View source for Physics:Quantum De Sitter invariant special relativity

{{Short description|Special relativity formed using the SO(4,1) symmetry group}}

{{Quantum book backlink|Advanced and frontier topics}}
In [[Wikipedia:mathematical physics|mathematical physics]], '''de Sitter invariant special relativity''' is the speculative idea that the fundamental [[Wikipedia:symmetry group|symmetry group]] of [[Wikipedia:spacetime|spacetime]] is the [[Indefinite orthogonal group]] SO(4,1), that of [[Wikipedia:de Sitter space|de Sitter space]]. In the standard theory of [[Wikipedia:general relativity|general relativity]], de Sitter space is a highly symmetrical special [[Wikipedia:vacuum solution|vacuum solution]], which requires a [[Wikipedia:cosmological constant|cosmological constant]] or the [[Wikipedia:Stress–energy tensor|stress–energy]] of a constant [[Wikipedia:scalar field|scalar field]] to sustain.
The idea of de Sitter invariant relativity is to require that the laws of physics are not fundamentally invariant under the [[Wikipedia:Poincaré group|Poincaré group]] of [[Wikipedia:special relativity|special relativity]], but under the symmetry group of de Sitter space instead. With this assumption, empty space automatically has de Sitter symmetry, and what would normally be called the cosmological constant in general relativity becomes a fundamental dimensional parameter describing the symmetry structure of spacetime.
First proposed by [[Wikipedia:Luigi Fantappiè|Luigi Fantappiè]] in 1954, the theory remained obscure until it was rediscovered in 1968 by [[Wikipedia:Henri Bacry|Henri Bacry]] and [[Wikipedia:Jean-Marc Lévy-Leblond|Jean-Marc Lévy-Leblond]]. In 1972, [[Wikipedia:Freeman Dyson|Freeman Dyson]] popularized it as a hypothetical road by which mathematicians could have guessed part of the structure of general relativity before it was discovered.<ref name=dyson>
{{cite journal
 |author=[[Wikipedia:Freeman Dyson|Freeman Dyson]]
 |year=1972
 |title=Missed opportunities
 |journal=Bull. Am. Math. Soc.
 |volume=78|issue=5|pages=635–652
 |doi=10.1090/S0002-9904-1972-12971-9
 |mr=0522147
 |url=https://projecteuclid.org/download/pdf_1/euclid.bams/1183533964
 |format=pdf
 |doi-access=free
}}</ref> The discovery of the [[Wikipedia:Accelerating universe|accelerating expansion of the universe]] has led to a revival of interest in de Sitter invariant theories, in conjunction with other speculative proposals for new physics, like [[Wikipedia:doubly special relativity|doubly special relativity]].

[[File:Penrose Diagrams of various black hole solutions.svg|thumb|450px|Penrose Diagrams of various black hole solutions]]
== Introduction ==
{{main|de Sitter space}}

[[Wikipedia:De Sitter|De Sitter]] suggested that spacetime curvature might not be due solely to gravity<ref>
{{cite journal
 |author=W. de Sitter
 |year=1917
 |title=On the curvature of space
 |journal=Proc. Roy. Acad. Sci. Amsterdam
 |volume=20 |pages=229–243
}}</ref> but he did not give any mathematical details of how this could be accomplished. In 1968 Henri Bacry and [[Wikipedia:Jean-Marc Lévy-Leblond|Jean-Marc Lévy-Leblond]] showed that the de Sitter group was the most general group compatible with isotropy, homogeneity and boost invariance.<ref name=posskinem/> Later, Freeman Dyson<ref name=dyson/> advocated this as an approach to making the [[Wikipedia:mathematical structure|mathematical structure]] of general relativity more self-evident.

[[Wikipedia:Hermann Minkowski|Minkowski]]'s unification of space and time within [[Wikipedia:special relativity|special relativity]] replaces the [[Wikipedia:Galilean group|Galilean group]] of [[Wikipedia:Newtonian mechanics|Newtonian mechanics]] with the [[Wikipedia:Lorentz group|Lorentz group]]. This is called a unification of space and time because the Lorentz group is [[Wikipedia:simple group|simple]], while the Galilean group is a [[Wikipedia:semi-direct product|semi-direct product]] of rotations and [[Wikipedia:Galilean transformation|Galilean boosts]]. This means that the Lorentz group mixes up space and time such that they cannot be disentangled, while the Galilean group treats time as a parameter with different units of measurement than space.

