Editing Kaluza–Klein theory (section)

{{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>