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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.

Let a dot denote differentiation with respect to an affine geodesic parameter $s$. The five-dimensional geodesic Lagrangian isBecause $z$ is cyclic, its Euler-Lagrange equation gives the conserved quantityThis is the conserved momentum per unit five-dimensional mass associated with the Killing vector field ${\partial }_z$. For the other coordinates, differentiate $\partial L/\partial {\dot{x}}^i={\gamma }_{ij}{\dot{x}}^j+k{A}_i$. Using $\dot{k}=0$ givesThe first two terms combine into ${\gamma }_{ij}({\ddot{x}}^j+{\Gamma }^j{}_{lm}{\dot{x}}^l{\dot{x}}^m)$, with the Christoffel symbols of $\gamma$. Define the four-dimensional electromagnetic field tensor $F_{ij}={\partial }_i{A}_j-{\partial }_j{A}_i$ and raise its index with $\gamma$. The complete reduced geodesic equations areThis is the Kaluza-Klein geodesic Lorentz force. The Christoffel symbols describe the gravitational motion in general relativity in the four-dimensional metric tensor, while the right-hand side is antisymmetric in its lowered indices and gives the relativistic Lorentz force. In particular,Thus the force preserves the four-dimensional velocity norm, as a Lorentz force must. A particle with $k=0$ is neutral and follows an ordinary four-dimensional geodesic. Nonzero compact momentum gives electric charge, with its sign determined by the direction of motion around the circle.

To identify the actual charge-to-mass ratio, suppose the five-dimensional particle has mass $M>0$ and $s$ is its five-dimensional proper time, so $g_5(\dot{X},\dot{X})=-1$. Then $\gamma (\dot{x},\dot{x})=-(1+k^2)$. Four-dimensional proper time obeys $d\tau /ds=\sqrt{1+k^2}$; since $k$ is constant, $u^i=d{x}^i/d\tau$ satisfiesHere $q$ is in the geometric normalization of the supplied potential $A$. The relation $m^2=M^2+p_z^2$ also follows directly from the five-dimensional mass shell, and continues to identify the effective mass when the higher-dimensional particle is massless. The distinction between the two proper times prevents mistaking $k$ itself for the physical four-dimensional charge-to-mass ratio. A rescaling of $A$ to a canonically normalized electromagnetic potential correspondingly rescales $q$.

For the wave calculation, take a scalar field of five-dimensional mass $M$ in units $\hslash =c=1$, obeying $({\square }_5-M^2)\Psi =0$. In the coordinate order $(x^i,z)$, direct inversion of the metric tensor givesThe determinant follows by the triangular change of coframe from $(d{x}^i,dz)$ to $(d{x}^i,dz+A_{i}d{x}^i)$, whose determinant is one. Write $J=\sqrt{|\det \gamma |}$. Since all coefficients are $z$-independent, the scalar covariant wave operator isExpanding the right-hand side verifies both cross terms $-2A^i{\partial }_i{\partial }_z\Psi$, the term $-J^{-1}{\partial }_i(J{A}^i){\partial }_z\Psi$, and the coefficient $1+A_i{A}^i$ of ${\partial }_z^2\Psi$.

Single-valuedness under the compact coordinate period requires the Fourier seriesFor the $n$th Kaluza-Klein mode, put $q_n=n/R$ and $D_i={\partial }_i-i{q}_n{A}_i$. The five-dimensional Klein-Gordon equation reduces toThis is the charged four-dimensional Klein-Gordon equation: the divergence expression includes both the spacetime connection and the gauge connection. Its interpretation as electric charge can also be checked directly. Under the fibre coordinate change $z^{\prime }=z-\chi (x)$, the metric retains its form with $A^{\prime }=A+d\chi$, and the scalar Fourier coefficient changes by ${\psi }_n^{\prime }=e^{i{q}_n\chi }{\psi }_n$. Hence $D_i^{\prime }{\psi }_n^{\prime }=e^{i{q}_n\chi }{D}_i{\psi }_n$, exactly the transformation law for a field of charge $q_n$.

The Kaluza-Klein charge quantization is thereforePositive and negative integers give opposite charges, and the zero mode is neutral. Restoring $\hslash$ gives compact momenta $p_z=n\hslash /R$, so their fundamental spacing is $\hslash /R$. These are charges in the metric's geometric normalization. If $A_i=\kappa {\mathcal{A}}_i$ for a physically normalized electromagnetic potential ${\mathcal{A}}_i$, the physical charge unit is $\kappa /R$ in natural units, or $\kappa \hslash /R$ with $\hslash$ restored. Determining $\kappa$ requires an electromagnetic action normalization not specified by the metric ansatz alone.