Editing Kaluza–Klein theory (section)

== Group theory interpretation ==
[[Image:Kaluza Klein compactification.svg|frame|left|The space {{nowrap|''M'' × ''C''}} is compactified over the compact set ''C'', and after Kaluza–Klein decomposition one has an [[effective field theory]] over ''M''.]]

In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small [[radius]], so that a [[Elementary particle|particle]] moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a [[compact set]], and construction of this compact dimension is referred to as [[compactification (physics)|compactification]].

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.  Once this geometrical interpretation is understood, it is relatively straightforward to replace U(1) by a general [[Lie group]].  Such generalizations are often called [[Yang–Mills theory|Yang–Mills theories]]. If a distinction is drawn, then it is that Yang–Mills theories occur on a flat spacetime, whereas Kaluza–Klein treats the more general case of curved spacetime. The base space of Kaluza–Klein theory need not be four-dimensional spacetime; it can be any ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian manifold]], or even a [[supersymmetry|supersymmetric]] manifold or [[orbifold]] or even a [[noncommutative space]].

The construction can be outlined, roughly, as follows.<ref>David Bleecker, "[https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf Gauge Theory and Variational Principles] {{Webarchive|url=https://web.archive.org/web/20210709185749/https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf |date=2021-07-09 }}" (1982) D. Reidel Publishing ''(See chapter 9'')</ref> One starts by considering a [[principal fiber bundle]] ''P'' with [[gauge group]] ''G'' over a [[manifold]] M. Given a [[connection (fibred manifold)|connection]] on the bundle, and a [[metric tensor|metric]] on the base manifold, and a gauge invariant metric on the tangent of each fiber, one can construct a [[metric (vector bundle)|bundle metric]] defined on the entire bundle. Computing the [[scalar curvature]] of this bundle metric, one finds that it is constant on each fiber: this is the "Kaluza miracle". One did not have to explicitly impose a cylinder condition, or to compactify: by assumption, the gauge group is already compact. Next, one takes this scalar curvature as the [[Lagrangian density]], and, from this, constructs the [[Einstein–Hilbert action]] for the bundle, as a whole.  The equations of motion, the [[Euler–Lagrange equations]], can be then obtained by considering where the action is [[stationary state|stationary]] with respect to variations of either the metric on the base manifold, or of the gauge connection.  Variations with respect to the base metric gives the [[Einstein field equations]] on the base manifold, with the [[energy–momentum tensor]] given by the [[curvature form|curvature]] ([[field strength]]) of the gauge connection. On the flip side, the action is stationary against variations of the gauge connection precisely when the gauge connection solves the [[Yang–Mills theory|Yang–Mills equation]]s. Thus, by applying a single idea: the [[principle of least action]], to a single quantity: the scalar curvature on the bundle (as a whole), one obtains simultaneously all of the needed field equations, for both the spacetime and the gauge field.

As an approach to the unification of the forces, it is straightforward to apply the Kaluza–Klein theory in an attempt to unify gravity with the [[strong force|strong]] and [[electroweak]] forces by using the symmetry group of the [[Standard Model]], [[SU(3)]] × [[SU(2)]] × [[U(1)]]. However, an attempt to convert this interesting geometrical construction into a bona-fide model of reality flounders on a number of issues, including the fact that the [[fermion]]s must be introduced in an artificial way (in nonsupersymmetric models). Nonetheless, KK remains an important [[Touchstone (metaphor)|touchstone]] in theoretical physics and is often embedded in more sophisticated theories. It is studied in its own right as an object of geometric interest in [[K-theory]].

Even in the absence of a completely satisfying theoretical physics framework, the idea of exploring extra, compactified, dimensions is of considerable interest in the [[experimental physics]] and [[astrophysics]] communities. A variety of predictions, with real experimental consequences, can be made (in the case of [[large extra dimension]]s and [[warped model]]s). For example, on the simplest of principles, one might expect to have [[standing wave]]s in the extra compactified dimension(s). If a spatial extra dimension is of radius ''R'', the invariant  [[mass]] of such  standing waves would be ''M''<sub>''n''</sub> = ''nh''/''Rc'' with ''n'' an [[integer]], ''h'' being the [[Planck constant]] and ''c'' the [[speed of light]]. This set of possible mass values is often called the '''Kaluza–Klein tower'''.  Similarly, in [[Thermal quantum field theory]] a compactification of the euclidean time dimension leads to the [[Matsubara frequency|Matsubara frequencies]] and thus to a discretized thermal energy spectrum.

However, Klein's approach to a quantum theory is flawed{{citation needed|date=February 2017}} and, for example, leads to a calculated electron mass in the order of magnitude of the [[Planck mass]].<ref>Ravndal, F., Oskar Klein and the fifth dimension, [https://arxiv.org/abs/1309.4113 arXiv:1309.4113] [physics.hist-ph]</ref>

Examples of experimental pursuits include work by the [[Collider Detector at Fermilab|CDF]] collaboration, which has re-analyzed [[particle collider]] data for the signature of effects associated with large extra dimensions/[[warped model]]s.{{citation needed|date=June 2024}}

[[Robert Brandenberger]] and [[Cumrun Vafa]] have speculated that in the early universe, [[cosmic inflation]] causes three of the space dimensions to expand to cosmological size while the remaining dimensions of space remained microscopic.{{citation needed|date=June 2024}}