View source for Physics:Complex spacetime

'''Complex spacetime''' is a mathematical framework that combines the concepts of [[Complex number|complex number]]s and [[Physics:Spacetime|spacetime]] in physics. In this framework, the usual real-valued coordinates of spacetime are replaced with complex-valued coordinates. This allows for the inclusion of imaginary components in the description of spacetime, which can have interesting implications in certain areas of physics, such as [[Physics:Quantum field theory|quantum field theory]] and [[Physics:String theory|string theory]].
The notion is entirely mathematical with no physics implied, but should be seen as a tool, for instance, as exemplified by the [[Physics:Wick rotation|Wick rotation]].

==Real and complex spaces==
===Mathematics===
The [[Complexification|complexification]] of a real vector space results in a complex vector space (over the [[Complex number|complex number]] [[Field (mathematics)|field]]). To "complexify" a space means extending ordinary [[Scalar multiplication|scalar multiplication]] of vectors by real numbers to scalar multiplication by [[Complex number|complex number]]s. For complexified [[Inner product space|inner product space]]s, the complex inner product on vectors replaces the ordinary real-valued inner product, an example of the latter being the [[Dot product|dot product]].

In mathematical physics, when we complexify a [[Real coordinate space|real coordinate space]] <math>\mathbb{R}^n</math> we create a complex [[Coordinate space|coordinate space]] <math>\mathbb{C}^n</math>, referred to in [[Differential geometry|differential geometry]] as a "[[Complex manifold|complex manifold]]". The space <math>\mathbb{C}^n</math> can be related to <math>\mathbb{R}^{2n}</math>, since every complex number constitutes two real numbers.

A '''complex spacetime ''geometry''''' refers to the [[Physics:Metric tensor|metric tensor]] being complex, not spacetime itself.

===Physics===
The [[Physics:Minkowski space|Minkowski space]] of [[Physics:Special relativity|special relativity]] (SR) and [[Physics:General relativity|general relativity]] (GR) is a 4&nbsp;dimensional [[Pseudo-Euclidean space|pseudo-Euclidean space]]. The [[Physics:Spacetime|spacetime]] underlying [[Physics:Einstein's field equations|Albert Einstein's field equations]], which mathematically describe gravitation, is a real 4&nbsp;dimensional [[Pseudo-Riemannian manifold|pseudo-Riemannian manifold]].

In quantum mechanics, [[Wave function|wave function]]s describing [[Physics:Particle|particle]]s are complex-valued functions of real space and time variables. The set of all wavefunctions for a given system is an infinite-dimensional complex [[Hilbert space]].

==History==
The notion of spacetime having more than four dimensions is of interest in its own mathematical right. Its appearance in physics can be rooted to attempts of unifying the [[Physics:Fundamental interaction|fundamental interaction]]s, originally [[Physics:Gravity|gravity]] and [[Physics:Electromagnetism|electromagnetism]]. These ideas prevail in [[Physics:String theory|string theory]] and beyond. The idea of ''complex'' spacetime has received considerably less attention, but it has been considered in conjunction with the Lorentz&ndash;Dirac equation and the Maxwell equations.<ref>{{cite journal|last=Trautman|first=A.|title=A discussion on the present state of relativity - Analytic solutions of Lorentz-invariant linear equations|journal=Proc. R. Soc. A|volume=270|issue=1342|pages=326–328|year=1962|doi=10.1098/rspa.1962.0222|bibcode=1962RSPSA.270..326T|s2cid=120301116}}</ref><ref>{{cite journal|last=Newman|first=E. T.|title=Maxwell's equations and complex Minkowski space|journal=J. Math. Phys.|year=1973|volume=14|issue=1|publisher=The American Institute of Physics|doi=10.1063/1.1666160|bibcode = 1973JMP....14..102N|pages=102–103}}</ref> Other ideas include mapping real spacetime into a complex representation space of {{math|SU(2, 2)}}, see [[Physics:Twistor theory|twistor theory]].<ref>{{citation|last=Penrose|first=Roger|title=Twistor algebra|url=http://link.aip.org/link/JMAPAQ/v8/i2/p345/s1|doi=10.1063/1.1705200|mr=0216828|year=1967|journal=[[Physics:Journal of Mathematical Physics|Journal of Mathematical Physics]]|volume=8|pages=345–366|bibcode=1967JMP.....8..345P|issue=2|access-date=2015-06-14|archive-url=https://archive.today/20130112095407/http://link.aip.org/link/JMAPAQ/v8/i2/p345/s1|archive-date=2013-01-12|url-status=dead}}</ref>

In 1919, [[Biography:Theodor Kaluza|Theodor Kaluza]] posted his 5-dimensional extension of [[Physics:General relativity|general relativity]], to [[Biography:Albert Einstein|Albert Einstein]],<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> who was impressed with how the equations of [[Physics:Electromagnetism|electromagnetism]] emerged from Kaluza's theory. In 1926, [[Biography:Oskar Klein|Oskar Klein]] suggested<ref>{{ cite journal|author=Oskar Klein |date=1926 |title=Quantentheorie und fünfdimensionale Relativitätstheorie |journal=[[Physics:Zeitschrift für Physik A|Zeitschrift für Physik A]] |volume=37 |issue=12 |pages=895–906 |doi=10.1007/BF01397481 |bibcode=1926ZPhy...37..895K }}</ref> that Kaluza's extra dimension might be "[[Physics:Dimension#Additional dimensions|curled up]]" into an extremely small circle, as if a [[Circle group|circular]] [[Topology|topology]] is hidden within every point in space. Instead of being another spatial dimension, the extra dimension could be thought of as an angle, which created a [[Physics:Dimension#Additional dimensions|hyper-dimension]] as it spun through 360°. This 5d theory is named [[Physics:Kaluza–Klein theory|Kaluza–Klein theory]].

