# de Sitter invariant special relativity

In mathematical physics, de Sitter invariant special relativity is the speculative idea that the fundamental symmetry group of spacetime is the indefinite orthogonal group SO(4,1), that of de Sitter space. In the standard theory of general relativity, de Sitter space is a highly symmetrical special vacuum solution, which requires a cosmological constant or the stress–energy of a constant 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 Poincaré group of 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 Luigi Fantappiè in 1954, the theory remained obscure until it was rediscovered in 1968 by Henri Bacry and Jean-Marc Lévy-Leblond. In 1972, 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.[1] The discovery of the 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 doubly special relativity.

## Introduction

De Sitter suggested that spacetime curvature might not be due solely to gravity[2] but he did not give any mathematical details of how this could be accomplished. In 1968 Henri Bacry and Jean-Marc Lévy-Leblond showed that the de Sitter group was the most general group compatible with isotropy, homogeneity and boost invariance.[3] Later, Freeman Dyson[1] advocated this as an approach to making the mathematical structure of general relativity more self-evident.

Minkowski's unification of space and time within special relativity replaces the Galilean group of Newtonian mechanics with the Lorentz group. This is called a unification of space and time because the Lorentz group is simple, while the Galilean group is a semi-direct product of rotations and 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 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 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 R2, 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 vector and not two scalars. 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 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 R2 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 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 boosts are non-commutative, 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 holonomy of 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 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 homogeneous and isotropic, the only option is that spacetime has a uniform 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 curvature implies a positive cosmological constant 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,[4] 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 vacuum energy, but at least partly due to the kinematics of the de Sitter group, which would replace the Poincaré group.

A modification of this idea allows ${\displaystyle \Lambda }$ to change with time, so that inflation may come from the cosmological constant being larger near the Big Bang than nowadays. It can also be viewed as a different approach to the problem of quantum gravity.[5]

### High energy

The Poincaré group contracts to the Galilean group for low-velocity kinematics, meaning that when all velocities are small the Poincaré group "morphs" into the Galilean group. (This can be made precise with İnönü and Wigner's concept of group contraction.[6])

Similarly, the de Sitter group 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.[5] 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 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 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 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 particle collider.[7]

### 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 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.[8][9] 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 mass, energy and momentum,[10] 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.[11] See also MacDowell–Mansouri action.

### Newton–Hooke: de Sitter special relativity in the limit v ≪ c

In the limit as vc, the de Sitter group contracts to the Newton–Hooke group.[12] 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 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.[13] 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 Luigi Fantappiè and Giuseppe Arcidiacono first published in 1954 by Fantappiè[14] and the same as another independent discovery in 1976.[15]
• In 1968 Henri Bacry and Jean-Marc Lévy-Leblond published a paper on possible kinematics[3]
• In 1972 Freeman Dyson[1] further explored this.
• In 1973 Eliano Pessa described how Fantappié–Arcidiacono projective relativity relates to earlier conceptions of projective relativity and to Kaluza Klein theory.[16]
• Han-Ying Guo, Chao-Guang Huang, Zhan Xu, Bin Zhou have used the term "de Sitter special relativity" from 2004 onwards.[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]
• 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".[10][32] This paper was based on previous work on amongst other things: the consequences of a non-vanishing cosmological constant,[33] on doubly special relativity[34] and on the Newton–Hooke group[3][35][36] and early work formulating special relativity with a de Sitter space[37][38][39]
• From 2006 onwards Ignazio Licata and Leonardo Chiatti have published papers on Fantappié–Arcidiacono theory of relativity pointing out that it is the same thing as de Sitter relativity[14][40][41][42][43]
• In 2008 S. Cacciatori, V. Gorini and A. Kamenshchik[4] published a paper about the kinematics of de Sitter relativity.
• Papers by other authors include: dSR and the fine structure constant;[44] dSR and dark energy;[45] dSR Hamiltonian Formalism;[46] and De Sitter Thermodynamics from Diamonds's Temperature,[47] Triply special relativity from six dimensions,[48] Deformed General Relativity and Torsion.[49]

### Quantum de Sitter special relativity

There are quantized or quantum versions of de Sitter special relativity.[50][51]

Early work on formulating a quantum theory in a de Sitter space includes:[52][53][54][55][56][57][58]

