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According to Klein's definition, "a geometry is the study of the invariant properties of a spacetime, under transformations within itself." Therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations. [11]
In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). In Kaluza–Klein theory this group suggests that gauge symmetry is the symmetry of circular compact dimensions.
Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms. [2]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot.
Quantum spacetimes are often described mathematically using the noncommutative geometry of Connes, quantum geometry, or quantum groups. Any noncommutative algebra with at least four generators could be interpreted as a quantum spacetime, but the following desiderata have been suggested:
Another kind of fifth force, which arises in Kaluza–Klein theory, where the universe has extra dimensions, or in supergravity or string theory is the Yukawa force, which is transmitted by a light scalar field (i.e. a scalar field with a long Compton wavelength, which determines the range).
Multiple independent timeframes, in which time passes at different rates, have long been a feature of stories. [15] Fantasy writers such as J. R. R. Tolkien and C. S. Lewis have made use of these and other multiple time dimensions, such as those proposed by Dunne, in some of their most well-known stories. [15]
[clarification needed] Similarly, modern quintessence would be the fifth known "dynamical, time-dependent, and spatially inhomogeneous" contribution to the overall mass–energy content of the universe. Of course, the other four components are not the ancient Greek classical elements, but rather "baryons, neutrinos, dark matter, [and] radiation."
Superspace is the coordinate space of a theory exhibiting supersymmetry.In such a formulation, along with ordinary space dimensions x, y, z, ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numbers rather than real numbers.