An analogous thing can be made to happen with the ordinary rotation group in three dimensions. If you imagine a nearly flat world, one in which pancake-like creatures wander around on a pancake flat world, their conventional unit of height might be the [[Wikipedia:micrometre|micrometre]] (μm), since that is how high typical structures are in their world, while their unit of distance could be the metre, because that is their body's horizontal extent.  Such creatures would describe the basic symmetry of their world as [[Wikipedia:SO(2)|SO(2)]], being the known rotations in the horizontal (x–y) plane. Later on, they might discover rotations around the x- and y-axes—and in their everyday experience such rotations might always be by an infinitesimal angle, so that these rotations would effectively commute with each other.

The rotations around the horizontal axes would tilt objects by an infinitesimal amount. The tilt in the x–z plane (the "x-tilt") would be one parameter, and the tilt in the y–z plane (the "y-tilt") another. The symmetry group of this pancake world is then SO(2) semidirect product with '''R'''<sup>2</sup>, meaning that a two-dimensional rotation plus two extra parameters, the x-tilt and the y-tilt. The reason it is a semidirect product is that, when you rotate, the x-tilt and the y-tilt rotate into each other, since they form a [[Wikipedia:Visualization software|vector]] and not two [[Wikipedia:scalar (physics)|scalar]]s. In this world, the difference in height between two objects at the same x, y would be a rotationally invariant quantity unrelated to length and width. The z-coordinate is effectively separate from x and y.

Eventually, experiments at large angles would convince the creatures that the symmetry of the world is [[Wikipedia:SO(3)|SO(3)]]. Then they would understand that z is really the same as x and y, since they can be mixed up by rotations. The SO(2) semidirect product R<sup>2</sup> limit would be understood as the limit that the free parameter ''μ'', the ratio of the height range ''μm'' to the length range ''m'', approaches 0. The Lorentz group is analogous—it is a simple group that turns into the Galilean group when the time range is made long compared to the space range, or where velocities may be regarded as infinitesimal, or equivalently, may be regarded as the limit {{nowrap|''c'' → ∞}}, where relativistic effects become observable "as good as at infinite velocity".

The symmetry group of special relativity is not entirely simple, due to translations. The Lorentz group is the set of the transformations that keep the origin fixed, but translations are not included. The full Poincaré group is the semi-direct product of translations with the Lorentz group. If translations are to be similar to elements of the Lorentz group, then as [[Wikipedia:Lorentz boost|boosts]] are [[Wikipedia:non-commutative|non-commutative]], [[Wikipedia:translation (physics)|translations]] would also be non-commutative.

In the pancake world, this would manifest if the creatures were living on an enormous sphere rather than on a plane.  In this case, when they wander around their sphere, they would eventually come to realize that translations are not entirely separate from rotations, because if they move around on the surface of a sphere, when they come back to where they started, they find that they have been rotated by the [[Wikipedia:holonomy|holonomy]] of [[Wikipedia:parallel transport|parallel transport]] on the sphere. If the universe is the same everywhere (homogeneous) and there are no preferred directions (isotropic), then there are not many options for the symmetry group: they either live on a flat plane, or on a sphere with a constant positive curvature, or on a [[Wikipedia:Lobachevski plane|Lobachevski plane]] with constant negative curvature.  If they are not living on the plane, they can describe positions using dimensionless angles, the same parameters that describe rotations, so that translations and rotations are nominally unified.

In relativity, if translations mix up nontrivially with rotations, but the universe is still [[Wikipedia:Homogeneity (physics)|homogeneous]] and [[Wikipedia:isotropic|isotropic]], the only option is that spacetime has a uniform [[Wikipedia:scalar curvature|scalar curvature]]. If the curvature is positive, the analog of the sphere case for the two-dimensional creatures, the spacetime is de Sitter space and its symmetry group is the de Sitter group rather than the Poincaré group.

De Sitter special relativity postulates that the empty space has de Sitter symmetry as a fundamental law of nature. This means that spacetime is slightly curved even in the absence of matter or energy. This residual [[Wikipedia:spacetime curvature|curvature]] implies a positive [[Wikipedia:cosmological constant|cosmological constant]] {{math|''Λ''}} to be determined by observation. Due to the small magnitude of the constant, special relativity with its Poincaré group is indistinguishable from de Sitter space for most practical purposes.

