Field Theory on a Circle

We consider a massless scalar field in $D$-dimensions satisfying the wave equation ${\partial }^{\mu }{\partial }_{\mu }\varphi =0$. Expanding in a complete set of periodic functions under $x^d\to x^d+2\pi R$, we find

$\varphi (x^{\mu })={\sum }_{n=-\infty }^{\infty }{\varphi }_n(x^m)e^{in{x}^d/R}$.

The momentum measured by $P_d=i{\partial }_d$, quantised as $p_d=n/R$ where the integer $n\in \mathbb{Z}$ is known as the Kaluza-Klein (KK) momentum. The modes ${\varphi }_n(x^m)$ therefore behave as $d$-dimensional fields with $m^2:=-p^m{p}_m={\left(\frac{n}{R}\right)}^2$.

In the limit $R\to \infty$, the massive states become light and numerous, and the momentum tends to a continuum, reproducing the $D$-dimensional theory. In the limit $R\to 0$, the modes with $n\ne 0$ become infinitely massive and decouple, leaving a $d$-dimensional theory of the constant massless mode.

We also introduce a non-trivial space-time metric $g_{\mu \nu }$. This has a decomposition into components $g_{mn}$, $A_m:=g_{md}$ and $\phi :=g_{dd}$. Infinitesimal coordinate transformations $x^{\mu }\to x^{\mu }+{ϵ}^{\mu }$ transform the metric by $g_{\mu \nu }\to g_{\mu \nu }-{\partial }_{\mu }{ϵ}_{\nu }-{\partial }_{\nu }{ϵ}_{\mu }$.

If we restrict ourselves to coordinate transformations of the form $x^d\to x^d+ϵ(x^m)$, then $A_m\to A_m-{\partial }_mϵ$ while $g_{mn}$ and $\phi$ are left invariant. Thus the component $A_m$ transforms as a $U(1)$ gauge field in $d$-dimensions where gauge transformations arise from coordinate transformations in higher dimensions.

Provided $\phi =1$, we can understand from the mode expansion that the $d$-dimensional fields are charged under the $U(1)$ gauge symmetry. Basically, there is a tower of massive fields ${\varphi }_n$ of mass $m^2=n^2/R^2$ and $U(1)$ charge $n$.