 Kaluza-Klein charge quantization

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 $d{s}_5^2={\gamma }_{ij}d{x}^{i}d{x}^j+(dz+A_{i}d{x}^i{)}^2$, the five-dimensional Klein-Gordon equation becomes $(D_i{D}^i-M^2-n^2/R^2){\psi }_n=0$ with $D_i={\nabla }_i-in{A}_i/R$. Thus the geometric gauge charge is quantized in units $1/R$ in units $\hslash =1$, or the conserved compact momentum in units $\hslash /R$. A rescaling to a canonically normalized electromagnetic potential changes the corresponding numerical unit of physical electric charge.