Editing Kaluza–Klein theory

{{Short description|Unified field theory}}
{{About|gravitation and electromagnetism|the mathematical generalization of [[K theory]]|KK-theory}}
{{Primary sources|date=January 2023}}
{{Beyond the Standard Model|expanded=Theories}}

In [[physics]], '''Kaluza–Klein theory''' ('''KK theory''') is a classical [[unified field theory]] of [[gravitation]] and [[electromagnetism]] built around the idea of a [[Five-dimensional space#Physics|fifth dimension]] beyond the common 4D of [[spacetime|space and time]] and considered an important precursor to [[string theory]]. In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle. [[Gunnar Nordström]] had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions.<ref name=nrd>{{cite journal |last1=Nordström |first1=Gunnar |year=1914 |title=Über die Möglichkeit, das elektromagnetische Feld und das Gravitationsfeld zu vereinigen |lang=de |trans-title=On the possibility of unifying the gravitational and electromagnetic fields |journal=Physikalische Zeitschrift |volume=15 |page=504 |url=https://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/17520}}</ref>

The five-dimensional (5D) theory developed in three steps. The original hypothesis came from [[Theodor Kaluza]], who sent his results to [[Albert Einstein]] in 1919<ref>{{ cite book |last=Pais |first=Abraham |date=1982 |title=Subtle is the Lord ...: The Science and the Life of Albert Einstein |publisher=Oxford University Press |location=Oxford |pages=329–330}}</ref> and published them in 1921.<ref name=kal>{{cite journal |last=Kaluza |first=Theodor |date=1921 |title=Zum Unitätsproblem in der Physik |lang=de |journal=Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) |pages=966–972 |bibcode=1921SPAW.......966K }}</ref> Kaluza presented a purely classical extension of [[general relativity]] to 5D, with a metric tensor of 15 components. Ten components are identified with the 4D spacetime metric, four components with the electromagnetic vector potential, and one component with an unidentified [[scalar field]] sometimes called the "[[Radion (physics)|radion]]" or the "dilaton". Correspondingly, the 5D Einstein equations yield the 4D [[Einstein field equations]], the [[Maxwell equations]] for the [[electromagnetic field]], and an equation for the scalar field. Kaluza also introduced the "cylinder condition" hypothesis, that no component of the five-dimensional metric depends on the fifth dimension. Without this restriction, terms are introduced that involve derivatives of the fields with respect to the fifth coordinate, and this extra degree of freedom makes the mathematics of the fully variable 5D relativity enormously complex. Standard 4D physics seems to manifest this "cylinder condition" and, along with it, simpler mathematics.

In 1926, [[Oskar Klein]] gave Kaluza's classical five-dimensional theory a quantum interpretation,<ref name=KZ>{{cite journal |last=Klein |first=Oskar |date=1926 |title=Quantentheorie und fünfdimensionale Relativitätstheorie |lang=de |journal=[[Zeitschrift für Physik A]] |volume=37 |issue=12 |pages=895–906 |doi=10.1007/BF01397481 |bibcode=1926ZPhy...37..895K }}</ref><ref name=KN>{{cite journal |last=Klein |first=Oskar |date=1926 |journal=Nature |volume=118 |issue=2971 |pages=516 |doi=10.1038/118516a0 |title=The Atomicity of Electricity as a Quantum Theory Law |bibcode= 1926Natur.118..516K |s2cid=4127863 |doi-access=free }}</ref> to accord with the then-recent discoveries of [[Werner Heisenberg]] and [[Erwin Schrödinger]]. Klein introduced the hypothesis that the fifth dimension was curled up and microscopic, to explain the cylinder condition. Klein suggested that the geometry of the extra fifth dimension could take the form of a circle, with the radius of {{val|e=-30|u=cm}}. More precisely, the radius of the circular dimension is 23 times the [[Planck units|Planck length]], which in turn is of the order of {{val|e=-33|u=cm}}.<ref name=KN /> Klein also made a contribution to the classical theory by providing a properly normalized 5D metric.<ref name=KZ /> Work continued on the Kaluza field theory during the 1930s by Einstein and colleagues at [[Princeton University]].

