= Maxwell reduction on a circle
{c}
{title2=$m_n=|n|/r$}
A real five-dimensional <Maxwell field> on a circle of radius $r$ has a <Fourier series> with coefficients $A_M^{(n)}$, $A_M^{(-n)}=(A_M^{(n)})^*$. Its mixed field-strength component is $F_{\mu5}^{(n)}=\partial_\mu A_5^{(n)}-inA_\mu^{(n)}/r$. 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.
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