Modern proponents of this idea, such as S. Cacciatori, V. Gorini and A. Kamenshchik,<ref name=c21st>
{{cite journal
 |author1=S. Cacciatori |author2=V. Gorini |author3=A. Kamenshchik |year=2008
 |title=Special Relativity in the 21st century
 |journal=Annalen der Physik
 |volume=17 |issue=9–10 |pages=728–768
 |arxiv=0807.3009
 |doi=10.1002/andp.200810321
 |bibcode = 2008AnP...520..728C |s2cid=119191753
}}</ref> have reinterpreted this theory as physics, not just mathematics. They postulate that the acceleration of the expansion of the universe is not entirely due to [[Wikipedia:Dark energy|vacuum energy]], but at least partly due to the kinematics of the [[Wikipedia:de Sitter|de Sitter]] [[Wikipedia:group (mathematics)|group]], which would replace the Poincaré group.

A modification of this idea allows <math>\Lambda</math> to change with time, so that [[Wikipedia:Inflation (cosmology)|inflation]] may come from the cosmological constant being larger near the [[Wikipedia:Big Bang|Big Bang]] than nowadays. It can also be viewed as a different approach to the problem of [[Wikipedia:quantum gravity|quantum gravity]].<ref name="newroad">
{{cite journal
 |author1=R. Aldrovandi |author2=J. G. Pereira |year=2009
 |title=de Sitter Relativity: a New Road to Quantum Gravity?
 |journal=Foundations of Physics
 |volume=39 |issue=2 |pages=1–19
 |arxiv=0711.2274
 |doi=10.1007/s10701-008-9258-5
 |bibcode=2009FoPh...39....1A|s2cid=15298756
}}</ref>

=== High energy ===
The [[Wikipedia:Poincaré group|Poincaré group]] [[Wikipedia:Galilean transformation#Origin in group contraction|contracts]] to the [[Wikipedia:Galilean group|Galilean group]] for low-velocity [[Wikipedia:kinematics|kinematics]], meaning that when all velocities are small the Poincaré group "morphs" into the Galilean group. (This can be made precise with [[Wikipedia:Erdal İnönü|İnönü]] and [[Wikipedia:Eugene Wigner|Wigner]]'s concept of [[Wikipedia:group contraction|group contraction]].<ref>
{{cite journal
 |author1=E. Inönü |author2=E.P. Wigner |year=1953
 |title=On the Contraction of Groups and Their Representations
 |journal=Proc. Natl. Acad. Sci. USA
 |volume=39 |issue=6 |pages=510–24
 |doi= 10.1073/pnas.39.6.510
 |pmc=1063815
 |pmid=16589298
 |bibcode = 1953PNAS...39..510I |doi-access=free
}}</ref>)

Similarly, the de Sitter group  [[Wikipedia:group contraction|contracts]] to the Poincaré group for short-distance kinematics, when the magnitudes of all translations considered are very small compared to the de Sitter radius.<ref name="newroad"/> In quantum mechanics, short distances are probed by high energies, so that for energies above a very small value related to the cosmological constant, the Poincaré group is a good approximation to the de Sitter group.

In de Sitter relativity, the cosmological constant is no longer a [[Wikipedia:free parameter|free parameter]] of the same type; it is determined by the de Sitter radius, a fundamental quantity that determines the commutation relation of translation with rotations/boosts. This means that the theory of de Sitter relativity might be able to provide insight on the value of the cosmological constant, perhaps explaining the [[Wikipedia:Dark energy#Quintessence|cosmic coincidence]]. Unfortunately, the de Sitter radius, which determines the cosmological constant, is an adjustable parameter in de Sitter relativity, so the theory requires a separate condition to determine its value in relation to the measurement scale.