In the 1940s, the classical theory was completed, and the full field equations including the scalar field were obtained by three independent research groups:<ref name=gon>{{cite journal |last=Goenner |first=H. |journal=General Relativity and Gravitation |date=2012 |volume=44 |issue=8 |pages=2077–2097 |arxiv=1204.3455 |bibcode=2012GReGr..44.2077G |doi=10.1007/s10714-012-1378-8 |title=Some remarks on the genesis of scalar–tensor theories |s2cid=13399708 }}</ref> Yves Thiry,<ref>{{cite journal |last1=Lichnerowicz| first1=A. |last2=Thiry |first2=M. Y. |date=1947 |journal=Compt. Rend. Acad. Sci. Paris |volume=224 |pages=529–531 |title=Problèmes de calcul des variations liés à la dynamique classique et à la théorie unitaire du champ |lang=fr}}</ref><ref name=thry>{{ cite journal |last=Thiry |first=M. Y. |date=1948 |journal=Compt. Rend. Acad. Sci. Paris |volume=226 |pages=216–218 |title=Les équations de la théorie unitaire de Kaluza |lang=fr}}</ref><ref>{{cite journal |last=Thiry |first=M. Y. |date=1948 |title=Sur la régularité des champs gravitationnel et électromagnétique dans les théories unitaires |lang=fr |journal=Compt. Rend. Acad. Sci. Paris |volume=226 |pages=1881–1882}}</ref> working in France on his dissertation under [[André Lichnerowicz]]; [[Pascual Jordan]], Günther Ludwig, and Claus Müller in Germany,<ref name=jor1>{{cite journal |last=Jordan |first=P. |journal=Naturwissenschaften |date=1946 |volume=11 |issue=8 |pages=250–251 |title=Relativistische Gravitationstheorie mit variabler Gravitationskonstante |lang=de |bibcode=1946NW.....33..250J |doi=10.1007/BF01204481 |s2cid=20091903 }}</ref><ref name=jor2>{{cite journal |last1=Jordan |first1=P. |last2=Müller |first2=C. |journal=Z. Naturforsch. |date=1947 |volume=2a |issue=1 |pages=1–2 |title=Über die Feldgleichungen der Gravitation bei variabler "Gravitationslonstante" |lang=de |bibcode=1947ZNatA...2....1J |doi=10.1515/zna-1947-0102 |s2cid=93849549 |doi-access=free }}</ref><ref>{{cite journal |last=Ludwig |first=G. |journal=Z. Naturforsch. |date=1947 |volume=2a |issue=1 |pages=3–5 |title=Der Zusammenhang zwischen den Variationsprinzipien der projektiven und der vierdimensionalen Relativitätstheorie |lang=de |bibcode=1947ZNatA...2....3L |doi=10.1515/zna-1947-0103 |s2cid=94454994 |doi-access=free }}</ref><ref name=jor3>{{cite journal |last=Jordan |first=P. |journal=Astron. Nachr. |date=1948 |volume=276 |issue=5–6 |pages=193–208 |bibcode=1948AN....276..193J |doi=10.1002/asna.19482760502 |title=Fünfdimensionale Kosmologie |lang=de}}</ref><ref>{{cite journal |last1=Ludwig |first1=G. |last2=Müller |first2=C. |journal=Annalen der Physik |date=1948 |volume=2 |issue=6 |pages=76–84 |doi=10.1002/andp.19484370106 |title=Ein Modell des Kosmos und der Sternentstehung |bibcode=1948AnP...437...76L|s2cid=120176841 }}</ref> with critical input from [[Wolfgang Pauli]] and [[Markus Fierz]]; and [[Paul Scherrer]]<ref>{{cite journal |last=Scherrer |first=W. |date=1941 |journal=Helv. Phys. Acta |volume=14 |issue=2 |pages=130 |title=Bemerkungen zu meiner Arbeit: "Ein Ansatz für die Wechselwirkung von Elementarteilchen" |lang=de}}</ref><ref>{{cite journal |last=Scherrer |first=W. |date=1949 |journal=Helv. Phys. Acta |volume=22 |pages=537–551 |title=Über den Einfluss des metrischen Feldes auf ein skalares Materiefeld }}</ref><ref>{{cite journal |last=Scherrer |first=W. |date=1950 |journal=Helv. Phys. Acta |volume=23 |pages=547–555 |title=Über den Einfluss des metrischen Feldes auf ein skalares Materiefeld (2. Mitteilung) |lang=de}}</ref> working alone in Switzerland. Jordan's work led to the scalar–tensor theory of [[Brans–Dicke theory|Brans–Dicke]];<ref>{{cite journal |last1=Brans |first1=C. H. |last2=Dicke |first2=R. H. |date= November 1, 1961 |title=Mach's Principle and a Relativistic Theory of Gravitation |journal=[[Physical Review]] |volume=124 |issue=3 |pages=925–935 |doi=10.1103/PhysRev.124.925 |bibcode=1961PhRv..124..925B }}</ref> [[Carl H. Brans]] and [[Robert H. Dicke]] were apparently unaware of Thiry or Scherrer. The full Kaluza equations under the cylinder condition are quite complex, and most English-language reviews, as well as the English translations of Thiry, contain some errors. The curvature tensors for the complete Kaluza equations were evaluated using [[Tensor software|tensor-algebra software]] in 2015,<ref name=LLW>{{cite journal |last1=Williams |first1=L. L. |year=2015 |title=Field Equations and Lagrangian for the Kaluza Metric Evaluated with Tensor Algebra Software |journal=Journal of Gravity |volume=2015 |page=901870 |doi=10.1155/2015/901870 |url=http://downloads.hindawi.com/archive/2015/901870.pdf |doi-access=free }}</ref> verifying results of J. A. Ferrari<ref name=fri>{{cite journal |last1=Ferrari |first1=J. A. |year=1989 |title=On an approximate solution for a charged object and the experimental evidence for the Kaluza-Klein theory |journal=Gen. Relativ. Gravit. |volume=21 |issue=7 |page=683 |doi=10.1007/BF00759078 |bibcode=1989GReGr..21..683F |s2cid=121977988 }}</ref> and R. Coquereaux & G. Esposito-Farese.<ref name=coq>{{cite journal |last1=Coquereaux |first1=R. |last2=Esposito-Farese |first2=G. |year=1990 |title=The theory of Kaluza–Klein–Jordan–Thiry revisited |journal=Annales de l'Institut Henri Poincaré |volume=52 |page=113 }}</ref> The 5D covariant form of the energy–momentum source terms is treated by L. L. Williams.<ref name=LLW2>{{cite journal |last1=Williams |first1=L. L. |year=2020 |title=Field Equations and Lagrangian of the Kaluza Energy-Momentum Tensor |journal=Advances in Mathematical Physics |volume=2020 |page=1263723 |doi=10.1155/2020/1263723 |doi-access=free }}</ref>

