# Positive Cosmological Constant and Quantum Theory

## Abstract

**:**

## 1. Introduction

## 2. Remarks on the Cosmological Constant Problem

## 3. Should Physical Theories Involve Spacetime Background?

## 4. Symmetry on Quantum Level

## 5. IRs of the dS Algebra

## 6. Absence of Weyl Particles in dS Invariant Theory

## 7. Other Implementations of IRs

## 8. Physical Interpretation of IRs of the dS Algebra

#### 8.1. Problems with Physical Interpretation of IRs

#### 8.2. Example of Transformation Mixing Particles and Antiparticles

#### 8.3. Summary

## 9. dS Quantum Mechanics and Cosmological Repulsion

## 10. Discussion and Conclusions

- Each state is either a particle or antiparticle only when one does not consider transformations mixing states on the upper and lower hyperboloids. Only in this case additive quantum numbers such as electric, baryon and lepton charges are conserved. In particular, they are conserved if Poincare approximation works with a high accuracy.

## Acknowledgements

## References and Notes

- Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurement of omega and lambda from H42 high-redshift supernovae. Astrophys. J.
**1999**, 517, 565–586. [Google Scholar] [CrossRef] - Melchiorri, A.; Ade, P.A.R.; de Bernardis, P.; Bock, J.J.; Borrill, J.; Boscaleri, A.; Crill, B.P.; De Troia, G.; Farese, P.; Ferreira, P.G.; et al. A Measurement of omega from the north American rest flight of boomerang. Astrophys. J.
**2000**, 536, L63–L66. [Google Scholar] [CrossRef] [PubMed] - Spergel, D.N.; Bean, R.; Dore, O.; Nolta, M.R.; Bennett, C.L.; Dunkley, J.; Hinshaw, G.; Jarosik, N.; Komatsu, E.; Page, L.; et al. Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology. Astrophys. J. Suppl.
**2007**, 170, 377–408. [Google Scholar] [CrossRef] - Nakamura, K. and Particle Data Group. See section “The Cosmological Parameters”, which also can be found at http://pdg.lbl.gov/2010/reviews/rpp2010-rev-cosmological-parameters.pdf.
- Lev, F.M. Could Only Fermions Be Elementary? J. Phys.
**2004**, A37, 3285–3304. [Google Scholar] [CrossRef] - Bianchi, E.; Rovelli, C. Why All These Prejudices Against a Constant? 2010; arXiv:1002.3966v3 (astro-ph.CO). [Google Scholar]
- Duff, M.J.; Okun, L.B.; Veneziano, G. Trialogue on the Number of Fundamental Constants. JHEP
**2002**, 3, 023. [Google Scholar] [CrossRef] - Weinberg, S.; Nielsen, H.B.; Taylor, J.G. Overview of Theoretical Prospects for Understanding the Values of Fundamental Constants. Phil. Trans. R. Soc. Lond.
**1983**, A310, 249–252. [Google Scholar] [CrossRef] - Akhmedov, E.T.; Roura, A.; Sadofyev, A. Classical Radiation by Free-falling Charges in de Sitter Spacetime. Phys. Rev.
**2010**, D82, 044035. [Google Scholar] [CrossRef] - Akhmedov, E.T.; Burda, P.A. Simple Way to Take Into Account Back Reaction on Pair Creation. arXiv:0912.3425 (hep-th). Phys. Lett.
**2010**, B687, 267–270. [Google Scholar] [CrossRef] - Akhmedov, E.T. Real or Imaginary? (On Pair Creation in de Sitter Space). 2010; arXiv:0909.3722 (hep-th). [Google Scholar]
- Rickles, D. Who’s Afraid of Background Independence? Philos. Found. Phys.
**2008**, 4, 133–152. [Google Scholar] - Rosen, J. Time, c and Nonlocality: A Glimpse Beneath the Surface. Phys. Essays
**1994**, 7, 335–339. [Google Scholar] [CrossRef] - Rosen, J. Symmetry Rules: How Science and Nature Are Founded on Symmetry; Springer: Berlin, Germany, 2008; p. 305. [Google Scholar]
- Rickles, D. Time and Structure in Canonical Gravity. In The Structural Foundations of Quantum Gravity; Rickles, D., French, S., Saatsi, J.T., Eds.; Oxford University Press: Oxford, New York, NY, USA, 2006; pp. 152–195. [Google Scholar]