When a cosmological constant is viewed as a kinematic parameter, the definitions of energy and momentum must be changed from those of special relativity. These changes could significantly modify the physics of the early universe if the cosmological constant was greater back then. Some speculate that a high energy experiment could modify the local structure of spacetime from [[Wikipedia:Minkowski space|Minkowski space]] to de Sitter space with a large cosmological constant for a short period of time, and this might eventually be tested in the existing or planned [[Wikipedia:particle collider|particle collider]].<ref>
{{cite journal
 |author=Freydoon Mansouri
 |year=2002
 |title=Non-Vanishing Cosmological Constant {{math|Λ}}, Phase Transitions, And {{math|Λ}}-Dependence Of High Energy Processes
 |journal=Phys. Lett. B
 |volume=538|issue=3–4|pages=239–245
 |arxiv=hep-th/0203150
 |doi=10.1016/S0370-2693(02)02022-1
 |bibcode = 2002PhLB..538..239M |s2cid=13986319
}}</ref>

=== Doubly special relativity ===
{{main|Doubly special relativity}}

Since the de Sitter group naturally incorporates an invariant length parameter, de Sitter relativity can be interpreted as an example of the so-called [[Wikipedia:doubly special relativity|doubly special relativity]]. There is a fundamental difference, though: whereas in all doubly special relativity models the Lorentz symmetry is violated, in de Sitter relativity it remains as a physical symmetry.<ref>
{{cite journal
 |arxiv=gr-qc/0702065
 |doi=10.1063/1.2752487
 |title=Some Implications of the Cosmological Constant to Fundamental Physics
 |journal=AIP Conference Proceedings
 |volume=910
 |pages=381–395
 |year=2007
 |last1=Aldrovandi |first1=R.
 |last2=Beltrán Almeida |first2=J. P.
 |last3=Pereira |first3=J. G.
 |hdl=11449/69891
 |bibcode=2007AIPC..910..381A
 |s2cid=16631274
}}</ref><ref>
{{cite arXiv
 |author1=R. Aldrovandi
 |author2=J.P. Beltran Almeida
 |author3=C.S.O. Mayor
 |author4=J.G. Pereira
 |year=2007 
 |title=Lorentz Transformations in de Sitter Relativity
 |class=gr-qc
 |eprint=0709.3947
}}</ref> A drawback of the usual doubly special relativity models is that they are valid only at the energy scales where ordinary special relativity is supposed to break down, giving rise to a patchwork relativity. On the other hand, de Sitter relativity is found to be invariant under a simultaneous re-scaling of [[Wikipedia:mass|mass]], [[Wikipedia:energy|energy]] and [[Wikipedia:momentum|momentum]],<ref name=dessrintro>
{{cite journal
 |author1=R. Aldrovandi
 |author2=J.P. Beltrán Almeida
 |author3=J.G. Pereira |year=2007
 |title=de Sitter Special Relativity
 |journal=Class. Quantum Grav.
 |volume=24 |issue=6 |pages=1385–1404
 |arxiv=gr-qc/0606122
 |doi=10.1088/0264-9381/24/6/002
 |bibcode = 2007CQGra..24.1385A |s2cid=11703342
}}</ref> and is consequently valid at all energy scales. A relationship between doubly special relativity, de Sitter space and general relativity is described by Derek Wise.<ref>
{{cite journal
 | author1=Wise
 | year=2010 
 | title=MacDowell–Mansouri Gravity and Cartan Geometry
 | doi=10.1088/0264-9381/27/15/155010
 | journal=Classical and Quantum Gravity
 | volume=27
 | issue=15
 | article-number=155010
 | arxiv=gr-qc/0611154
 | bibcode = 2010CQGra..27o5010W | s2cid=16706599 
}}</ref> See also [[Wikipedia:MacDowell–Mansouri action|MacDowell–Mansouri action]].

=== Newton–Hooke: de Sitter special relativity in the limit ''v'' ≪ ''c'' ===
In the limit as {{nowrap|''v'' ≪ ''c''}}, the [[Wikipedia:De Sitter space|de Sitter group]] contracts to the Newton–Hooke group.<ref>
{{cite journal
 | author1=Aldrovandi
 | author2=Barbosa
 | author3=Crispino
 | author4=Pereira
 | year=1999 
 | title=Non–Relativistic Spacetimes with Cosmological Constant
 | doi=10.1088/0264-9381/16/2/013
 | journal=Classical and Quantum Gravity
 | volume=16
 | issue=2
 | pages=495–506
 | arxiv=gr-qc/9801100
 | bibcode = 1999CQGra..16..495A | citeseerx=10.1.1.339.919
 | s2cid=16691405
}}</ref> This has the effect that in the nonrelativistic limit, objects in de Sitter space have an extra "repulsion" from the origin: objects have a tendency to move away from the center with an outward pointing [[Wikipedia:fictitious force|fictitious force]] proportional to their distance from the origin.