== Kaluza hypothesis ==
In his 1921 article,<ref name="kal"/> Kaluza established all the elements of the classical five-dimensional theory: the [[Kaluza–Klein metric]], the [[Kaluza–Klein–Einstein field equations]], the equations of motion, the stress–energy tensor, and the cylinder condition. With no [[free parameter]]s, it merely extends general relativity to five dimensions. One starts by hypothesizing a form of the five-dimensional Kaluza–Klein metric <math> \widetilde{g}_{ab}</math>, where Latin indices span five dimensions. Let one also introduce the four-dimensional spacetime metric <math> {g}_{\mu\nu}</math>, where Greek indices span the usual four dimensions of space and time; a 4-vector <math> A^\mu </math> identified with the electromagnetic vector potential; and a scalar field <math>\phi</math>. Then decompose the 5D metric so that the 4D metric is framed by the electromagnetic vector potential, with the scalar field at the fifth diagonal. This can be visualized as
: <math>
\widetilde{g}_{ab} \equiv \begin{bmatrix}
 g_{\mu\nu} + \phi^2 A_\mu A_\nu & \phi^2 A_\mu \\
 \phi^2 A_\nu & \phi^2\end{bmatrix}.
</math>

One can write more precisely
: <math>
\widetilde{g}_{\mu\nu} \equiv g_{\mu\nu} + \phi^2 A_\mu A_\nu, \qquad
\widetilde{g}_{5\nu} \equiv \widetilde{g}_{\nu 5} \equiv \phi^2 A_\nu, \qquad
\widetilde{g}_{55} \equiv \phi^2,
</math>
where the index <math>5</math> indicates the fifth coordinate by convention, even though the first four coordinates are indexed with 0, 1, 2, and 3. The associated inverse metric is
: <math>
\widetilde{g}^{ab} \equiv \begin{bmatrix}
 g^{\mu\nu} & -A^\mu \\
 -A^\nu & g_{\alpha\beta} A^\alpha A^\beta + \frac{1}{\phi^2}
\end{bmatrix}.
</math>

This decomposition is quite general, and all terms are dimensionless. Kaluza then applies the machinery of standard [[general relativity]] to this metric. The field equations are obtained from five-dimensional [[Einstein equations]], and the equations of motion from the five-dimensional geodesic hypothesis. The resulting field equations provide both the equations of general relativity and of electrodynamics; the equations of motion provide the four-dimensional [[geodesic equation]] and the [[Lorentz force law]], and one finds that electric charge is identified with motion in the fifth dimension.

The hypothesis for the metric implies an invariant five-dimensional length element <math>ds</math>:
: <math>
ds^2 \equiv \widetilde{g}_{ab}\,dx^a\,dx^b = g_{\mu\nu}\,dx^\mu\,dx^\nu + \phi^2 (A_\nu\,dx^\nu + dx^5)^2.
</math>

== Field equations from the Kaluza hypothesis ==
The [[Kaluza–Klein–Einstein field equations]] of the five-dimensional theory were never adequately provided by Kaluza or Klein because they ignored the [[scalar field]]. The full Kaluza field equations are generally attributed to Thiry,<ref name="thry"/> who obtained vacuum field equations, although Kaluza<ref name=kal/> originally provided a stress–energy tensor for his theory, and Thiry included a stress–energy tensor in his thesis. But as described by Gonner,<ref name="gon"/> several independent groups worked on the field equations in the 1940s and earlier. Thiry is perhaps best known only because an English translation was provided by Applequist, Chodos, & Freund in their review book.<ref>{{cite book |title=Modern Kaluza–Klein Theories |last=Appelquist |first=Thomas |author2=Chodos, Alan |author3=Freund, Peter G. O. |date=1987 |publisher=Addison–Wesley |location=Menlo Park, Cal. |isbn=978-0-201-09829-7 }}</ref> Applequist et al. also provided an English translation of Kaluza's article. Translations of the three (1946, 1947, 1948) Jordan articles can be found on the [[ResearchGate]] and [[Academia.edu]] archives.<ref name="jor1"/><ref name="jor2"/><ref name="jor3"/> The first correct English-language Kaluza field equations, including the scalar field, were provided by Williams.<ref name=LLW />

To obtain the 5D Kaluza–Klein–Einstein field equations, the 5D [[Kaluza–Klein–Christoffel symbol|Kaluza–Klein–Christoffel symbols]] <math>\widetilde{\Gamma}^a_{bc}</math> are calculated from the 5D Kaluza–Klein metric <math>\widetilde{g}_{ab}</math>, and the 5D [[Kaluza–Klein–Ricci tensor]] <math>\widetilde{R}_{ab}</math> is calculated from the 5D [[Connection (mathematics)|connections]].

The classic results of Thiry and other authors presume the cylinder condition:
: <math>\frac{\partial \widetilde{g}_{ab}}{\partial x^5} = 0.</math>
Without this assumption, the field equations become much more complex, providing many more degrees of freedom that can be identified with various new fields. Paul Wesson and colleagues have pursued relaxation of the cylinder condition to gain extra terms that can be identified with the matter fields,<ref>{{cite book |title=Space–Time–Matter, Modern Kaluza–Klein Theory |last=Wesson |first=Paul S. |date=1999 |publisher=World Scientific |location=Singapore |isbn=978-981-02-3588-8 |url-access=registration |url=https://archive.org/details/spacetimematterm0000wess }}</ref> for which Kaluza<ref name=kal /> otherwise inserted a stress–energy tensor by hand.

It has been an objection to the original Kaluza hypothesis to invoke the fifth dimension only to negate its dynamics. But Thiry argued<ref name=gon/> that the interpretation of the Lorentz force law in terms of a five-dimensional [[geodesic]] militates strongly for a fifth dimension irrespective of the cylinder condition. Most authors have therefore employed the cylinder condition in deriving the field equations. Furthermore, vacuum equations are typically assumed for which
: <math>\widetilde{R}_{ab} = 0,</math>
where
: <math>\widetilde{R}_{ab} \equiv \partial_c \widetilde{\Gamma}^c_{ab} - \partial_b \widetilde{\Gamma}^c_{ca} + \widetilde{\Gamma}^c_{cd}\widetilde{\Gamma}^d_{ab} - \widetilde{\Gamma}^c_{bd}\widetilde{\Gamma}^d_{ac}</math>
and
: <math>\widetilde{\Gamma}^a_{bc} \equiv \frac{1}{2} \widetilde{g}^{ad}(\partial_b \widetilde{g}_{dc} + \partial_c \widetilde{g}_{db} - \partial_d \widetilde{g}_{bc}).</math>
The vacuum field equations obtained in this way by Thiry<ref name=thry/> and Jordan's group<ref name=jor1 /><ref name=jor2 /><ref name=jor3 /> are as follows.

