Complex spacetime: Difference between revisions
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{{ cite journal | author=Hubert F. M. Goenner | date=2014 | title=On the History of Unified Field Theories Part II (ca. 1930 — ca. 1965) | journal=Living Reviews in Relativity | volume=17 | issue=5 | page=| url=http://www.livingreviews.org/lrr-2004-2 }} |
*{{ cite journal | author=Hubert F. M. Goenner | date=2014 | title=On the History of Unified Field Theories Part II (ca. 1930 — ca. 1965) | journal=Living Reviews in Relativity | volume=17 | issue=5 | page=| url=http://www.livingreviews.org/lrr-2004-2 }} |
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*{{ Citation | author=Walter, Scott | year=1999 | contribution=The non-Euclidean style of Minkowskian relativity | editor=J. Gray | title=The Symbolic Universe: Geometry and Physics 1890-1930 | pages=91—127 | publisher=Oxford University Press | contribution-url=http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf }} |
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[[Category:Physics]] |
[[Category:Physics]] |
Revision as of 23:05, 16 June 2015
![]() | This article may present fringe theories, without giving appropriate weight to the mainstream view and explaining the responses to the fringe theories. (June 2015) |
In mathematics and mathematical physics, complex spacetime extends the traditional notion of spacetime described by real-valued space and time coordinates to complex-valued space and time coordinates.
Real and complex spaces
Mathematics
The complexification of a real vector space results in a complex vector space (over the complex number field). To "complexify" a space means that we extend ordinary scalar multiplication of vectors by real numbers to scalar multiplication by complex numbers. For complexified inner product spaces, the complex inner product on vectors replaces the ordinary real-valued inner product, an example of the latter being the dot product.
In mathematical physics, when we complexify a real coordinate space Rn we create a complex coordinate space Cn, referred to in differential geometry as a "complex manifold". The space Cn can be related to R2n, since every complex number constitutes two real numbers.
A complex spacetime geometry refers to the metric tensor being complex, not spacetime itself.
Physics
The Minkowski space of special relativity (SR) and general relativity (GR) is a 4-dimensional "pseudo-Euclidean space" vector space. The spacetime underlying Einstein's field equations, which mathematically describe gravitation, is a real 4-dimensional "Pseudo-Riemannian manifold".
In QM, wave functions describing particles 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 fundamental interactions, originally gravity and electromagnetism. These ideas prevail in string theory and beyond. The idea of complex spacetime has received considerably less attention, but it has been considered in conjunction with the Lorentz–Dirac equation and the Maxwell equations.[1] Other ideas include mapping real spacetime into a complex representation space of SU(2, 2), see twistor theory.[2]
In 1919, Theodor Kaluza posted his 5-dimensional extension of general relativity, to Albert Einstein,[3] who was impressed with how the equations of electromagnetism emerged from Kaluza's theory. In 1926, Oskar Klein suggested[4] that Kaluza's extra dimension might be "curled up" into an extremely small circle, as if a circular 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 hyper-dimension as it spun through 360°. This 5d theory is named Kaluza–Klein theory.
In 1932, Hsin P. Soh of MIT, advised by Arthur Eddington, published a theory attempting to unifying gravitation and electromagnetism within a complex 4-dimensional Reimannian geometry. The line element ds2 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.[5]
For several decades after publishing his general theory of relativity in 1915, Einstein tried to unify gravity with electromagnetism, to create a unified field theory explaining both interactions. In the latter years of WWII, Einstein began considering complex spacetime geometries of various kinds.[6]
In 1953, Wolfgang Pauli generalised[7] the Kaluza-Klein theory to a six-dimensional space, and (using dimensional reduction) derived the essentials of an SU(2) gauge theory (applied in QM to the electroweak interaction), as if Klein's "curled up" circle had become the surface of an infinitesimal hypersphere.
There have been attempts to formulate the Dirac equation in complex spacetime by analytic continuation.[8]
See also
References
Notes
- ^ Newman, E. T. (1973). "Maxwell's equations and complex Minkowski space". J. Math. Phys. 14 (1). The American Institute of Physics. doi:10.1063/1.1666160.
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suggested) (help) - ^ Penrose, Roger (1967), "Twistor algebra", Journal of Mathematical Physics, 8 (2): 345–366, Bibcode:1967JMP.....8..345P, doi:10.1063/1.1705200, MR 0216828
- ^ Pais, Abraham (1982). Subtle is the Lord ...: The Science and the Life of Albert Einstein. Oxford: Oxford University Press. pp. 329–330.
- ^ Oskar Klein (1926). "Quantentheorie und fünfdimensionale Relativitätstheorie". Zeitschrift für Physik A. 37 (12): 895–906. Bibcode:1926ZPhy...37..895K. doi:10.1007/BF01397481.
- ^ Soh, H. P. (1932). "A Theory of Gravitation and Electricity". J. Math. Phys. (MIT) (12): 298—305.
- ^ Einstein, A. (1945), "A Generalization of the Relativistic Theory of Gravitation", Ann. of Math., 46 (4): 578–584 – via JSTOR
- ^ N. Straumann (2000). "On Pauli's invention of non-abelian Kaluza-Klein Theory in 1953" (PDF).
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(help) - ^ Mark Davidson (2012). "A study of the Lorentz-Dirac equation in complex space-time for clues to emergent quantum mechanics". Journal of Physics: Conference Series.
Further reading
- Hubert F. M. Goenner (2014). "On the History of Unified Field Theories Part II (ca. 1930 — ca. 1965)". Living Reviews in Relativity. 17 (5).
- Walter, Scott (1999), "The non-Euclidean style of Minkowskian relativity" (PDF), in J. Gray (ed.), The Symbolic Universe: Geometry and Physics 1890-1930, Oxford University Press, pp. 91–127