While it looks as though this might pick out a preferred point in space—the center of repulsion, it is more subtly isotropic. Moving to the uniformly accelerated frame of reference of an observer at another point, all accelerations appear to have a repulsion center at the new point.

What this means is that in a spacetime with non-vanishing curvature, gravity is modified from Newtonian gravity.<ref>
{{cite journal
 | author1=Yu Tian
 | author2=Han-Ying Guo
 | author3=Chao-Guang Huang
 | author4=Zhan Xu
 | author5=Bin Zhou
 | year=2004 
 | title=Mechanics and Newton–Cartan-Like Gravity on the Newton–Hooke Space–time
 | doi=10.1103/PhysRevD.71.044030
 | journal=Physical Review D
 | volume=71
 | issue=4
 | article-number=44030
 | arxiv=hep-th/0411004
 |bibcode = 2005PhRvD..71d4030T | s2cid=119378100
}}</ref> At distances comparable to the radius of the space, objects feel an additional linear repulsion from the center of coordinates.

=== History of de Sitter invariant special relativity ===
* "de Sitter relativity" is the same as the theory of "projective relativity" of [[Wikipedia:Luigi Fantappiè|Luigi Fantappiè]] and [[Wikipedia:Giuseppe Arcidiacono|Giuseppe Arcidiacono]] first published in 1954 by Fantappiè<ref name=Licata_Chiatti2008>
{{cite journal
 |first1=Ignazio |last1=Licata
 |author2=Leonardo Chiatti
 |year=2009
 |title=The archaic universe: Big Bang, cosmological term, and the quantum origin of time in projective cosmology
 |doi=10.1007/s10773-008-9874-z
 |journal=International Journal of Theoretical Physics
 |volume=48
 |issue=4
 |pages=1003–1018
 |arxiv=0808.1339
 |bibcode = 2009IJTP...48.1003L |s2cid=119262177
}}</ref> and the same as another independent discovery in 1976.<ref>
{{cite journal
 |last1= Dey |first1= Anind K.
 |year= 2001
 |title=An extension of the concept of inertial frame and of Lorentz transformation
 |journal=Proc. Natl. Acad. Sci. USA
 |volume=73 |issue=5 |pages= 1418–21
 | bibcode=1976PNAS...73.1418K
 |doi= 10.1073/pnas.73.5.1418
 |pmid= 16592318
 |pmc= 430307
 |doi-access= free
}}</ref>
* In 1968 [[Wikipedia:Henri Bacry|Henri Bacry]] and Jean-Marc Lévy-Leblond published a paper on possible kinematics<ref name=posskinem>
{{cite journal
 |author1=Henri Bacry
 |author2= Jean-Marc Lévy-Leblond
 |year=1968
 |title=Possible Kinematics
 |journal=Journal of Mathematical Physics
 |volume=9 |issue=10 |page=1605
 |doi=10.1063/1.1664490
 |bibcode = 1968JMP.....9.1605B
}}</ref>
* In 1972 Freeman Dyson<ref name=dyson/> further explored this.
* In 1973 Eliano Pessa described how Fantappié–Arcidiacono projective relativity relates to earlier conceptions of projective relativity and to [[Wikipedia:Kaluza Klein theory|Kaluza Klein theory]].<ref>[http://www.imub.ub.es/collect/accdocg/COLLECTANEAMATHEMATICA_1973_24_02_05.pdf The De Sitter Universe and general relativity]</ref>
* R. Aldrovandi, J.P. Beltrán Almeida and J.G. Pereira have used the terms "de Sitter special relativity" and "de Sitter relativity" starting from their 2007 paper "de Sitter special relativity".<ref name=dessrintro/><ref>
{{cite journal
 |author1=R. Aldrovandi
 |author2=J. G. Pereira
 |year=2009
 |title=De Sitter Special Relativity: Effects on Cosmology
 |doi=10.1134/S020228930904001X
 |journal=Gravitation and Cosmology
 |volume=15
 |issue=4
 |pages=287–294
 |arxiv=0812.3438
 |bibcode = 2009GrCo...15..287A |s2cid=18473868
}}</ref> This paper was based on previous work on amongst other things: the consequences of a non-vanishing cosmological constant,<ref>
{{cite journal
 |author1=R. Aldrovandi |author2=J.P. Beltran Almeida |author3=J.G. Pereira |year=2004
 |title=Cosmological Term and Fundamental Physics
 |journal=Int. J. Mod. Phys. D
 |volume=13 |issue= 10|pages=2241–2248
 |arxiv=gr-qc/0405104
 |doi=10.1142/S0218271804006279
 |bibcode = 2004IJMPD..13.2241A |s2cid=118889785
}}</ref> on doubly special relativity<ref>
{{cite journal
 |author=Giovanni Amelino-Camelia
 |year=2001
 |title=Testable scenario for Relativity with minimum-length
 |journal=Phys. Lett. B
 |volume=510|issue=1–4|pages=255–263
 |arxiv=hep-th/0012238
 |doi=10.1016/S0370-2693(01)00506-8
 |bibcode = 2001PhLB..510..255A |s2cid=119447462
