Einstein Returns, 1938–1943

Einstein did not leave the Kaluza idea alone for good. After about five years away, he and Peter Bergmann returned to it in 1938, and he kept working on it until 1943.

What he wanted

Einstein's lifelong target was a pure field theory in which particles are not extra ingredients but smooth lumps of the field, solutions without singularities. In their paper he and Bergmann note that many attempts to build such a particle in Kaluza's theory had failed, and that they were convinced no such solution exists in the vacuum version.

The change

They relaxed the cylinder condition. Fields could now depend on the fifth coordinate, but only periodically, on a circle of small circumference b. Their own picture is a strip curved into a tube. If b is small compared with how quickly things change along the other directions, then a function on the tube looks almost the same all around the circle and can be treated as a function of the ordinary coordinates alone. That is the answer they gave to why we see four dimensions.

Allowing the fifth-direction wiggle brought matter back in. A field that varies around the circle carries momentum in the fifth direction, which is charge, so charged matter could arise from the geometry itself, which is what Einstein had wanted. According to van Dongen, they did not write out a Fourier expansion (that came from later literature), but they tacitly assumed that the parts of the fields that vary around the circle are small compared with the constant part.

What they could not do

Klein had fixed the size of the circle by quantizing, using Planck's constant and the de Broglie relation. Einstein and Bergmann worked classically, with no ħ, and so could not compute b from the electron's charge. The scale of the extra dimension stayed a free parameter. They also did not find the non-singular particle they sought. Van Dongen discusses how Einstein hoped a classical field theory could eventually account for quantum behaviour rather than start from it.

[STATUS]Everything on this page about the 1938–43 work is taken from van Dongen's paper (§4). A delegated search could not reach the original Einstein–Bergmann paper (Ann. Math. 39, 1938, on JSTOR behind a paywall) and found only secondary descriptions, so I have still not read it. Van Dongen also reports that the coupling constants in their field equations stayed arbitrary because five-dimensional general covariance was not imposed; a second paper imposed it, which fixed the constants, but then the electromagnetic and gravitational interactions came out of the same order of magnitude. A reader who quotes the 1938 paper should read it first.

For a general reader the lesson is not that Einstein was wrong. A small compact direction is what many later higher-dimensional theories use. He reached it from a different starting point than Klein, and without the quantum step that makes its size computable.