## References

1. ^ a b c F. J. Dyson (1972). "Missed opportunities" (pdf). Bull. Am. Math. Soc. 78 (5): 635–652. doi:10.1090/S0002-9904-1972-12971-9. MR 0522147.
2. ^ W. de Sitter (1917). "On the curvature of space". Proc. Roy. Acad. Sci. Amsterdam. 20: 229–243.
3. ^ a b c Henri Bacry; Jean-Marc Lévy-Leblond (1968). "Possible Kinematics". Journal of Mathematical Physics. 9 (10): 1605. Bibcode:1968JMP.....9.1605B. doi:10.1063/1.1664490.
4. ^ a b S. Cacciatori; V. Gorini; A. Kamenshchik (2008). "Special Relativity in the 21st century". Annalen der Physik. 17 (9–10): 728–768. arXiv:0807.3009. Bibcode:2008AnP...520..728C. doi:10.1002/andp.200810321. S2CID 119191753.
5. ^ a b R. Aldrovandi; J. G. Pereira (2009). "de Sitter Relativity: a New Road to Quantum Gravity?". Foundations of Physics. 39 (2): 1–19. arXiv:0711.2274. Bibcode:2009FoPh...39....1A. doi:10.1007/s10701-008-9258-5. S2CID 15298756.
6. ^ E. Inönü; E.P. Wigner (1953). "On the Contraction of Groups and Their Representations". Proc. Natl. Acad. Sci. USA. 39 (6): 510–24. Bibcode:1953PNAS...39..510I. doi:10.1073/pnas.39.6.510. PMC 1063815. PMID 16589298.
7. ^ Freydoon Mansouri (2002). "Non-Vanishing Cosmological Constant Λ, Phase Transitions, And Λ-Dependence Of High Energy Processes". Phys. Lett. B. 538 (3–4): 239–245. arXiv:hep-th/0203150. Bibcode:2002PhLB..538..239M. doi:10.1016/S0370-2693(02)02022-1. S2CID 13986319.
8. ^ Aldrovandi, R.; Beltrán Almeida, J. P.; Pereira, J. G. (2007). "Some Implications of the Cosmological Constant to Fundamental Physics". AIP Conference Proceedings. 910: 381–395. arXiv:gr-qc/0702065. Bibcode:2007AIPC..910..381A. doi:10.1063/1.2752487. hdl:11449/69891. S2CID 16631274.
9. ^ R. Aldrovandi; J.P. Beltran Almeida; C.S.O. Mayor; J.G. Pereira (2007). "Lorentz Transformations in de Sitter Relativity". arXiv:0709.3947 [gr-qc].
10. ^ a b R. Aldrovandi; J.P. Beltrán Almeida; J.G. Pereira (2007). "de Sitter Special Relativity". Class. Quantum Grav. 24 (6): 1385–1404. arXiv:gr-qc/0606122. Bibcode:2007CQGra..24.1385A. doi:10.1088/0264-9381/24/6/002. S2CID 11703342.
11. ^ Wise (2010). "MacDowell–Mansouri Gravity and Cartan Geometry". Classical and Quantum Gravity. 27 (15): 155010. arXiv:gr-qc/0611154. Bibcode:2010CQGra..27o5010W. doi:10.1088/0264-9381/27/15/155010. S2CID 16706599.
12. ^ Aldrovandi; Barbosa; Crispino; Pereira (1999). "Non–Relativistic Spacetimes with Cosmological Constant". Classical and Quantum Gravity. 16 (2): 495–506. arXiv:gr-qc/9801100. Bibcode:1999CQGra..16..495A. CiteSeerX 10.1.1.339.919. doi:10.1088/0264-9381/16/2/013. S2CID 16691405.
13. ^ Yu Tian; Han-Ying Guo; Chao-Guang Huang; Zhan Xu; Bin Zhou (2004). "Mechanics and Newton–Cartan-Like Gravity on the Newton–Hooke Space–time". Physical Review D. 71 (4): 44030. arXiv:hep-th/0411004. Bibcode:2005PhRvD..71d4030T. doi:10.1103/PhysRevD.71.044030. S2CID 119378100.
14. ^ a b Licata, Ignazio; Leonardo Chiatti (2009). "The archaic universe: Big Bang, cosmological term, and the quantum origin of time in projective cosmology". International Journal of Theoretical Physics. 48 (4): 1003–1018. arXiv:0808.1339. Bibcode:2009IJTP...48.1003L. doi:10.1007/s10773-008-9874-z. S2CID 119262177.