}}</ref> and on the Newton–Hooke group<ref name=posskinem/><ref>
{{cite journal
 |author1=G.W. Gibbons |author2=C.E. Patricot |year=2003
 |title=Newton–Hooke spacetimes, Hpp-waves and the cosmological constant
 |journal=Class. Quantum Grav.
 |volume=20|issue=23|page=5225
 |arxiv=hep-th/0308200
 |doi=10.1088/0264-9381/20/23/016
 |bibcode = 2003CQGra..20.5225G |s2cid=26557629
}}</ref><ref>
{{cite journal
 |author1=Yu Tian |author2=Han-Ying Guo |author3=Chao-Guang Huang |author4=Zhan Xu |author5=Bin Zhou |year=2005
 |title=Mechanics and Newton–Cartan-Like Gravity on the Newton–Hooke Space–time
 |journal=Phys. Rev. D
 |volume=71|issue=4|article-number=044030
 |arxiv=hep-th/0411004
 |doi=10.1103/PhysRevD.71.044030
 |bibcode = 2005PhRvD..71d4030T |s2cid=119378100
}}</ref> and early work formulating special relativity with a de Sitter space<ref>F. G. Gursey, "Introduction to the de Sitter group", Group Theoretical Concepts and Methods in Elementary Particle Physics edited by F. G. Gursey (Gordon and Breach, New York, 1965)</ref><ref>
{{cite journal
 |author1=L. F. Abbott |author2=S. Deser |year=1982
 |title=Stability of gravity with a cosmological constant
 |journal=Nucl. Phys. B
 |volume=195 |issue= 1|pages=76–96
 |doi=10.1016/0550-3213(82)90049-9
 |bibcode = 1982NuPhB.195...76A
 |url=https://cds.cern.ch/record/134600
 |type=Submitted manuscript
}}</ref><ref>
{{cite journal
 |author1=J. Kowalski-Glikman |author2=S. Nowak |year=2003
 |title=Doubly special relativity and de Sitter space
 |journal=Class. Quantum Grav.
 |volume=20|issue=22|pages=4799–4816
 |doi=10.1088/0264-9381/20/22/006
 |arxiv = hep-th/0304101 |bibcode = 2003CQGra..20.4799K |s2cid=16875852 }}</ref>
* In 2008 S. Cacciatori, V. Gorini and A. Kamenshchik<ref name=c21st/> published a paper about the kinematics of de Sitter relativity.
* Papers by other authors include: dSR and the fine structure constant;<ref>
{{cite journal
 |author1=Shao-Xia Chen 
 |author2=Neng-Chao Xiao 
 |author3=Mu-Lin Yan 
 |year=2008 
 |title=Variation of the Fine-Structure Constant from the de Sitter Invariant Special Relativity 
 |url=http://mp.ihep.ac.cn/qikan/epaper/zhaiyao.asp?bsid=7371 
 |journal=Chinese Physics C 
 |volume=32 
 |issue=8 
 |pages=612–616 
 |arxiv=astro-ph/0703110 
 |doi=10.1177/0022343307082058 
 |archive-url=https://web.archive.org/web/20110707013012/http://mp.ihep.ac.cn/qikan/epaper/zhaiyao.asp?bsid=7371 
 |archive-date=2011-07-07 
 |bibcode=2008ChPhC..32..612C 
 |s2cid=143773103 
}}</ref> dSR and dark energy;<ref>
{{cite journal
 |author1=C G Bohmer |author2=T Harko |year= 2008
 |title=Physics of dark energy particles
 |journal=Foundations of Physics
 |volume=38 |issue=3 |pages=216–227
 |arxiv=gr-qc/0602081
 |doi= 10.1007/s10701-007-9199-4
 |bibcode = 2008FoPh...38..216B |s2cid=16361512
}}</ref> dSR Hamiltonian Formalism;<ref>
{{cite journal
 | author1=Mu-Lin Yan
 | author2=Neng-Chao Xiao
 | author3=Wei Huang
 | author4=Si Li
 | year=2007
 | title=Hamiltonian Formalism of the de-Sitter Invariant Special Relativity
 | pages=27–36
 | volume=48
 | issue=1
 | journal=Communications in Theoretical Physics
 | arxiv=hep-th/0512319 
 |bibcode = 2007CoTPh..48...27Y |doi = 10.1088/0253-6102/48/1/007 | s2cid=250880550
}}</ref> and De Sitter Thermodynamics from Diamonds's Temperature,<ref>
{{cite journal
 | author1=Yu Tian
 | year=2005 
 | title=De Sitter Thermodynamics from Diamonds's Temperature
 | doi=10.1088/1126-6708/2005/06/045
 | journal=Journal of High Energy Physics
 | volume=2005
 | issue=6
 | page=045
 | arxiv=gr-qc/0504040v3
 |bibcode = 2005JHEP...06..045T | s2cid=119399508 
}}</ref> Triply special relativity from six dimensions,<ref>
{{cite arXiv
 |author1=S. Mignemi
 |year=2008
 |title=Triply special relativity from six dimensions
 |class=gr-qc
 |eprint=0807.2186
}}</ref> Deformed General Relativity and Torsion.<ref>
{{cite journal
 |last1=Gibbons |first1=Gary W.
 |last2=Gielen |first2=Steffen
 |year=2009
 |title=Deformed General Relativity and Torsion
 |doi=10.1088/0264-9381/26/13/135005
 |journal=Classical and Quantum Gravity
 |volume=26
 |issue=13
 |article-number=135005
 |arxiv=0902.2001
 |bibcode = 2009CQGra..26m5005G |s2cid=119296100
}}</ref>