- Newton, T.D.; Wigner, E.P. Localized States for Elementary Systems. Rev. Mod. Phys.
**1949**, 21, 400–405. [Google Scholar] [CrossRef] - Berestetsky, V.B.; Lifshits, E.M.; Pitaevsky, L.P. Relativistic Quantum Theory; Nauka: Moscow, Russia, 1968; Volume IV, Part 1, p. 480. [Google Scholar]
- Weinberg, S. The Quantum Theory of Fields; Cambridge University Press: Cambridge, UK, 1999; Volume I, p. 602. [Google Scholar]
- Landau, L.D.; Lifshits, E.M. Classical Field Theory; Nauka: Moscow, Russia, 1973; p. 504. [Google Scholar]
- Dirac, P.A.M. Forms of Relativistic Dynamics. Rev. Mod. Phys.
**1949**, 21, 392–399. [Google Scholar] [CrossRef] - Colosi, D.; Rovelli, C. What is a Particle? Class. Quantum Grav.
**2009**, 26, 025002. [Google Scholar] [CrossRef] - Weinberg, S. What is Quantum Field Theory, and What Did We Think It Is? hep-th/9702027. 1997. [Google Scholar]
- Wigner, E.P. On Unitary Representations of the Inhomogeneous Lorentz Group. Ann. Math.
**1939**, 40, 149–204. [Google Scholar] [CrossRef] - Lev, F.M. Some Group-Theoretical Aspects of SO(1,4)-invariant Theory. J. Phys.
**1988**, A21, 599–615. [Google Scholar] - Lev, F.M. The Problem of Interactions in de Sitter Invariant Theories. J. Phys.
**1999**, A32, 1225–1239. [Google Scholar] [CrossRef] - Lev, F.M. Finiteness of Physics and Its Possible Consequences. J. Math. Phys.
**1993**, 34, 490–527. [Google Scholar] [CrossRef] - Mensky, M.B. Theory of Induced Representations, Space-Time and Concept of Particles; Nauka: Moscow, Russia, 1976; p. 288. [Google Scholar]
- Mielke, E.W. Quantenfeldtheorie im de Sitter-Raum. Fortschr. Phys.
**1977**, 25, 401–457. [Google Scholar] [CrossRef] - Mackey, G.W. Induced Representatios of Locally Compact Groups. Ann. Math.
**1952**, 55, 101–139. [Google Scholar] [CrossRef] - Naimark, M.A. Normalized Rings; Nauka: Moscow, Russia, 1968; p. 664. [Google Scholar]
- Dixmier, J. von Neumann algebras; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1977; Volume 27, p. 437. [Google Scholar]
- Barut, A.O.; Raczka, R. Theory of Group Representations and Applications; Polish Scientific Publishers: Warsaw, Poland, 1977; p. 710. [Google Scholar]
- Dobrev, V.K.; Mack, G.G.; Petkova, V.B.; Petrova, S.; Todorov, I.T. On the Clebsch-Gordan Expansion for the Lorentz Group in n Dimensions. Rep. Math. Phys.
**1976**, 9, 219–246. [Google Scholar] [CrossRef] - Dobrev, V.K.; Mack, G.G.; Petkova, V.B.; Petrova, S.; Todorov, I.T. Harmonic Analysis on the n-Dimensional Lorentz group and Its Application to Conformal Quantum Field Theory; Springer Verlag: Berlin, Germany, 1977; Volume 63, p. 280. [Google Scholar]
- Ikeda, N.; Fukuyuama, T. Fermions in (Anti) de Sitter Gravity in Four Dimensions. Prog. Theor. Phys.
**2009**, 122, 339–353. [Google Scholar] [CrossRef] - Lev, F. Massless Elementary Particles in a Quantum Theory Over a Galois Field. Theor. Math. Phys.
**2004**, 138, 208–225. [Google Scholar] [CrossRef] - Evans, N.T. Discrete Series for the Universal Covering Group of the 3+2 de Sitter Group. J. Math. Phys.
**1967**, 8, 170–184. [Google Scholar] [CrossRef] - Braden, B. Restricted Representatins of Classical Lie Algebras of Types A
_{2}and B_{2}. Bull. Amer. Math. Soc.**1967**, 73, 482–486. [Google Scholar] [CrossRef] - Kondratyuk, L.A.; Terent’ev, M.V. The Scattering Problem for Relativistic Systems with a Fixed Number of Particles in Light-front Dynamics. Sov. J. Nucl. Phys.
