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

Main article: de Sitter space

De Sitter himself suggested that space-time 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 so 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 \mu m, since that's how high typical structures are in their world, while their x and y axis could be the meter, because that's the size of their body. Such creatures would describe the basic symmetry structure of their world as SO(2), rotations in the x-y plane. Later on, they might discover rotations into the z axis— and in their every-day experience such rotations would always be by an infinitesimal angle, so that these z-rotations would commute with each other.

The rotations into the z-axis would tilt objects by an infinitesimal amount. The tilt in the x-z plane would be one parameter, and the tilt in the y-z plane another. The symmetry group of this pancake world is 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 completely separate from x and y.

But eventually, experiments at large angles would convince the creatures that the actual 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 \mu, the ratio of the height-unit \mu m to the length-unit m, approaches 0. The Lorentz group is analogous— it is a simple group that turns into the Galilean group when the unit of time is made long compared to the unit of space, which is the limit c\rightarrow\infty.

But the symmetry group of special relativity is not entirely simple because there are still translations. The Lorentz group are 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. But if you take the unification idea to its logical conclusion then not only are boosts non-commutative but translations should be non-commutative too.

In the pancake world, this would happen if the creatures were living on an enormous sphere, not 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 everywhere 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 options are that space-time has a uniform scalar curvature. If the curvature is positive, the analog of the sphere case for the two-dimensional creatures, the space-time is de Sitter and the symmetry group of spacetime is a 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 is caused by a positive cosmological constant \Lambda to be determined by observation. Due to the small magnitude of the constant, then special relativity with the Poincaré group is more than accurate enough for all 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 believe that the acceleration of the expansion of the universe is not all due to vacuum energy, but at least partly due to the kinematics of the de Sitter group, which in their view is the correct symmetry group of space time, replacing the Lorentz group.

A modification of this idea allows \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 generalizes the Galilean group for high–velocity kinematics, meaning that when all velocities are small the Lorentz group 'becomes' the Galilean group. (This can be made precise with İnönü and Wigner's concept of group contraction.[6]) Similarly, the de Sitter group generalizes Poincaré for long distance kinematics, meaning that when magnitudes of all translations are small compared to the de Sitter radius, the de Sitter group becomes the Lorentz group.[5] In quantum mechanics, short distances are probed by high energies, so that for energies larger than a very small cosmological scale, 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 is interchangeable with the cosmological constant, is an adjustable parameter in de Sitter relativity, so the theory requires a separate condition to determine its value.

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 modify significantly the physics of the early universe, if the cosmological constant was bigger 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 colliders.[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 as v≪c

In the limit as v≪c, the de Sitter group contracts to the Newton–Hooke group.[12] This implies 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 this setup looks like it picks out a preferred point in space— the center of repulsion— it is more subtly isotropic. If you move to another point, you should transform to the uniformly accelerated frame of reference of an observer at this point, which changes all accelerations to shift the repulsion center to the new origin of coordinates.

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

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]

