Kaluza-Klein compactification

If the compactized dimension becomes sufficiently small, a theory that is defined on becomes an effective theory about

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{\ displaystyle M \ times C}

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Under Kaluza-Klein compactification (also briefly compactification if no likelihood of confusion with the mathematical term is) is understood in theoretical physics , the reduction of a higher-dimensional theory to a niedrigerdimensionale.

The name goes back to the Kaluza-Klein theory , in which both the four-dimensional Einstein field equations and the Maxwell equations arise from a five-dimensional theory . The term is used in particular when reducing the 26-dimensional bosonic or the 10-dimensional supersymmetric string theory to a four-dimensional effective theory or when reducing the 11-dimensional supergravity to a 10-dimensional theory.

In the Kaluza-Klein compactification, the superfluous dimensions z. B. compactified by adding a point in the topological sense . Then these compact dimensions are "shrunk" so that they reach the order of magnitude of the Planck length . The resulting effective theory then only perceives the number of turns of these dimensions . ${\ displaystyle \ infty}$

String theory

In string theory , one speaks of compactized or rolled up dimensions. This means that the topological structure of this dimension is a circle, i.e. a circle . A two-dimensional space with a compact dimension would then be, so to speak, an infinitely long cylinder , mathematically represented as . Similarly, in (supersymmetric) string theory, one has to imagine spacetime as a kind of 10-dimensional cylinder, of which 6 dimensions are like a circle. ${\ displaystyle S ^ {1}}$ ${\ displaystyle \ mathbb {R} \ times S ^ {1}}$

The mathematical compactification with an infinitely distant point suggests that this dimension is very large. However, the correct view is that the extent of such a dimension is more likely to be found in the area of Planck length. As with mathematical compactification, one reaches the same point again after going around the compact dimension. This mathematical background leads to the descriptive way of speaking of rolling up the dimensions into a circle.

literature

Brian Greene: The Elegant Universe . Goldmann, Munich 2006, ISBN 978-3-442-15374-9 .