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Kaluza-Klein Theory

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Wave Equations in Higher Dimensions

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Abstract

The model of higher-dimensional space-time is a powerful ingredient to be needed to unify the interactions of various fields in nature. In this Chapter, we shall briefly review the development of the high-dimensional Kaluza-Klein theory, in particular, the (4+D) dimensional Kaluza-Klein theories and the particle spectrum of Kaluza-Klein theory for fermions.

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Notes

  1. 1.

    Up to now, there exist two main types of compactifications of an extra dimension. The first is a circular compactification. With this extra dimension the entire space can be denoted M 4×S 1, where M 4 is the regular Minkowski space and S 1 is the circle with a radius R. It is chosen to be small enough to avoid detection. Another type is an orbifold compactification by imposing the discrete Z 2 symmetry, where y→−y on the circle.

  2. 2.

    The large extra dimension scenario of ADD was proposed as a potential solution to the hierarchy problem, i.e., the question of why the reduced Planck scale \(\bar {M}_{\mathrm{pl}}\simeq 2. 4\cdot 10^{18}\) is so much larger than the weak scale ∼1 TeV. They propose that we live on an assumed to be rigid 4D hypersurface (also named a wall or brane). The gravity is allowed to propagate in a (4+D)-dimensional n-torus T n, whose radii are equal to R.

  3. 3.

    In modern geometry the extra 5th dimension can be understood as a circle group U(1) since the electromagnetism can be formulated essentially as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). If one is able to understand this geometrical interpretation, it is relatively straightforward to replace U(1) by a general Lie group. Such generalizations are often called Yang-Mills theories. If a distinction is drawn between them, then the Yang-Mills theories occur on a flat space-time, while Kaluza-Klein treats a more general case of curved space-time. The base space of Kaluza-Klein theory need not be four-dimensional space-time; it can be any pseudo-Riemannian manifold, or even a supersymmetric manifold or orbifold.

  4. 4.

    The choice of a time-like extra dimension shall lead to the tachyon. Tachyons are well known to be very dangerous in most theories. This seems to imply that we should only pick the space-like solution.

  5. 5.

    In fact in (4+D) dimensions a gauge field can be decomposed into a 4D gauge Kaluza-Klein tower plus D distinct scalar towers.

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  1. Escuela Superior de Física y Matemáticas, Unidad Profesional Adolfo López Mateos, Instituto Politécnico Nacional, Edificio 9, Mexico DF, 07738, Mexico

    Shi-Hai Dong

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Dong, SH. (2011). Kaluza-Klein Theory. In: Wave Equations in Higher Dimensions. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1917-0_17

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