15. ^ Dey, Anind K. (2001). "An extension of the concept of inertial frame and of Lorentz transformation". Proc. Natl. Acad. Sci. USA. 73 (5): 1418–21. Bibcode:1976PNAS...73.1418K. doi:10.1073/pnas.73.5.1418. PMC 430307. PMID 16592318.
16. ^ The De Sitter Universe and general relativity
17. ^ Han-Ying Guo; Chao-Guang Huang; Zhan Xu; Bin Zhou (2004). "On Special Relativity with Cosmological Constant". Phys. Lett. A. 331 (1–2): 1–7. arXiv:hep-th/0403171. Bibcode:2004PhLA..331....1G. doi:10.1016/j.physleta.2004.08.036. S2CID 119425901.
18. ^ Guo, Han-Ying; Huang, Chao-Guang; Tian, Yu; Xu, Zhan; Zhou, Bin (2004). "On de Sitter Invariant Special Relativity and Cosmological Constant as Origin of Inertia". arXiv:hep-th/0405137.
19. ^ Han-Ying Guo; Chao-Guang Huang; Hong-Tu Wu (2008). "Yang's Model as Triply Special Relativity and the Snyder's Model—de Sitter Special Relativity Duality". Physics Letters B. 663 (3): 270–274. arXiv:0801.1146. Bibcode:2008PhLB..663..270G. doi:10.1016/j.physletb.2008.04.012. S2CID 118643874.
20. ^ Han-Ying Guo; Chao-Guang Huang; Yu Tian; Hong-Tu Wu; Zhan Xu; Bin Zhou (2007). "Snyder's Model — de Sitter Special Relativity Duality and de Sitter Gravity". Class. Quantum Grav. 24 (16): 4009–4035. arXiv:gr-qc/0703078. Bibcode:2007CQGra..24.4009G. doi:10.1088/0264-9381/24/16/004. S2CID 118977864.
21. ^ Wu Hong-Tu; Huang Chao-Guang; Guo Han-Ying (2008). "From the Complete Yang Model to Snyder's Model, de Sitter Special Relativity and Their Duality". Chinese Phys. Lett. 25 (8): 2751–2753. arXiv:0809.3560. Bibcode:2008ChPhL..25.2751W. doi:10.1088/0256-307X/25/8/005. S2CID 119258431.
22. ^ Han-Ying Guo (2007). "On Principle of Inertia in Closed Universe". Phys. Lett. B. 653: 88–94. arXiv:hep-th/0611341. Bibcode:2007PhLB..653...88G. doi:10.1016/j.physletb.2007.05.006. S2CID 119508570.
23. ^ Han-Ying Guo (2008). "Special Relativity and Theory of Gravity via Maximum Symmetry and Localization". Science China Mathematics. 51 (4): 568–603. arXiv:0707.3855. Bibcode:2008ScChA..51..568G. doi:10.1007/s11425-007-0166-5. S2CID 116889828.
24. ^ "Our Universe Prefers The De Sitter Special Relativity And Its Localization"
25. ^ H.-Y. Guo; C.-G. Huang; Z. Xu; B. Zhou (2004). "On beltrami model of de Sitter spacetime". Mod. Phys. Lett. A. 19 (22): 1701–1709. arXiv:hep-th/0311156. Bibcode:2004MPLA...19.1701G. doi:10.1142/S0217732304014033. S2CID 119405913.
26. ^ H.-Y. Guo, B. Zhou, Y. Tian, and Z. Xu (2007). "The triality of conformal extensions of three kinds of special relativity". Physical Review D. 75 (2): 026006. arXiv:hep-th/0611047. Bibcode:2007PhRvD..75b6006G. doi:10.1103/PhysRevD.75.026006. S2CID 119450917.CS1 maint: multiple names: authors list (link)
27. ^ Chang Zhe; Chen Shao-Xia; Huang Chao-Guang (2005). "Absence of GZK Cutoff and Test of de Sitter Invariant Special Relativity". Chinese Phys. Lett. 22 (4): 791–794. Bibcode:2005ChPhL..22..791C. doi:10.1088/0256-307X/22/4/003.
28. ^ Guo, H.-Y; Huang, C.-G; Zhou, B (2005). "Temperature at horizon in de Sitter spacetime". Europhysics Letters. 72 (6): 1045–1051. arXiv:hep-th/0404010. Bibcode:2005EL.....72.1045G. doi:10.1209/epl/i2005-10327-4. S2CID 17958395.
