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Einstein’s idea, which gave rise to general relativity, was the proposal that spacetime is nontrivially curved and that the curvature is responsible for the gravity. The Kaluza-Klein idea takes this one step further, proposing that there are more than one time and three space dimensions and that the curvature of the higher-dimensional spacetime in the low energy approximation is perceived in the effective ordinary four dimensions as a unified theory of gravity and gauge fields. In 1921, Kaluza had suggested that gravitation and electromagnetism could be unified in a five-dimensional theory of gravity. In the classic Kaluza-Klein theory, the fifth coordinate was made invisible through a process of dimensional reduction. This idea has been revived several times, but recently it has gained popularity because of the concept of spontaneous compactification of extra dimensions. The starting point of all Kaluza-Klein theories is a theory of gravitation coupled to some matter fields in d = 4 + K dimensions. The classical field equations of the metric and the matter fields induce compactification of the extra space dimensions such that a 4 + K-dimensional background manifold, M 1 x M 2, emerges out as a product of a compact space of K dimensions, M 2, and an ordinary four-dimensional spacetime M 1. This is called spontaneous compactification of spacetime. The next step in the Kaluza-Klein programme is to expand the metric and the matter fields around this particular solution using ‘harmonic functions’ for the invariance group G of the compact space of K dimensions.
KeywordsDirac Operator Zero Mode Compact Space Gauge Field Isometry Group
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Introduction to Kaluza-Klein Theory
- Th. Kaluza, Sitzungsber. Preuss. Ahad. Wiss. Berlin, Math. Phys. K1 (1921), 966; O. Klein, Z. Phys. 37 (1926). English translations of these two papers by T. Muta are available in An Introduction to Kaluza-Klein Theories (H.C. Lee (ed.)), (World Scientific, Singapore, 1984).Google Scholar
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Chiral Fermions in Kaluza-Klein Theories
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