Kaluza-Klein theory

The theory of Kaluza and Klein [89, 90] is based on an observation that of two macroscopic forces of Nature only gravitation can be ascribed to geometric features of four-dimensional space-time. In order to incorporate another interaction the logical development would be to consider an additional dimension and to examine if extra degrees of freedom provided by 15 covariant components of the five-dimensional symmetric tensor needed to specify the line element 

In this theory the equations of motion of an electrified particle are geodesics referred to a line element 

If q is an integral multiple of some elementary electronic charge e, p = Ne/f3c. If five-dimensional space is assumed to be closed in the direction of x4 with a period /, p is quantized according to 

N becoming a quantum number, which may be a positive or negative integer according to the sense of motion in the direction of the klein dimension R is the radius of this compacted dimension. The relationship 

5 An intriguing correspondence exists between the klein radius and vortices that feature in theories of Winterberg and Meno. 

Another attempt by Kaluza and Klein to incorporate electromagnetic interaction in the held equations of general relativity was to generate extra degrees of freedom by increasing the number of space-time dimensions from four to hve, such that the line element is  

Four coordinates, x fi yt 4) are used to characterize regular space-time and is assumed independent of x. To explain why the hfth dimension is not observed, space was proposed to have cylindrical symmetry with respect to x,  

On this basis it was possible to obtain equations which agreed with known relativistic equations of the gravitational held and generalized Maxwell equations of electromagnetism. Writing (4.16) in the form 

This is analogous to a gauge transformation, suggesting identihcation of 7 4 with the electromagnetic potential tpi, apart from a proportionality constant. The antisymmetric tensor 

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Kaluza-Klein theory (named after two European scientists) suggests the existence of additional dimensions that are rolled up or compactified in such a way that they are undetectable at macroscopic levels. 

A five-dimensional wave equation based on Kaluza-Klein theory in the absence of gravitation has the form... 

P.S. Wesson, Space, Time, Matter Modern Kaluza-Klein Theory, 1999, World Scientific, Singapore. 

Additional spatial dimensions beyond the three we move in (anywhere from one extra, as in traditional Kaluza-Klein theory, up to 11 or thereabouts) imply the possibility of extra symmetries, extra conserved quantities, and a lowest-mass particle that cannot decay without violating that conservation law and which is, therefore, a possible DM candidate. Some names we caught were ... 

Apart from interesting speculations the modern theory produces little more than classical Kaluza-Klein theory. 

Ponce de Leon, J. (1988) Cosmological models in Kaluza-Klein theory with variable rest mass, Gen. Rel. Grav., 1988 (20) 539-550. 

It seems to me that the appropriate computational structure for our purpose was that briefly presented by B. Hoffmann and myself in the Physical Review (1930) in connection with earlier work in the Quarterly Journal of Mathematics (Oxford Series 1930). The correspondence with the formalism of the Kaluza-Klein theory is so complete that Hoffmann and I described our theory as the geometrical meaning of Kaluza-Klein. However, it must be emphasized that our theory starts from a totally different physical and geometrical point of view as Kaluza. In particular, we infer no link between electric charge and a fifth coordinate our theory is rather four-dimensional throughout. 

Gauge Field Theories, 1986 High Energy Astrophysics, 1987 Experimental Techniques in High Energy Physics, 1987 Modem Kaluza-Klein Theories, 1987... 

Strings associated with bosons are only consistent as quantum theories in a 26-dimensional space-time those associated with fermions are only consistent as quantum theories in 10-dimensional space-time. It is thought that four macroscopic dimensions arise by a Kaluza-Klein theory mechanism, with the remaining dimensions being curled up to become very small, although other possibUlties for the higher dimensions have been put forward. 

On the theoretical side, simple cosmological models with a varying fine structure constant have now been developed (see, for example, Sandvik et al., 2002 Barrow etal., 2002b). They share some properties with Kaluza-Klein-type models in which a varies at the same rate as the extra dimensions of space (see, for example, Damour and Polyakov, 1994) and with varying-speed-of-light theories (see, for example, Albrecht and Magueijo, 1999 Barrow and Magueijo, 2000). 

Although the analysis of CMBR excludes the domination of hot dark matter, i.e., relativistic weakly interacting particles, like light neutrinos, there stfll exists a plethora of more exotic propositions for the constituents of dark matter, not yet accessible for experimental verification, like primordial black holes, nonthermal WIMPzdlas, and the so-called Kaluza-Klein excitations of higher-dimensional theories. 

Kaluza and Klein managed to formulate a unified theory of gravitation and electromagnetism in terms of Einstein s field equations in five-dimensional space, but with the metric tensor defined to be independent of the fourth space dimension. Without this restriction, solution of the equations in apparent 5D vacuum ... 

The Kaluza theory was simplified by 0. Klein (Bibl. 1926,5 1927,11) in a way that the Einstein-Maxwell theory was not approximated, but came out in its exact shape. Since then several mathematical physicists investigated the theory and found its formal shape alluring. However, there remains a fundamental question without a satisfactory answer What does the fifth dimension mean Nobody has found convincing ground to doubt our conviction that the physical world is four dimensional. Theorists, especially Einstein himself, conducted several investigations with the intention to create a four-dimensional theory. Some of these works are referenced in the bibliography. 

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