A Toy Model for Effective Field Theory and Extra Dimensions

I wanted to see how the Fourier transform can turn field theory into many-particle mechanics. This is just silly fooling around, so you shouldn't take what follows too seriously (there are much better models of extra dimensions, to be sure!).

Take $\phi(t, s)$ to be a field on a cylinder of radius $R$. We consider the action

$$S = \frac{1}{R} \int_{-\infty}^\infty \int_0^{R} |\nabla \phi|^2 ds dt$$

Expand $\phi(t, s)$ in Fourier series:
$$\phi(t,s ) = \sum_n \phi_n(t) e^{2 \pi i n s / R}$$
Then in Lorentzian signature, we have
$$\int_0^{R} |\nabla \phi|^2 d\theta = R \sum_n \dot{\phi}_n^2 - \left(\frac{2\pi n}{R}\right)^2 \phi_n^2.$$

Putting this back into the action, we find
$$S = \sum_n \int_{-\infty}^\infty \dot{\phi}_n^2- \left(\frac{2\pi n}{R}\right)^2 \phi_n^2 dt.$$

This is the action for infinitely many harmonic oscillators, with frequencies $\omega_n = 2\pi |n| / R$. Recall that the energy levels of the harmonic oscillator are $k\omega$ for $k = 0, 1, \ldots$. So supposing that only a finite energy $E$ is accessible in some particular experiment, we can only excite those modes $\phi_n$ for which
$$\frac{2\pi |n|}{R} < E.$$
In particular, only finitely many $\phi_n$ may be excited at energies below $E$, effectively reducing the field theory on the cylinder to many-particle quantum mechanics.