The field equation for <math>\phi</math> is obtained from
: <math>\widetilde{R}_{55} = 0 \Rightarrow \Box \phi = \frac{1}{4} \phi^3 F^{\alpha\beta} F_{\alpha\beta},</math>
where <math>F_{\alpha\beta} \equiv \partial_\alpha A_\beta - \partial_\beta A_\alpha,</math> <math>\Box \equiv g^{\mu\nu} \nabla_\mu \nabla_\nu,</math> and <math>\nabla_\mu</math> is a standard, 4D [[covariant derivative]]. It shows that the electromagnetic field is a source for the [[scalar field]]. Note that the scalar field cannot be set to a constant without constraining the electromagnetic field. The earlier treatments by Kaluza and Klein did not have an adequate description of the scalar field and did not realize the implied constraint on the electromagnetic field by assuming the scalar field to be constant.

The field equation for <math>A^\nu</math> is obtained from
: <math>\widetilde{R}_{5\alpha} = 0 = \frac{1}{2\phi} g^{\beta\mu} \nabla_\mu(\phi^3 F_{\alpha\beta}) - A_\alpha\phi\Box\phi.</math>
It has the form of the vacuum Maxwell equations if the scalar field is constant.

The field equation for the 4D Ricci tensor <math>R_{\mu\nu}</math> is obtained from
: <math>\begin{align}
  \widetilde{R}_{\mu\nu} - \frac{1}{2} \widetilde{g}_{\mu\nu} \widetilde{R} &= 0 \Rightarrow \\
  R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R &= \frac{1}{2} \phi^2 \left(g^{\alpha\beta} F_{\mu\alpha} F_{\nu\beta} - \frac{1}{4} g_{\mu\nu} F_{\alpha\beta} F^{\alpha\beta}\right) + \frac{1}{\phi} (\nabla_\mu \nabla_\nu \phi - g_{\mu\nu} \Box\phi),
\end{align}</math>
where <math>R</math> is the standard 4D Ricci scalar.

This equation shows the remarkable result, called the "Kaluza miracle", that the precise form for the [[electromagnetic stress–energy tensor]] emerges from the 5D vacuum equations as a source in the 4D equations: field from the vacuum. This relation allows the definitive identification of <math>A^\mu</math> with the electromagnetic vector potential. Therefore, the field needs to be rescaled with a conversion constant <math>k</math> such that <math>A^\mu \to kA^\mu</math>.

The relation above shows that we must have
: <math>\frac{k^2}{2} = \frac{8\pi G}{c^4} \frac{1}{\mu_0} = \frac{2G}{c^2} 4\pi\epsilon_0,</math>
where <math>G</math> is the [[gravitational constant]], and <math>\mu_0</math> is the [[permeability of free space]]. In the Kaluza theory, the gravitational constant can be understood as an electromagnetic coupling constant in the metric. There is also a stress–energy tensor for the scalar field. The scalar field behaves like a variable gravitational constant, in terms of modulating the coupling of electromagnetic stress–energy to spacetime curvature. The sign of <math>\phi^2</math> in the metric is fixed by correspondence with 4D theory so that electromagnetic energy densities are positive. It is often assumed that the fifth coordinate is spacelike in its signature in the metric.

In the presence of matter, the 5D vacuum condition cannot be assumed. Indeed, Kaluza did not assume it. The full field equations require evaluation of the 5D [[Kaluza–Klein–Einstein tensor]]
: <math>\widetilde{G}_{ab} \equiv \widetilde{R}_{ab} - \frac{1}{2} \widetilde{g}_{ab}\widetilde{R},</math>
as seen in the recovery of the electromagnetic stress–energy tensor above. The 5D curvature tensors are complex, and most English-language reviews contain errors in either <math>\widetilde{G}_{ab}</math> or <math>\widetilde{R}_{ab}</math>, as does the English translation of Thiry.<ref name=thry /> In 2015, a complete set of 5D curvature tensors under the cylinder condition, evaluated using tensor-algebra software, was produced.<ref name="LLW"/>

== Equations of motion from the Kaluza hypothesis ==
The equations of motion are obtained from the five-dimensional geodesic hypothesis<ref name=kal/> in terms of a 5-velocity <math>\widetilde{U}^a \equiv dx^a/ds</math>:
: <math>
\widetilde{U}^b \widetilde{\nabla}_b \widetilde{U}^a = \frac{d\widetilde{U}^a}{ds} + \widetilde{\Gamma}^a_{bc} \widetilde{U}^b \widetilde{U}^c = 0.
</math>

