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Use a flat five-dimensional metric of signature $(+----)$ and a circle coordinate $y\sim y+2\pi r$. Start with the canonically normalized Maxwell actionExpand the real Maxwell field in a Fourier series:The circle integral makes different modes orthogonal. Writing $a^{(n)}=A_5^{(n)}$ and $k_n=n/r$, the resulting four-dimensional action isThe scalar sign is positive because the fifth direction is spacelike in the chosen mostly-minus metric. This expression follows directly by separating the $\mu \nu$ and the two identical $\mu 5$ contributions in $F_{MN}{F}^{MN}$.

A five-dimensional gauge transformation $A_M\mapsto A_M+{\partial }_M\Lambda$ acts on its modes byFor $n\ne 0$, define the gauge-invariant vectorThen $F_{\mu 5}^{(n)}=-i{k}_n{B}_{\mu }^{(n)}$, so its term is a Proca field mass term $+\frac{1}{2}{k}_n^2{B}_{\mu }^{(n)}{B}^{\mu (-n)}$. Equivalently, a periodic mode-dependent gauge transformation sets $a^{(n)}=0$. The nonzero modes are massive vectors whose longitudinal polarization is supplied by $a^{(n)}$ through the Stueckelberg mechanism:For each positive $n$ there is one complex vector, equivalently the two real sine/cosine vectors, each with three physical polarizations. The negative mode is its conjugate, not another independent complex field. This is the infinite Kaluza-Klein tower of the Maxwell reduction on a circle.

At $n=0$, $\delta a^{(0)}=0$ under periodic infinitesimal gauge transformations. The zero modes are a massless vector with two polarizations and a real massless scalar with one. The scalar is the circle component, or Wilson line degree of freedom; a generic constant value cannot be removed by a periodic gauge parameter. Both have $m_0=0$. At energies $E\ll 1/r$, the free theory's low-energy effective action is thereforeThere is no scalar potential in this free Maxwell theory. If the original action instead has coefficient $1/g_5^2$ and unrescaled constant zero-mode fields are used, integrating the circle gives $g_4^2=g_5^2/(2\pi r)$; canonical field rescaling recovers the normalization displayed above.

For the gravitational part, allow a circle radius field $R(x)$ with vacuum value $r$, and use a dimensionless angle $\chi \sim \chi +2\pi$ in the Kaluza-Klein theory ansatzThe vector comes from off-diagonal metric components; $R$ is the radion. Perform the coordinate transformation ${\chi }^{\prime }=\chi -\kappa \lambda (x)$, leaving $x$ unchanged. Since $d\chi =d{\chi }^{\prime }+\kappa {\partial }_{\mu }\lambda d{x}^{\mu }$, the metric has the same form withThus a higher-dimensional diffeomorphism becomes an Abelian gauge transformation in the reduced theory. Changing the sign convention for the internal coordinate shift reverses the displayed sign without changing its content.

More generally, continuous internal isometries give lower-dimensional gauge symmetries, with vector fields from metric components along the corresponding Killing vector fields. The circle translation group gives $U(1)$. A scalar mode ${\psi }_n(x)e^{in\chi }$ transforms as ${\psi }_n^{\prime }=e^{in\kappa \lambda }{\psi }_n$, so $D_{\mu }{\psi }_n=({\partial }_{\mu }-in\kappa A_{\mu }){\psi }_n$ transforms covariantly. This derives Kaluza-Klein charge quantization in the chosen vector normalization. The metric vector is a graviphoton; if a five-dimensional Maxwell field is present as well, its zero-mode photon is a separate field originating from that Maxwell gauge symmetry.

Radius stabilization fixes the size of a compactification dynamically rather than choosing it as an undetermined parameter. If the four-dimensional low-energy effective action contains a radion $R$ with positive kinetic coefficient $Z(R)$ and scalar potential $V(R)$, a stable radius $R_0$ requires $V^{\prime }(R_0)=0$ and $m_R^2=V^{\prime \prime }(R_0)/Z(R_0)>0$. This matters for the hierarchy problem: the volume or warp factor explains a hierarchy only if a consistent dynamics selects and maintains the required geometry.