29. ^ Guo, Han-Ying; Huang, Chao-Guang; Wu, Hong-Tu; Zhou, Bin (2005). "Three Kinds of Special Relativity via Inverse Wick Rotation". Chinese Phys. Lett. 22 (10): 2477–2480. arXiv:hep-th/0508094. Bibcode:2005ChPhL..22.2477G. doi:10.1088/0256-307X/22/10/006. S2CID 119464602.
30. ^ Guo, Han-Ying; Huang, Chao-Guang; Wu, Hong-Tu; Zhou, Bin (2010). "The Principle of Relativity, Kinematics and Algebraic Relations". Science China Physics, Mechanics and Astronomy. 53 (4): 591–597. arXiv:0812.0871. Bibcode:2010SCPMA..53..591G. doi:10.1007/s11433-010-0162-6. S2CID 118464788.
31. ^ Guo, Han-Ying; Wu, Hong-Tu; Zhou, Bin (2008). "The Principle of Relativity and the Special Relativity Triple". Physics Letters B. 670 (4–5): 437. arXiv:0809.3562. Bibcode:2009PhLB..670..437G. doi:10.1016/j.physletb.2008.11.027. S2CID 115169240.
32. ^ R. Aldrovandi; J. G. Pereira (2009). "De Sitter Special Relativity: Effects on Cosmology". Gravitation and Cosmology. 15 (4): 287–294. arXiv:0812.3438. Bibcode:2009GrCo...15..287A. doi:10.1134/S020228930904001X. S2CID 18473868.
33. ^ R. Aldrovandi; J.P. Beltran Almeida; J.G. Pereira (2004). "Cosmological Term and Fundamental Physics". Int. J. Mod. Phys. D. 13 (10): 2241–2248. arXiv:gr-qc/0405104. Bibcode:2004IJMPD..13.2241A. doi:10.1142/S0218271804006279. S2CID 118889785.
34. ^ Giovanni Amelino-Camelia (2001). "Testable scenario for Relativity with minimum-length". Phys. Lett. B. 510 (1–4): 255–263. arXiv:hep-th/0012238. Bibcode:2001PhLB..510..255A. doi:10.1016/S0370-2693(01)00506-8.
35. ^ G.W. Gibbons; C.E. Patricot (2003). "Newton–Hooke spacetimes, Hpp-waves and the cosmological constant". Class. Quantum Grav. 20 (23): 5225. arXiv:hep-th/0308200. Bibcode:2003CQGra..20.5225G. doi:10.1088/0264-9381/20/23/016. S2CID 26557629.
36. ^ Yu Tian; Han-Ying Guo; Chao-Guang Huang; Zhan Xu; Bin Zhou (2005). "Mechanics and Newton–Cartan-Like Gravity on the Newton–Hooke Space–time". Phys. Rev. D. 71 (4): 044030. arXiv:hep-th/0411004. Bibcode:2005PhRvD..71d4030T. doi:10.1103/PhysRevD.71.044030. S2CID 119378100.
37. ^ 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)
38. ^ L. F. Abbott; S. Deser (1982). "Stability of gravity with a cosmological constant". Nucl. Phys. B (Submitted manuscript). 195 (1): 76–96. Bibcode:1982NuPhB.195...76A. doi:10.1016/0550-3213(82)90049-9.
39. ^ J. Kowalski-Glikman; S. Nowak (2003). "Doubly special relativity and de Sitter space". Class. Quantum Grav. 20 (22): 4799–4816. arXiv:hep-th/0304101. Bibcode:2003CQGra..20.4799K. doi:10.1088/0264-9381/20/22/006. S2CID 16875852.
40. ^ Ignazio Licata (2007). "Universe Without Singularities. A Group Approach to De Sitter Cosmology" (PDF). Electronic Journal of Theoretical Physics. 3: 211–224. arXiv:0704.0563. Bibcode:2007arXiv0704.0563L.
41. ^ Leonardo Chiatti (2007). "Fantappié–Arcidiacono theory of relativity versus recent cosmological evidences : a preliminary comparison" (PDF). Endocrinology. 15 (4): 17–36. arXiv:physics/0702178. Bibcode:2007physics...2178C. doi:10.1210/en.138.7.3069.
42. ^ Chiatti, Leonardo (2009). "The Fundamental Equations of Point, Fluid and Wave Dynamics in the De Sitter–Fantappie–Arcidiacono Projective Relativity Theory". arXiv:0901.3616 [physics.gen-ph].
43. ^ Chiatti, Leonardo (2009). "Choosing the Right Relativity for QFT". arXiv:0902.1393 [physics.gen-ph].