This equation can be recast in several ways, and it has been studied in various forms by authors including Kaluza,<ref name=kal/> Pauli,<ref>{{cite book |last1=Pauli |first1=Wolfgang |title=Theory of Relativity |url=https://archive.org/details/theoryofrelativi00paul |url-access=registration |date=1958 |publisher=Pergamon Press |location=New York |pages=Supplement 23 |edition=translated by George Field}}</ref> Gross & Perry,<ref>{{cite journal |last1=Gross |first1=D. J. |last2=Perry |first2=M. J. |journal=Nucl. Phys. B |date=1983 |volume=226 |issue=1 |pages=29–48 |doi=10.1016/0550-3213(83)90462-5 |title=Magnetic monopoles in Kaluza–Klein theories |bibcode = 1983NuPhB.226...29G }}</ref> Gegenberg & Kunstatter,<ref>{{cite journal |last1=Gegenberg |first1=J. |last2=Kunstatter |title=The motion of charged particles in Kaluza–Klein space–time |first2=G. |journal=Phys. Lett. |date=1984 |volume=106A |issue=9 |page=410 |bibcode=1984PhLA..106..410G |doi=10.1016/0375-9601(84)90980-0}}</ref> and Wesson & Ponce de Leon,<ref>{{cite journal |last1=Wesson |first1=P. S. |last2=Ponce de Leon |first2=J. |title=The equation of motion in Kaluza–Klein cosmology and its implications for astrophysics |journal=Astronomy and Astrophysics |date=1995 |volume=294 |page=1 |bibcode=1995A&A...294....1W }}</ref> but it is instructive to convert it back to the usual 4-dimensional length element <math>c^2\,d\tau^2 \equiv g_{\mu\nu}\,dx^\mu\,dx^\nu</math>, which is related to the 5-dimensional length element <math>ds</math> as given above:
: <math>
ds^2 = c^2\,d\tau^2 + \phi^2 (kA_\nu\,dx^\nu + dx^5)^2.
</math>

Then the 5D geodesic equation can be written<ref>{{cite conference |last=Williams |first=Lance L. |conference=48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 30 July 2012 – 01 August 2012. Atlanta, Georgia |chapter=Physics of the Electromagnetic Control of Spacetime and Gravity |date=2012 |title=Proceedings of 48th AIAA Joint Propulsion Conference |volume=AIAA 2012-3916 |doi=10.2514/6.2012-3916 |isbn=978-1-60086-935-8 |s2cid=122586403 }}</ref> for the spacetime components of the 4-velocity:
:<math>
  U^\nu \equiv \frac{dx^\nu}{d\tau},
</math>
:<math>
  \frac{dU^\nu}{d\tau} + \widetilde{\Gamma}^\mu_{\alpha\beta} U^\alpha U^\beta + 2 \widetilde{\Gamma}^\mu_{5\alpha} U^\alpha U^5 + \widetilde{\Gamma}^\mu_{55} (U^5)^2 + U^\mu \frac{d}{d\tau} \ln \frac{c\,d\tau}{ds} = 0.
</math>

The term quadratic in <math>U^\nu</math> provides the 4D [[Geodesics in general relativity|geodesic equation]] plus some electromagnetic terms:
: <math>
\widetilde{\Gamma}^\mu_{\alpha\beta} = \Gamma^\mu_{\alpha\beta} + \frac{1}{2} g^{\mu\nu} k^2 \phi^2 (A_\alpha F_{\beta\nu} + A_\beta F_{\alpha\nu} - A_\alpha A_\beta \partial_\nu \ln \phi^2).
</math>

The term linear in <math>U^\nu</math> provides the [[Lorentz force law]]:
: <math>
\widetilde{\Gamma}^\mu_{5\alpha} = \frac{1}{2} g^{\mu\nu} k \phi^2 (F_{\alpha\nu} - A_\alpha \partial_\nu \ln \phi^2).
</math>

This is another expression of the "Kaluza miracle". The same hypothesis for the 5D metric that provides electromagnetic stress–energy in the Einstein equations, also provides the Lorentz force law in the equation of motions along with the 4D geodesic equation. Yet correspondence with the Lorentz force law requires that we identify the component of 5-velocity along the fifth dimension with electric charge:
:<math>
kU^5 = k \frac{dx^5}{d\tau} \to \frac{q}{mc},
</math>
where <math>m</math> is particle mass, and <math>q</math> is particle electric charge. Thus electric charge is understood as motion along the fifth dimension. The fact that the Lorentz force law could be understood as a geodesic in five dimensions was to Kaluza a primary motivation for considering the five-dimensional hypothesis, even in the presence of the aesthetically unpleasing cylinder condition.

Yet there is a problem: the term quadratic in <math>U^5</math>,
: <math>
\widetilde{\Gamma}^\mu_{55} = -\frac{1}{2} g^{\mu\alpha} \partial_\alpha \phi^2.
</math>
If there is no gradient in the scalar field, the term quadratic in <math>U^5</math> vanishes. But otherwise the expression above implies
: <math>U^5 \sim c \frac{q/m}{G^{1/2}}.
</math>
For elementary particles, <math>U^5 > 10^{20} c</math>. The term quadratic in <math>U^5</math> should dominate the equation, perhaps in contradiction to experience. This was the main shortfall of the five-dimensional theory as Kaluza saw it,<ref name=kal/> and he gives it some discussion in his original article.{{clarify|date=September 2023}}

The equation of motion for <math>U^5</math> is particularly simple under the cylinder condition. Start with the alternate form of the geodesic equation, written for the covariant 5-velocity:
: <math>
\frac{d\widetilde{U}_a}{ds} = \frac{1}{2} \widetilde{U}^b \widetilde{U}^c \frac{\partial\widetilde{g}_{bc}}{\partial x^a}.
</math>

This means that under the cylinder condition, <math>\widetilde{U}_5</math> is a constant of the five-dimensional motion:
: <math>
\widetilde{U}_5 = \widetilde{g}_{5a} \widetilde{U}^a = \phi^2 \frac{c\,d\tau}{ds} (kA_\nu U^\nu + U^5) = \text{constant}.
</math>

== Kaluza's hypothesis for the matter stress–energy tensor ==
Kaluza proposed<ref name=kal/> a five-dimensional matter [[stress–energy tensor|stress tensor]] <math>\widetilde{T}_M^{ab}</math> of the form
: <math>\widetilde{T}_M^{ab} = \rho \frac{dx^a}{ds} \frac{dx^b}{ds},</math>
where <math>\rho</math> is a density, and the length element <math>ds</math> is as defined above.

