Editing Kaluza–Klein theory (section)

== Quantum interpretation of Klein ==
Kaluza's original hypothesis was purely classical and extended discoveries of general relativity. By the time of Klein's contribution, the discoveries of Heisenberg, Schrödinger, and [[Louis de Broglie]] were receiving a lot of attention. Klein's ''[[Nature (journal)|Nature]]'' article<ref name="KN"/> suggested that the fifth dimension is closed and periodic, and that the identification of electric charge with motion in the fifth dimension can be interpreted as standing waves of wavelength <math>\lambda^5</math>, much like the electrons around a nucleus in the [[Bohr model]] of the atom. The quantization of electric charge could then be nicely understood in terms of integer multiples of fifth-dimensional momentum. Combining the previous Kaluza result for <math>U^5</math> in terms of electric charge, and a [[de Broglie relation]] for momentum <math>p^5 = h/\lambda^5</math>, Klein obtained<ref name=KN /> an expression for the 0th mode of such waves:
: <math>
mU^5 = \frac{cq}{G^{1/2}} = \frac{h}{\lambda^5} \quad \Rightarrow \quad \lambda^5 \sim \frac{hG^{1/2}}{cq},
</math>
where <math>h</math> is the [[Planck constant]]. Klein found that <math>\lambda^5 \sim 10^{-30}</math>&nbsp;cm, and thereby an explanation for the cylinder condition in this small value.

Klein's ''[[Zeitschrift für Physik]]'' article of the same year,<ref name="KZ"/> gave a more detailed treatment that explicitly invoked the techniques of Schrödinger and de Broglie. It recapitulated much of the classical theory of Kaluza described above, and then departed into Klein's quantum interpretation. Klein solved a Schrödinger-like wave equation using an expansion in terms of fifth-dimensional waves resonating in the closed, compact fifth dimension.