# Kaluza-Klein theory

Kaluza-Klein theory (or KK theory, for short) is a model which sought to unify classical gravity and electromagnetism, first published in 1921. It was discovered by the mathematician Theodor Kaluza that if general relativity is extended to a five-dimensional spacetime, the equations can be separated out into ordinary four-dimensional gravitation plus an extra set, which is equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton". In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of very small radius, i.e. 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, in fact, also gives rise to quantization of charge, as waves directed along a finite axis can only occupy discrete frequencies. (This occurs because electromagnetism is a U(1) symmetry theory and U(1) is simply the group of rotations around a circle).

Kaluza-Klein theory can be extended to cover the other fundamental forces - namely, the weak and strong nuclear forces - but a straightforward approach, if done using an odd dimensional manifold runs into difficulties involving chirality. The problem is that all neutrinos appear to be left-handed, meaning that they are spinning in the direction of the fingers of the left hand when they are moving in the direction of the thumb. All anti-neutrinos appear to be right-handed. Somehow particle reactions are asymmetric when it comes to spin and it is not straightforward to build this into a Kaluza-Klein theory since the extra dimensions of physical space are symmetric with respect to left-hand spinning and right-hand spinning particles.

But see Orbifold for one possible solution.

Some physicists have speculated that in the early universe, cosmic inflation causes three of the space dimensions to expand to cosmological size while the remaining dimensions of space remained microscopic.

The theory as it stands does not deal with quantum effects. However, the same approach to unification of forces is taken by some more modern theories, notably string theory and the related M-theory.

### Compact dimensions

In string theory, a model used in theoretical physics, a compact dimension is curled up in itself and very small (Planck length). Anything moving along this dimension's direction would return to its starting point almost instantaneously, and the fact that the dimension is smaller than the smallest particle means that it cannot be observed by conventional means.

Main article: Why 10 dimensions?

• Hyperspace : A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension - Michio Kaku, Robert O'Keefe
• Kaluza-Klein Theory in Perspective (http://xxx.lanl.gov/abs/hep-th/9410046) by M. J. Duff

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