Then the spacetime component gives a typical [[Dust solution|"dust"]] stress–energy tensor:
: <math>
\widetilde{T}_M^{\mu\nu} = \rho \frac{dx^\mu}{ds} \frac{dx^\nu}{ds}.
</math>

The mixed component provides a 4-current source for the Maxwell equations:
: <math>
\widetilde{T}_M^{5\mu} = \rho \frac{dx^\mu}{ds} \frac{dx^5}{ds} = \rho U^\mu \frac{q}{kmc}.
</math>

Just as the five-dimensional metric comprises the four-dimensional metric framed by the electromagnetic vector potential, the five-dimensional stress–energy tensor comprises the four-dimensional stress–energy tensor framed by the vector 4-current.

== Quantum interpretation of Klein ==
Kaluza's original hypothesis was purely classical and extended discoveries of general relativity. By the time of Klein's contribution, the discoveries of Heisenberg, Schrödinger, and [[Louis de Broglie]] were receiving a lot of attention. Klein's ''[[Nature (journal)|Nature]]'' article<ref name="KN"/> suggested that the fifth dimension is closed and periodic, and that the identification of electric charge with motion in the fifth dimension can be interpreted as standing waves of wavelength <math>\lambda^5</math>, much like the electrons around a nucleus in the [[Bohr model]] of the atom. The quantization of electric charge could then be nicely understood in terms of integer multiples of fifth-dimensional momentum. Combining the previous Kaluza result for <math>U^5</math> in terms of electric charge, and a [[de Broglie relation]] for momentum <math>p^5 = h/\lambda^5</math>, Klein obtained<ref name=KN /> an expression for the 0th mode of such waves:
: <math>
mU^5 = \frac{cq}{G^{1/2}} = \frac{h}{\lambda^5} \quad \Rightarrow \quad \lambda^5 \sim \frac{hG^{1/2}}{cq},
</math>
where <math>h</math> is the [[Planck constant]]. Klein found that <math>\lambda^5 \sim 10^{-30}</math>&nbsp;cm, and thereby an explanation for the cylinder condition in this small value.

Klein's ''[[Zeitschrift für Physik]]'' article of the same year,<ref name="KZ"/> gave a more detailed treatment that explicitly invoked the techniques of Schrödinger and de Broglie. It recapitulated much of the classical theory of Kaluza described above, and then departed into Klein's quantum interpretation. Klein solved a Schrödinger-like wave equation using an expansion in terms of fifth-dimensional waves resonating in the closed, compact fifth dimension.

== Quantum field theory interpretation ==
{{empty section|date=February 2015}}

== 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}}

== Space–time–matter theory ==
One particular variant of Kaluza–Klein theory is '''space–time–matter theory''' or '''induced matter theory''', chiefly promulgated by [[Paul S. Wesson|Paul Wesson]] and other members of the Space–Time–Matter Consortium.<ref>[http://5dstm.org 5Dstm.org]</ref> In this version of the theory, it is noted that solutions to the equation
: <math>\widetilde{R}_{ab}=0</math>
may be re-expressed so that in four dimensions, these solutions satisfy [[Einstein's equation]]s
: <math>G_{\mu\nu} = 8\pi T_{\mu\nu}\,</math>
with the precise form of the ''T''<sub>''μν''</sub> following from the [[Ricci-flat condition]] on the five-dimensional space. In other words, the cylinder condition of the previous development is dropped, and the stress–energy now comes from the derivatives of the 5D metric with respect to the fifth coordinate. Because the [[energy–momentum tensor]] is normally understood to be due to concentrations of matter in four-dimensional space, the above result is interpreted as saying that four-dimensional matter is induced from geometry in five-dimensional space.

In particular, the [[soliton]] solutions of <math>\widetilde{R}_{ab}=0</math> can be shown to contain the [[Friedmann–Lemaître–Robertson–Walker metric]] in both radiation-dominated (early universe) and matter-dominated (later universe) forms.  The general equations can be shown to be sufficiently consistent with classical [[tests of general relativity]] to be acceptable on physical principles, while still leaving considerable freedom to also provide interesting [[cosmological model]]s.

== Geometric interpretation ==
The Kaluza–Klein theory has a particularly elegant presentation in terms of geometry. In a certain sense, it looks just like ordinary gravity in [[free space]], except that it is phrased in five dimensions instead of four.

=== Einstein equations ===
The equations governing ordinary gravity in free space can be obtained from an [[action (physics)|action]], by applying the [[variational principle]] to a certain [[action (physics)|action]]. Let ''M'' be a ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian manifold]], which may be taken as the [[spacetime]] of [[general relativity]]. If ''g'' is the [[Metric (mathematics)|metric]] on this manifold, one defines the [[action (physics)|action]] ''S''(''g'') as
: <math>S(g) = \int_M R(g) \operatorname{vol}(g),</math>
where ''R''(''g'') is the [[scalar curvature]], and vol(''g'') is the [[volume element]]. By applying the [[variational principle]] to the action
: <math>\frac{\delta S(g)}{\delta g} = 0,</math>
one obtains precisely the [[Einstein equation]]s for free space:
: <math>R_{ij} - \frac{1}{2} g_{ij} R = 0,</math>
where ''R''<sub>''ij''</sub> is the [[Ricci tensor]].

=== Maxwell equations ===
By contrast, the [[Maxwell equation]]s describing [[electromagnetism]] can be understood to be the [[de Rham cohomology|Hodge equations]] of a [[principal bundle|principal U(1)-bundle]] or [[circle bundle]] <math>\pi:P\to M</math> with fiber [[U(1)]].  That is, the [[electromagnetic field]] <math>F</math> is a [[harmonic form|harmonic 2-form]] in the space <math>\Omega^2(M)</math> of differentiable [[2-form]]s on the manifold <math>M</math>. In the absence of charges and currents, the free-field Maxwell equations are
: <math>\mathrm{d}F = 0 \quad\text{and}\quad \mathrm{d}{\star}F = 0,</math>
where <math>\star</math> is the [[Hodge star operator]].