In 1932, Hsin P. Soh of [[Organization:Massachusetts Institute of Technology|MIT]], advised by [[Biography:Arthur Eddington|Arthur Eddington]], published a theory attempting to unify gravitation and electromagnetism within a complex 4-dimensional [[Riemannian geometry]]. The [[Line element|line element]] ''ds''<sup>2</sup> is complex-valued, so that the real part corresponds to mass and gravitation, while the imaginary part with charge and electromagnetism. The usual space ''x'', ''y'', ''z'' and time ''t'' coordinates themselves are real and spacetime is not complex, but tangent spaces are allowed to be.<ref>{{ cite journal | author=Soh, H. P. | date=1932 | title=A Theory of Gravitation and Electricity | journal=J. Math. Phys. (MIT) | volume=12 | issue=1–4 | pages=298–305 | doi=10.1002/sapm1933121298}}</ref>

For several decades after publishing his general theory of relativity in 1915, Albert Einstein tried to unify [[Physics:Gravity|gravity]] with [[Physics:Electromagnetism|electromagnetism]], to create a [[Physics:Unified field theory|unified field theory]] explaining both interactions. In the latter years of [[Social:World War II|World War II]], Albert Einstein began considering complex spacetime geometries of various kinds.<ref>{{citation|last=Einstein|first=A.|title=A Generalization of the Relativistic Theory of Gravitation|year=1945|volume=46|issue=4|pages=578&ndash;584|journal=Ann. of Math.|jstor=1969197|doi=10.2307/1969197}}</ref>

In 1953, [[Biography:Wolfgang Pauli|Wolfgang Pauli]] generalised<ref>{{cite journal| author = N. Straumann | year = 2000 | title =On Pauli's invention of non-abelian Kaluza–Klein Theory in 1953 | arxiv = gr-qc/0012054 | bibcode = 2000gr.qc....12054S }}</ref> the [[Physics:Kaluza–Klein theory|Kaluza–Klein theory]] to a six-dimensional space, and (using [[Physics:Dimensional reduction|dimensional reduction]]) derived the essentials of an  {{math|[[Special unitary group#The group SU(2)|SU(2)]]}} [[Physics:Gauge theory|gauge theory]] (applied in quantum mechanics to the [[Physics:Electroweak interaction|electroweak interaction]]), as if Klein's "curled up" circle had become the surface of an infinitesimal [[N-sphere|hypersphere]].

In 1975, Jerzy Plebanski published "Some Solutions of Complex Albert Einstein Equations".<ref>{{cite journal | author=Plebański, J. | s2cid=122814301 | title=Some solutions of complex Einstein equations | journal=[[Physics:Journal of Mathematical Physics|Journal of Mathematical Physics]] | volume=16 | issue=12 |pages=2395–2402 | year=1975 |bibcode = 1975JMP....16.2395P |doi = 10.1063/1.522505 | doi-access=free }}</ref>

There have been attempts to formulate the [[Physics:Dirac equation|Dirac equation]] in complex spacetime by [[Analytic continuation|analytic continuation]].<ref>{{cite journal| author = Mark Davidson | year = 2012 | title =A study of the Lorentz–Dirac equation in complex space-time for clues to emergent quantum mechanics|journal=Journal of Physics: Conference Series| doi=10.1088/1742-6596/361/1/012005| volume=361| issue = 1 | pages=012005|bibcode = 2012JPhCS.361a2005D | doi-access =free}}</ref>

==See also==
*[[Physics:Construction of a complex null tetrad|Construction of a complex null tetrad]]
*[[Physics:Four-vector|Four-vector]]
*[[Hilbert space]]
*[[Twistor space]]
*[[Spherical basis]]
*[[Physics:Riemann–Silberstein vector|Riemann–Silberstein vector]]

==References==
{{reflist|25em}}

==Further reading==
* {{ cite journal
 | first=Hubert F.M. |last=Goenner
 | year=2014
 | title=On the history of unified field theories Part&nbsp;II (ca.&nbsp;1930 — ca.&nbsp;1965)
 | journal = Living Reviews in Relativity
 | volume=17 | issue=5 | pages=5
 | doi=10.12942/lrr-2014-5 | pmid=28179849
 | pmc=5255905 |bibcode = 2014LRR....17....5G
 | url=http://www.livingreviews.org/lrr-2004-2
}}
* {{cite arXiv
 |last=Kaiser |first=Gerald
 |year=2009
 |title=Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis
 |class=math-ph |eprint=0910.0352
}}

[[Category:Spacetime]]
[[Category:Theory of relativity]]

{{Sourceattribution|Complex spacetime}}