=== Quantum de Sitter special relativity===
There are quantized or quantum versions of de Sitter special relativity.<ref>
{{cite journal
 |author1=Ashok Das |author2=Otto C. W. Kong |year=2006
 |title=Physics of Quantum Relativity through a Linear Realization
 |journal=Phys. Rev. D
 |volume=73|issue=12|article-number=124029
 |arxiv=gr-qc/0603114
 |doi=10.1103/PhysRevD.73.124029
 |bibcode = 2006PhRvD..73l4029D |s2cid=30161988
}}</ref><ref>
{{cite journal
 |author1=Han-Ying Guo
 |author2=Chao-Guang Huang
 |author3=Yu Tian
 |author4=Zhan Xu
 |author5=Bin Zhou
 |year=2007
 |title=Snyder's Quantized Space–time and De Sitter Special Relativity
 |journal=Front. Phys. China
 |volume=2 |issue= 3|pages=358–363
 |arxiv=hep-th/0607016
 |doi=10.1007/s11467-007-0045-0
 |bibcode = 2007FrPhC...2..358G |s2cid=119368124
}}</ref>

Early work on formulating a quantum theory in a de Sitter space includes:<ref>
{{cite book
 |author1=N. D. Birrell
 |author2=P. C. W. Davies
 |year=1982
 |title=Quantum fields in curved space
 |publisher=Cambridge University Press
 |isbn=978-0-521-23385-9
}}</ref><ref>
{{cite journal
 |author1=J. Bros |author2=U. Moschella |year=1996
 |title=Two-point functions and quantum fields in de Sitter universe
 |journal=Rev. Math. Phys.
 |volume=8 |issue= 3|pages=327–392
 |doi=10.1142/S0129055X96000123
|arxiv = gr-qc/9511019 |bibcode = 1996RvMaP...8..327B |s2cid=17974712 }}</ref><ref>
{{cite journal
 |author1=J. Bros |author2=H. Epstein |author3=U. Moschella |year=1998
 |title=Analyticity properties and thermal effects for general quantum field theory on de Sitter space–time
 |journal=Commun. Math. Phys.
 |volume=196 |issue= 3|pages=535–570
 |doi=10.1007/s002200050435
 |arxiv = gr-qc/9801099 |bibcode = 1998CMaPh.196..535B |s2cid=2027732
}}</ref><ref>
{{cite journal
 |author1=J. Bros |author2=H. Epstein |author3=U. Moschella |title=Lifetime of a massive particle in a de Sitter universe
 |journal=Transactions of the American Fisheries Society
 |volume=137 |issue= 6|page= 1879
 |year=2008
 |doi=10.1577/T07-141.1
 |arxiv=hep-th/0612184 |bibcode=2008JCAP...02..003B
}}</ref><ref>U. Moschella (2006), "The de Sitter and anti-de Sitter sightseeing tour", in Einstein, 1905–2005 (T. Damour, O. Darrigol, B. Duplantier, and V. Rivesseau, eds.), ''Progress in Mathematical Physics'', Vol. 47, Basel: Birkhauser, 2006.</ref><ref>{{cite journal | author = Moschella U | year = 2007 | title = Particles and fields on the de Sitter universe | doi = 10.1063/1.2752488 | journal = AIP Conference Proceedings | volume = 910 | pages = 396–411 | bibcode = 2007AIPC..910..396M }}</ref><ref>