See also

References

  1. ↑ 1.0 1.1 1.2 F. J. Dyson (1972). "Missed opportunities". 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. ↑ 3.0 3.1 3.2 Henri Bacry, Jean-Marc Lévy-Leblond; Lévy-Leblond, Jean-Marc (1968). "Possible Kinematics". Journal of Mathematical Physics 9 (10): 1605. Bibcode:1968JMP.....9.1605B. doi:10.1063/1.1664490.
  4. ↑ 4.0 4.1 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.
  5. ↑ 5.0 5.1 R. Aldrovandi, J. G. Pereira (2008). "de Sitter Relativity: a New Road to Quantum Gravity?". Foundations of Physics 57 (2): 221–225. arXiv:0711.2274. doi:10.1006/gyno.1995.1129. PMID 7729738.
  6. ↑ E. Inönü, E.P. Wigner (1953). "On the Contraction of Groups and Their Representations". Proc. Natl. Acad. Sci. U.S.A. 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 \Lambda, Phase Transitions, And \Lambda-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.
  8. ↑ Aldrovandi, R.; Beltrán Almeida, J. P.; Pereira, J. G. (2007). "AIP Conference Proceedings" 910. p. 381. arXiv:gr-qc/0702065. doi:10.1063/1.2752487. |chapter= ignored (help)
  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. ↑ 10.0 10.1 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.
  11. ↑ Wise (2006). "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.
  12. ↑ Aldrovandi; Barbosa; Crispino; Pereira (1998). "Non–Relativistic Spacetimes with Cosmological Constant". Classical and Quantum Gravity 16 (2): 495–506. arXiv:gr-qc/9801100. Bibcode:1999CQGra..16..495A. doi:10.1088/0264-9381/16/2/013.
  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.
  14. ↑ 14.0 14.1 Licata, Ignazio; Leonardo Chiatti (2008). "The archaic universe: Big Bang, cosmological term, and the quantum origin of time in projective cosmology". International Journal of Theoretical Physics 48 (4): 1003. arXiv:0808.1339. Bibcode:2009IJTP...48.1003L. doi:10.1007/s10773-008-9874-z.
  15. ↑ Dey, Anind K. (2001). "An extension of the concept of inertial frame and of Lorentz transformation" (PDF). Proc. Natl. Acad. Sci. U.S.A. 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: 1–7. arXiv:hep-th/0403171. Bibcode:2004PhLA..331....1G. doi:10.1016/j.physleta.2004.08.036.
  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:0405137 Check |arxiv= value (help) [hep-th].
  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.
  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.
  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.
  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.
  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.
  24. ↑ "Our Universe Prefers The De Sitter Special Relativity And Its Localization"
  25. ↑ H.-Y. Guo, C.-G. Huang, Z. Xu, and B. Zhou (2004). "On beltrami model of de Sitter spacetime". Mod. Phys. Lett. A 19 (20): 1701–1710. doi:10.1021/ie900292p.
  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.
  27. ↑ Chang Zhe, Chen Shao-Xia, and 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 (EPL) 72 (6): 1045. arXiv:hep-th/0404010. Bibcode:2005EL.....72.1045G. doi:10.1209/epl/i2005-10327-4.
  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.
  30. ↑ Guo, Han-Ying; Huang, Chao-Guang; Wu, Hong-Tu; Zhou, Bin (2008). "The Principle of Relativity, Kinematics and Algebraic Relations". arXiv:0812.0871 [hep-th].
  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.
  32. ↑ R. Aldrovandi; J. G. Pereira (2008). "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.
  33. ↑ R. Aldrovandi, J.P. Beltran Almeida, and 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.
  34. ↑ Giovanni Amelino-Camelia (2001). "Testable scenario for Relativity with minimum-length". Phys. Lett. B 510: 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. Quant. Grav. 20 (23): 5225. arXiv:hep-th/0308200. Bibcode:2003CQGra..20.5225G. doi:10.1088/0264-9381/20/23/016.
  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.
  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 and S. Deser (1982). "Stability of gravity with a cosmological constant". Nucl. Phys. B 195: 76–96. Bibcode:1982NuPhB.195...76A. doi:10.1016/0550-3213(82)90049-9.
  39. ↑ J. Kowalski-Glikman and S. Nowak (2003). "Doubly special relativity and de Sitter space". Class. Quant. Grav. 20 (22): 4799–4816. arXiv:hep-th/0304101. Bibcode:2003CQGra..20.4799K. doi:10.1088/0264-9381/20/22/006.
  40. ↑ Ignazio Licata (2007). "Universe Without Singularities. A Group Approach to De Sitter Cosmology" (PDF). Electronics 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. 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. doi:10.1177/0022343307082058.
  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.
  46. ↑ Mu-Lin Yan; Neng-Chao Xiao; Wei Huang; Si Li (2005). "Hamiltonian Formalism of the de-Sitter Invariant Special Relativity". Commun.Theor.Phys. 48: 27–36. arXiv:hep-th/0512319.
  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.
  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.
  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.
  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.
  52. ↑ N. D. Birrell and P. C. W. Davies (1982). Quantum fields in curved space. Cambridge University Press. ISBN 978-0521233859.
  53. ↑ J. Bros and 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.
  54. ↑ J. Bros, H. Epstein, and 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.
  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. 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.
  58. ↑ E. Benedetto (2009). "Fantappiè-Arcidiacono Spacetime and Its Consequences in Quantum Cosmology". Int J Theor Phys 48 (6): 1603. Bibcode:2009IJTP...48.1603B. doi:10.1007/s10773-009-9933-0.

Further reading