44. ^ Shao-Xia Chen; Neng-Chao Xiao; Mu-Lin Yan (2008). "Variation of the Fine-Structure Constant from the de Sitter Invariant Special Relativity". Chinese Physics C. 32 (8): 612–616. arXiv:astro-ph/0703110. Bibcode:2008ChPhC..32..612C. doi:10.1177/0022343307082058. S2CID 143773103. Archived from the original on 2011-07-07.
45. ^ C G Bohmer; T Harko (2008). "Physics of dark energy particles". Foundations of Physics. 38 (3): 216–227. arXiv:gr-qc/0602081. Bibcode:2008FoPh...38..216B. doi:10.1007/s10701-007-9199-4. S2CID 16361512.
46. ^ Mu-Lin Yan; Neng-Chao Xiao; Wei Huang; Si Li (2007). "Hamiltonian Formalism of the de-Sitter Invariant Special Relativity". Communications in Theoretical Physics. 48 (1): 27–36. arXiv:hep-th/0512319. Bibcode:2007CoTPh..48...27Y. doi:10.1088/0253-6102/48/1/007.
47. ^ Yu Tian (2005). "De Sitter Thermodynamics from Diamonds's Temperature". Journal of High Energy Physics. 2005 (6): 045. arXiv:gr-qc/0504040v3. Bibcode:2005JHEP...06..045T. doi:10.1088/1126-6708/2005/06/045. S2CID 119399508.
48. ^ S. Mignemi (2008). "Triply special relativity from six dimensions". arXiv:0807.2186 [gr-qc].
49. ^ Gibbons, Gary W.; Gielen, Steffen (2009). "Deformed General Relativity and Torsion". Classical and Quantum Gravity. 26 (13): 135005. arXiv:0902.2001. Bibcode:2009CQGra..26m5005G. doi:10.1088/0264-9381/26/13/135005. S2CID 119296100.
50. ^ Ashok Das; Otto C. W. Kong (2006). "Physics of Quantum Relativity through a Linear Realization". Phys. Rev. D. 73 (12): 124029. arXiv:gr-qc/0603114. Bibcode:2006PhRvD..73l4029D. doi:10.1103/PhysRevD.73.124029. S2CID 30161988.
51. ^ Han-Ying Guo; Chao-Guang Huang; Yu Tian; Zhan Xu; Bin Zhou (2007). "Snyder's Quantized Space–time and De Sitter Special Relativity". Front. Phys. China. 2 (3): 358–363. arXiv:hep-th/0607016. Bibcode:2007FrPhC...2..358G. doi:10.1007/s11467-007-0045-0. S2CID 119368124.
52. ^ N. D. Birrell; P. C. W. Davies (1982). Quantum fields in curved space. Cambridge University Press. ISBN 978-0521233859.
53. ^ J. Bros; U. Moschella (1996). "Two-point functions and quantum fields in de Sitter universe". Rev. Math. Phys. 8 (3): 327–392. arXiv:gr-qc/9511019. Bibcode:1996RvMaP...8..327B. doi:10.1142/S0129055X96000123. S2CID 17974712.
54. ^ J. Bros; H. Epstein; U. Moschella (1998). "Analyticity properties and thermal effects for general quantum field theory on de Sitter space–time". Commun. Math. Phys. 196 (3): 535–570. arXiv:gr-qc/9801099. Bibcode:1998CMaPh.196..535B. doi:10.1007/s002200050435. S2CID 2027732.
55. ^ J. Bros; H. Epstein; U. Moschella (2008). "Lifetime of a massive particle in a de Sitter universe". Transactions of the American Fisheries Society. 137 (6): 1879. arXiv:hep-th/0612184. Bibcode:2008JCAP...02..003B. doi:10.1577/T07-141.1.
56. ^ 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.
57. ^ Moschella U (2007). "Particles and fields on the de Sitter universe". AIP Conference Proceedings. 910: 396–411. Bibcode:2007AIPC..910..396M. doi:10.1063/1.2752488.
58. ^ E. Benedetto (2009). "Fantappiè–Arcidiacono Spacetime and Its Consequences in Quantum Cosmology". Int J Theor Phys. 48 (6): 1603–1621. Bibcode:2009IJTP...48.1603B. doi:10.1007/s10773-009-9933-0. S2CID 121015516.