a Fourier series with coefficients $A_M^{(n)}$, $A_M^{(-n)}=(A_M^{(n)}{)}^{\ast }$. Its mixed field-strength component is $F_{\mu 5}^{(n)}={\partial }_{\mu }{A}_5^{(n)}-in{A}_{\mu }^{(n)}/r$. The nonzero modes transform as $\delta A_{\mu }^{(n)}={\partial }_{\mu }{\Lambda }^{(n)}$, $\delta A_5^{(n)}=in{\Lambda }^{(n)}/r$. Thus $A_5^{(n)}$ is a Stueckelberg mechanism field and the gauge-invariant massive vector is $A_{\mu }^{(n)}-r{\partial }_{\mu }{A}_5^{(n)}/(in)$. Its mass is $|n|/r$ and it has three physical polarizations. The zero modes instead consist of a massless four-dimensional Maxwell field and a real massless scalar field; the latter cannot be removed by a periodic infinitesimal gauge transformation. A positive Fourier level and its negative level are conjugate, not independent real fields.