**1980**, 31, 1087–1117. [Google Scholar] - Fuda, M.G. Poincare Invariant Potential Model. Phys. Rev.
**1987**, C36, 1489–1506. [Google Scholar] [CrossRef] - Fuda, M.G. A New Picture For Light Front Dynamics. Ann. Phys.
**1990**, 197, 265–299. [Google Scholar] [CrossRef] - Lev, F. Exact Construction of the Electromagnetic Current Operator in Relativistic Quantum Mechanics. Ann. Phys.
**1995**, 237, 355–419. [Google Scholar] [CrossRef] - Hughes, W.B. SU(2) X SU(2) Shift Operators and Representations of SO(5). J. Math. Phys.
**1983**, 24, 1015–1020. [Google Scholar] [CrossRef] - Fetter, A.L.; Walecka, J.D. Quantum Theory of Many-Particle Systems; Dover Publications Inc.: Mineola, NY, USA, 2003; p. 645. [Google Scholar]
- Giulini, D. Superselection Rules. 2009; arXiv:0710.1516v2 (quant-ph). [Google Scholar]
- Akhmedov, E.T.; Buividovich, P.V. Interacting Field Theories in de Sitter Space are Non-Unitary. Phys. Rev.
**2008**, D78, 104005. [Google Scholar] - Akhmedov, E.T.; Buividovich, P.V.; Singleton, D.A. De Sitter Space and Perpetuum Mobile. 2009; arXiv:0905.2742 (gr-qc). [Google Scholar]
- Volovik, G.E. Particle Decay in de Sitter Spacetime Via Quantum Tunneling. JETP Lett.
**2009**, 90, 1–4. [Google Scholar] [CrossRef] - Witten, E. Quantum Gravity In De Sitter Space. hep-th/0106109. 2001. [Google Scholar]
- Polyakov, A.M. Decay of Vacuum Energy. Nucl. Phys.
**2010**, B834, 316–329. [Google Scholar] [CrossRef] - Flato, M.; Fronsdal, C. One Massles Particle Equals Two Dirac Singletons. Lett. Math. Phys.
**1978**, 2, 421–426. [Google Scholar] [CrossRef] - Castell, L.; Heidenreich, W. SO(3,2) Invariant Scattering and Dirac Singletons. Phys. Rev.
**1981**, D24, 371–377. [Google Scholar] [CrossRef] - Fronsdal, C. Dirac supermultiplet. Phys. Rev.
**1982**, D26, 1988–1995. [Google Scholar] [CrossRef] - Lev, F.M. Modular Representations as a Possible Basis of Finite Physics. J. Math. Phys.
**1989**, 30, 1985–1998. [Google Scholar] [CrossRef] - Lev, F.M. Introduction to a Quantum Theory over a Galois Field. Symmetry
**2010**, 2, 1810–1845. [Google Scholar] [CrossRef] - Lev, F.M. Why is Quantum Theory Based on Complex Numbers? Finite Field. Appli.
**2006**, 12, 336–356. [Google Scholar] [CrossRef] - Dyson, F.G. Missed Opportunities. Bull. Amer. Math. Soc.
**1972**, 78, 635–652. [Google Scholar] [CrossRef] - Lev, F.M. Is Gravity an Interaction? Phys. Essays
**2010**, 23, 355–362. [Google Scholar] [CrossRef] - Bogolubov, N.N.; Logunov, A.A.; Oksak, A.I.; Todorov, I.T. General Principles of Quantum Field Theory; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1987; p. 682. [Google Scholar]
- Lev, F.M.; Mirmovich, E.G. Some Aspects of de Sitter Invariant Theory. VINITI, 1984; No 6099 Dep. [Google Scholar]

© 2010 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

## Share and Cite

**MDPI and ACS Style**

Lev, F.M.
Positive Cosmological Constant and Quantum Theory. *Symmetry* **2010**, *2*, 1945-1980.
https://doi.org/10.3390/sym2041945

**AMA Style**

Lev FM.
Positive Cosmological Constant and Quantum Theory. *Symmetry*. 2010; 2(4):1945-1980.
https://doi.org/10.3390/sym2041945

**Chicago/Turabian Style**

Lev, Felix M.
2010. "Positive Cosmological Constant and Quantum Theory" *Symmetry* 2, no. 4: 1945-1980.
https://doi.org/10.3390/sym2041945