Klein dimension

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... 

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. 

Below we present the idea of self-interacting dark matter and gravitationally-produced WIMPZILLAs. Other interesting candidates have been proposed recently in models with extra dimensions, such as Kaluza-Klein dark matter ( Cheng, Feng Matchev (2002)) and branons (Cembranos, Dobado Maroto (2003)). 

Mobius strip. In this way the antimatter mystery disappears matter and antimatter are one and the same thing, which merely appear to be different depending on their position in the double cover. In more dimensions the Mobius model is replaced by a projective plane, obtained from an open hemisphere on identifying points on opposite sides of the circular edge. Topologically equivalent constructs are known as a Roman surface or a Klein bottle. 

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... 

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 ... 

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). 

This assumes the independence of the three relative second-order densities fWlK By combining the last two equations, a closed continuity equation for the pair density is obtained. Unfortunately, the superposition hypothesis is not valid when extended to nonequilibrium states of the system. Klein and Prigogine24 have carried out a rigorous development of Eqs 48 and 49 for a onedimensional system with nearest neighbor interactions only. Their results show that, while the superposition hypothesis is valid for equilibrium systems, it cannot be used to describe transport processes and fails badly for the case of thermal conduction. While the treatment presented is limited to a one-dimensional system, it can be generalized to three dimensions with equivalent results. The essential difficulty of Eq. 48 is that, like the Liouville equation for N molecules, it is completely reversible in time so that it is intrinsically incapable of treating unidirectional dissipative processes. 

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 ... 

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. 

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. 

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