This note is devoted to a historical detail concerning the paper of Albert Einstein and Peter Bergmann, published in 1938, about unified theories of electromagnetism and gravitation derived from five dimensions [1]. ...Wikipedia explains Kaluza–Klein theory:

A major influence in Einstein's efforts to unify electromagnetism and gravitation was the proposal made by Theodore Kaluza [2] around 1921, later independently discovered and extended by Oskar Klein [3] and commonly called Kaluza-Klein theory. In this proposal, in addition to the four dimensions of conventional relativity theory (three space dimensions and a fourth dimension of time) there is a fifth dimension; electromagnetism results from a gravitational field that is "polarized" in the fifth dimension. ...

The main novelty of Einstein and Bergmann was to take the fifth dimension seriously as a physical entity, not just an excuse to combine the metric tensor and the electromagnetic potential as different components of a 5x5 matrix.

A splitting of five-dimensional spacetime into the Einstein equations and Maxwell equations in four dimensions was first discovered by Gunnar NordstrÃ¶m in 1914, in the context of his theory of gravity, but subsequently forgotten. Kaluza published his derivation in 1921 as an attempt to unify electromagnetism with Einstein's general relativity.Hermann Weyl constructed a similar theory in 1918, and that was the foundation of gauge theory. Gauge theory is now used for all the forces in the Standard Model.

In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and the phenomenon of having a space-time with compact dimensions is referred to as compactification.

In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). In Kaluza–Klein theory this group suggests that gauge symmetry is the symmetry of circular compact dimensions.

Thus you can think of the modern theory of gravity and electromagnetism as a unified geometrical theory on a 5-dimensional manifold, as was understood by experts around 1920 or so. The 5th dimension is not something that we can spatially measure with a meter stick, but it is physical in the sense that we need it to understand light. You can think of it as an electromagnetic phase that is only indirectly observable.

Einstein wanted to find an extra physical meaning to the 5th dimension in 1938, and Witten wants to find more meaning to more dimensions today. None of this research has ever amounted to anything.

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