=== Kaluza–Klein geometry ===
To build the Kaluza–Klein theory, one picks an invariant metric on the circle <math>S^1</math> that is the fiber of the U(1)-bundle of electromagnetism. In this discussion, an ''invariant metric'' is simply one that is invariant under rotations of the circle. Suppose that this metric gives the circle a total length <math>\Lambda</math>. One then considers metrics <math>\widehat{g}</math> on the bundle <math>P</math> that are consistent with both the fiber metric, and the metric on the underlying manifold <math>M</math>. The consistency conditions are:
* The projection of <math>\widehat{g}</math> to the [[vertical bundle|vertical subspace]] <math>\operatorname{Vert}_p P \subset T_p P</math> needs to agree with metric on the fiber over a point in the manifold <math>M</math>.
* The projection of <math>\widehat{g}</math> to the [[horizontal bundle|horizontal subspace]] <math>\operatorname{Hor}_p P \subset T_p P</math> of the [[tangent space]] at point <math>p \in P</math> must be isomorphic to the metric <math>g</math> on <math>M</math> at <math>\pi(P)</math>.

The Kaluza–Klein action for such a metric is given by
: <math>S(\widehat{g}) = \int_P R(\widehat{g}) \operatorname{vol}(\widehat{g}).</math>

The scalar curvature, written in components, then expands to
: <math>R(\widehat{g}) = \pi^*\left(R(g) - \frac{\Lambda^2}{2} |F|^2\right),</math>
where <math>\pi^*</math> is the [[pullback (differential geometry)|pullback]] of the fiber bundle projection <math>\pi: P \to M</math>. The connection <math>A</math> on the fiber bundle is related to the electromagnetic field strength as

:<math>\pi^*F = dA.</math>

That there always exists such a connection, even for fiber bundles of arbitrarily complex topology, is a result from [[homology (mathematics)|homology]] and specifically, [[K-theory]]. Applying [[Fubini's theorem]] and integrating on the fiber, one gets
: <math>S(\widehat{g}) = \Lambda \int_M \left(R(g) - \frac{1}{\Lambda^2} |F|^2\right) \operatorname{vol}(g).</math>

Varying the action with respect to the component <math>A</math>, one regains the Maxwell equations. Applying the variational principle to the base metric <math>g</math>, one gets the Einstein equations
: <math>R_{ij} - \frac{1}{2} g_{ij} R = \frac{1}{\Lambda^2} T_{ij}</math>
with the [[electromagnetic stress–energy tensor]] being given by
: <math>T^{ij} = F^{ik} F^{jl} g_{kl} - \frac{1}{4} g^{ij} |F|^2.</math>

The original theory identifies <math>\Lambda</math> with the fiber metric <math>g_{55}</math> and allows <math>\Lambda</math> to vary from fiber to fiber. In this case, the coupling between gravity and the electromagnetic field is not constant, but has its own dynamical field, the [[Radion (physics)|radion]].

=== Generalizations ===
In the above, the size of the loop <math>\Lambda</math> acts as a coupling constant between the gravitational field and the electromagnetic field.  If the base manifold is four-dimensional, the Kaluza–Klein manifold ''P'' is five-dimensional. The fifth dimension is a [[compact space]] and is called the '''compact dimension'''. The technique of introducing compact dimensions to obtain a higher-dimensional manifold is referred to as [[compactification (physics)|compactification]]. Compactification does not produce group actions on [[Chirality (physics)|chiral]] [[fermions]] except in very specific cases: the dimension of the total space must be 2 mod 8, and the G-index of the Dirac operator of the compact space must be nonzero.<ref>L. Castellani et al., Supergravity and superstrings, vol.&nbsp;2, ch.&nbsp;V.11.</ref>

The above development generalizes in a more-or-less straightforward fashion to general [[principal G-bundle|principal ''G''-bundles]] for some arbitrary [[Lie group]] ''G'' taking the place of [[U(1)]]. In such a case, the theory is often referred to as a [[Yang–Mills theory]] and is sometimes taken to be synonymous. If the underlying manifold is [[supersymmetric]], the resulting theory is a super-symmetric Yang–Mills theory.

== Empirical tests ==
No experimental or observational signs of extra dimensions have been officially reported. Many theoretical search techniques for detecting Kaluza–Klein resonances have been proposed using the mass couplings of such resonances with the [[top quark]]. An analysis of results from the LHC in December 2010 severely constrains theories with [[large extra dimensions]].<ref name="arxiv.org 1">{{cite journal |title=Search for microscopic black hole signatures at the Large Hadron Collider |year=2011 |doi=10.1016/j.physletb.2011.02.032 |arxiv=1012.3375 |last1=Khachatryan |first1=V. |last2=Sirunyan |first2=A. M. |last3=Tumasyan |first3=A. |last4=Adam |first4=W. |last5=Bergauer |first5=T. |last6=Dragicevic |first6=M. |last7=Erö |first7=J. |last8=Fabjan |first8=C. |last9=Friedl |first9=M. |last10=Frühwirth |first10=R. |last11=Ghete |first11=V.M. |last12=Hammer |first12=J. |last13=Hänsel |first13=S. |last14=Hartl |first14=C. |last15=Hoch |first15=M. |last16=Hörmann |first16=N. |last17=Hrubec |first17=J. |last18=Jeitler |first18=M. |last19=Kasieczka |first19=G. |last20=Kiesenhofer |first20=W. |last21=Krammer |first21=M. |last22=Liko |first22=D. |last23=Mikulec |first23=I. |last24=Pernicka |first24=M. |last25=Rohringer |first25=H. |last26=Schöfbeck |first26=R. |last27=Strauss |first27=J. |last28=Taurok |first28=A. |last29=Teischinger |first29=F. |last30=Waltenberger |first30=W. |journal=Physics Letters B |volume=697 |issue=5 |pages=434–453 |bibcode=2011PhLB..697..434C |s2cid=122803232 |display-authors=1 |collaboration=CMS Collaboration}}</ref>

