= Kaluza-Klein charge quantization
{c}
{title2=$q_n=n/R,\quad m_n^2=M^2+n^2/R^2$}
For a periodic scalar on a circle of circumference $2\pi R$, its <Fourier series> has modes $\psi_n(x)e^{inz/R}$. In the fixed-radius metric $ds_5^2=\gamma_{ij}dx^idx^j+(dz+A_idx^i)^2$, the five-dimensional <Klein-Gordon equation> becomes $(D_iD^i-M^2-n^2/R^2)\psi_n=0$ with $D_i=\nabla_i-inA_i/R$. Thus the geometric gauge charge is quantized in units $1/R$ in units $\hbar=1$, or the conserved compact momentum in units $\hbar/R$. A rescaling to a canonically normalized electromagnetic potential changes the corresponding numerical unit of physical electric charge.
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