{{cite journal
 |author=E. Benedetto
 |year=2009
 |title=Fantappiè–Arcidiacono Spacetime and Its Consequences in Quantum Cosmology
 |journal=Int J Theor Phys
 |volume=48|issue=6|pages=1603–1621
 |doi=10.1007/s10773-009-9933-0
 |bibcode = 2009IJTP...48.1603B |s2cid=121015516
}}</ref>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
{{Portal|Physics}}
* [[Wikipedia:Noncommutative geometry|Noncommutative geometry]]
* [[Wikipedia:Quantum field theory in curved spacetime|Quantum field theory in curved spacetime]]

== References ==
{{reflist|3}}

== Further reading ==
* {{cite journal
 |author1=R. Aldrovandi
 |author2=J. G. Pereira
 |year=2009
 |title=Is Physics Asking for a New Kinematics?
 |doi=10.1142/S0218271808013972
 |journal=International Journal of Modern Physics D
 |volume=17
 |issue=13 & 14
 |pages=2485–2493
 |arxiv=0805.2584
 |bibcode = 2008IJMPD..17.2485A |s2cid=14403086
}}
* {{cite journal
 |author1=S Cacciatori
 |author2=V Gorini
 |author3=A Kamenshchik
 |author4=U Moschella
 |year=2008
 |title=Conservation laws and scattering for de Sitter classical particles
 |journal=Class. Quantum Grav.
 |volume=25 |issue=7 |article-number=075008
 |arxiv=0710.0315
 |doi=10.1088/0264-9381/25/7/075008
 |bibcode = 2008CQGra..25g5008C |s2cid=118544579
}}
* {{cite arXiv
 |author1=S Cacciatori
 |year=2009
 |title=Conserved quantities for the Sitter particles
 |class=gr-qc
 |eprint=0909.1074
}}
* {{cite journal
 |year=2007
 |journal=AIP Conference Proceedings
 |author1=Aldrovandi
 |author2=Beltran Almeida
 |author3=Mayor
 |author4=Pereira
 |doi=10.1063/1.2827302
 |title=de Sitter Relativity and Quantum Physics
 |last5=Adenier |first5=Guillaume
 |last6=Khrennikov |first6=Andrei Yu.
 |last7=Lahti |first7=Pekka
 |last8=Man'Ko |first8=Vladimir I.
 |last9=Nieuwenhuizen |first9=Theo M.
 |volume=962
 |pages=175–184
 |arxiv=0710.0610
 |hdl=11449/70009
 |bibcode=2007AIPC..962..175A
 |s2cid=1178656
 }}
* {{cite book
 |author1=Claus Lämmerzahl
 |author2=Jürgen Ehlers
 |year=2005
 |title=Special Relativity: Will it Survive the Next 101 Years?
 |publisher=Springer
 |isbn=978-3-540-34522-0
}}
* {{cite book
 |author=Giuseppe Arcidiacono
 |year=1986
 |title=Projective Relativity, Cosmology, and Gravitation
 |publisher=Hadronic Press
 |isbn=978-0-911767-39-1 
}}

[[Category:Special relativity]]
[[Category:General relativity]]
[[Category:Physical cosmology]]
[[Category:Quantum gravity]]
[[Category:Kinematics]]
[[Category:Riemannian geometry]]
[[Category:Group theory]]
{{Author|Harold Foppele}}

{{Sourceattribution|De Sitter invariant special relativity|1}}