The observation of a [[Higgs boson|Higgs]]-like boson at the LHC establishes a new empirical test which can be applied to the search for Kaluza–Klein resonances and supersymmetric particles.
The loop [[Feynman diagram]]s that exist in the Higgs interactions allow any particle with electric charge and mass to run in such a loop. Standard Model particles besides the [[top quark]] and [[W and Z bosons|W boson]] do not make big contributions to the cross-section observed in the {{nowrap|H → γγ}} decay, but if there are new particles beyond the Standard Model, they could potentially change the ratio of the predicted Standard Model {{nowrap|H → γγ}} cross-section to the experimentally observed cross-section. Hence a measurement of any dramatic change to the {{nowrap|H → γγ}} cross-section predicted by the Standard Model is crucial in probing the physics beyond it.

An article from July 2018<ref name="arxiv.org 2">{{cite journal |title=Limits on the number of spacetime dimensions from [[GW170817]] |year=2018 |doi=10.1088/1475-7516/2018/07/048 |arxiv=1801.08160 |last1=Pardo |first1=Kris |last2=Fishbach |first2=Maya |last3=Holz |first3=Daniel E. |last4=Spergel |first4=David N. |journal=Journal of Cosmology and Astroparticle Physics |volume=2018 |issue=7 |page=048 |bibcode=2018JCAP...07..048P |s2cid=119197181 }}</ref> gives some hope for this theory; in the article they dispute that gravity is leaking into higher dimensions as in [[brane theory]]. However, the article does demonstrate that electromagnetism and gravity share the same number of dimensions, and this fact lends support to Kaluza–Klein theory; whether the number of dimensions is really 3&nbsp;+&nbsp;1 or in fact 4&nbsp;+&nbsp;1 is the subject of further debate.

== See also ==
{{cols}}
* [[Classical theories of gravitation]]
* [[Complex spacetime]]
* [[DGP model]]
* [[Quantum gravity]]
* [[Compactification (physics)]]
* [[Randall–Sundrum model]]
* [[Matej Pavšič]]
* [[String theory]]
* [[Supergravity]]
* [[Superstring theory]]
* [[Non-relativistic gravitational fields]]
* [[Teleparallelism]]
{{colend}}

== Notes ==
{{Reflist|30em}}

== References ==

* {{cite journal |last=Kaluza |first=Theodor |date=1921 |title=Zum Unitätsproblem in der Physik |journal=Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) |pages=966–972 |bibcode=1921SPAW.......966K }}  https://archive.org/details/sitzungsberichte1921preussi
* {{cite journal |last=Klein |first=Oskar |date=1926 |title=Quantentheorie und fünfdimensionale Relativitätstheorie |journal=[[Zeitschrift für Physik A]] |volume=37 |issue=12 |pages=895–906 |doi=10.1007/BF01397481 |bibcode = 1926ZPhy...37..895K }}
* {{cite journal |last=Witten |first=Edward |date=1981 |title=Search for a realistic Kaluza–Klein theory |journal=[[Nuclear Physics B]] |volume=186 |issue=3 |pages=412–428 |doi=10.1016/0550-3213(81)90021-3 |bibcode = 1981NuPhB.186..412W }}
* {{cite book |title=Modern Kaluza–Klein Theories |last=Appelquist |first=Thomas |author2=Chodos, Alan |author3=Freund, Peter G. O.  |date=1987 |publisher=Addison–Wesley |location=Menlo Park, Cal. |isbn=978-0-201-09829-7 }} ''(Includes reprints of the above articles as well as those of other important papers relating to Kaluza–Klein theory.)''
* {{cite book |title=Proceedings of the Symposium 'The Oskar Klein Centenary' |chapter=Kaluza–Klein Theory in Perspective |last=Duff |first=M. J. |editor=Lindström, Ulf|date=1994 |publisher=World Scientific |location=Singapore |isbn=978-981-02-2332-8 |pages=22–35 }}
* {{cite journal |last=Overduin |first=J. M. |author2=Wesson, P. S. |date=1997 |title=Kaluza–Klein Gravity |journal=Physics Reports |volume=283 |issue=5 |pages=303–378 |doi=10.1016/S0370-1573(96)00046-4 |arxiv = gr-qc/9805018 |bibcode = 1997PhR...283..303O |s2cid=119087814 }}
* {{cite book |title=Five-Dimensional Physics: Classical and Quantum Consequences of Kaluza–Klein Cosmology |last=Wesson |first=Paul S. |date=2006 |publisher=World Scientific |location=Singapore |isbn=978-981-256-661-4 |bibcode=2006fdpc.book.....W |url-access=registration |url=https://archive.org/details/fivedimensionalp0000wess }}

== Further reading ==

* The CDF Collaboration, ''[http://www-cdf.fnal.gov/PES/kkgrav/kkgrav.html Search for Extra Dimensions using Missing Energy at CDF]'', (2004) ''(A simplified presentation of the search made for extra dimensions at the [[Collider Detector at Fermilab]] (CDF) particle physics facility.)''
* John M. Pierre, ''[http://www.sukidog.com/jpierre/strings/extradim.htm SUPERSTRINGS! Extra Dimensions]'', (2003).
* Chris Pope, ''[http://faculty.physics.tamu.edu/pope/ihplec.pdf Lectures on Kaluza–Klein Theory]''.
* Edward Witten (2014). "A Note On Einstein, Bergmann, and the Fifth Dimension", {{ArXiv|1401.8048}}

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