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Introduction

  • Chapter
Wave Equations in Higher Dimensions

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Abstract

The exact solutions of wave equations with a spherically symmetric potential have become an important subject in quantum mechanics. It should be noticed that many works along this line have been carried out in the usual three dimensional space. However, what extra dimensions could there possibly be if we never see them? It turns out that we do not really know yet how many dimensions our world has. Nevertheless, all that our current observations tell us is that the world around us is at least (3+1) dimensional space-time as illustrated in general relativity.

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Notes

  1. 1.

    The theory was first proposed by the Finnish physicist Gunnar Nordström in 1914. Before Einstein’s general relativity theory was presented, Nordström proposed a relativistic theory for gravity. He unified his gravity theory with Maxwell’s electromagnetism through introducing a 5-vector gauge field where the first four components are identified with Maxwell’s vector potential A μ and the 5th component with the scalar gravity field. After that in 1919 a German mathematician Theodor Kaluza performed similar calculations but with Einstein’s gravity theory and Maxwell’s electromagnetism. In terms of a circular extra dimension Kaluza obtained a 4-dimensional action from a 5-dimensional one. The 4-action contained a graviton, an Abelian gauge boson identified as the photon and a scalar field that Kaluza put to be constant. The resulting equations can be separated out into further sets of equations, one of which is equivalent to Einstein’s field equations, another set equivalent to Maxwell’s equations for electromagnetic field and the final part an extra scalar field now termed the “radian”. In 1926, it was Swedish physicist Oskar Klein who focused on the resulting higher modes of the particles and the size of the extra dimension [7].

  2. 2.

    This is just like an insect crawling on a sheet of paper. For this insect, the universe is pretty much two-dimensional since it cannot leave the surface of that paper. As a result, the insect only knows the surface, but up and down does not make any sense as long as it has to stay on the sheet of that paper. These extra spatial dimensions, if they really exist, are thought to be curled-up, or “compactified”.

  3. 3.

    It was first proposed by Levinson in 1949 [109] and reviewed by Ma [110]. The Levinson theorem establishes the relation between the number of the bound states and the phase shift of the scattering states at the zero momentum.

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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). Introduction. In: Wave Equations in Higher Dimensions. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1917-0_1

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