# Quantum Cosmologies under Geometrical Unification of Gravity and Dark Energy

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Lagrangian for FRWq System

## 3. Quintessence and Fermat-Like Lagrangian

## 4. Quantization

#### 4.1. Quantization of the FRWq System as a Klein–Gordon Particle

#### 4.2. Quantization of the FRWq System à la Dirac-Pauli

#### 4.3. Quantization of the FRWq System à la Majorana

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Dirac Equation in Curved Spacetimes

## References

- Tsujikawa, S. Quintessence: A review. Class. Quantum Grav.
**2013**, 30, 214003. [Google Scholar] [CrossRef] - Copeland, E.J.; Sami, M.; Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys. D
**2006**, 15, 1753. [Google Scholar] [CrossRef] - Capozziello, S.; Roshan, M. Exact cosmological solutions from Hojman conservation quantities. Phys. Lett. B
**2013**, 726, 471. [Google Scholar] [CrossRef] - Steinhardt, P.J. A quintessential introduction to dark energy. Philos. Trans. R. Soc. Lond. A
**2003**, 361, 2497. [Google Scholar] [CrossRef] [PubMed] - Bojowald, M. Quantum cosmology: A review. Rep. Prog. Phys.
**2015**, 78, 023901. [Google Scholar] [CrossRef] [PubMed] - Hojman, S.A.; Asenjo, F.A. Supersymmetric Majorana quantum cosmologies. Phys. Rev. D
**2015**, 92, 083518. [Google Scholar] [CrossRef][Green Version] - Breit, G. An Interpretation of Dirac’s Theory of the Electron. Proc. Natl. Acad. Sci. USA
**1928**, 14, 553. [Google Scholar] [CrossRef] - Ryden, B. Introduction to Cosmology; Addison Wesley: Boston, MA, USA, 2003. [Google Scholar]
- Hojman, S.A.; Chayet, S.; Núñez, D.; Roque, M.A. An algorithm to relate general solutions of different bidimensional problems. J. Math. Phys.
**1991**, 32, 1491. [Google Scholar] [CrossRef] - Luneburg, R.K. Mathematical Theory of Optics; University of California at Berkeley: Berkeley, CA, USA, 1948. [Google Scholar]
- DeWitt, B.S. Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev.
**1967**, 160, 1113. [Google Scholar] [CrossRef] - Halliwell, J.J.; Hawking, S.W. Origin of structure in the Universe. Phys. Rev. D
**1985**, 31, 1777. [Google Scholar] [CrossRef] - Alvarenga, F.G.; Lemos, N.A. Dynamical vacuum in quantum cosmology. Gen. Rel. Grav.
**1998**, 30, 681. [Google Scholar] [CrossRef] - Oliveira-Neto, G. No-boundary wave function of the anti–de Sitter space-time and the quantization of Λ. Phys. Rev. D
**1998**, 58, 107501. [Google Scholar] [CrossRef] - Lemos, N.A. Radiation-dominated quantum Friedmann models. J. Math. Phys.
**1996**, 37, 1449. [Google Scholar] [CrossRef] - Monerat, G.A.; Corrêa Silva, E.V.; Oliveira-Neto, G.; Ferreira Filho, L.G.; Lemos, N.A. Quantization of Friedmann-Robertson-Walker spacetimes in the presence of a negative cosmological constant and radiation. Phys. Rev. D
**2006**, 73, 044022. [Google Scholar] [CrossRef][Green Version] - Oliveira-Neto, G.; Monerat, G.A.; Corrêa Silva, E.V.; Neves, C.; Ferreira Filho, L.G. Quantization of Friedmann-Robertson-Walker Spacetimes in the presence of a cosmological constant and stiff matter. Int. J. Theor. Phys.
**2013**, 52, 2991. [Google Scholar] [CrossRef] - Vakili, B. Noether symmetric minisuperspace model of f(R) cosmology. Ann. Phys. (Berlin)
**2010**, 19, 359. [Google Scholar] [CrossRef] - Huang, R.-N. The Wheeler-DeWitt equation of f(R, L
_{m}) gravity in minisuperspace. arXiv**2013**, arXiv:1304.5309v2. [Google Scholar] - Barbour, J.B.; Murchadha, N.O. Classical and quantum gravity on conformal superspace. arXiv
**1999**, arXiv:gr-qc/9911071v1. [Google Scholar] - Hawking, S.W.; Wu, Z.C. Numerical calculations of minisuperspace cosmological models. Phys. Lett.
**1985**, 151B, 15. [Google Scholar] [CrossRef] - Saa, A. Canonical quantization of the relativistic particle in static spacetimes. Class. Quantum Grav.
**1996**, 13, 553. [Google Scholar] [CrossRef] - Hanson, A.; Regge, T.; Teitelboim, C. Constrained Hamiltonian Systems; Accademia Nazionale dei Lincei: Roma, Italy, 1976. [Google Scholar]
- Hojman, S.; Montemayor, R. s-Equivalent Lagrangians for free particles and canonical quantization. Hadron. J.
**1980**, 3, 1644. [Google Scholar] - Gavrilov, S.P.; Gitman, D.M. Quantization of the relativistic particle. Class. Quantum Grav.
**2000**, 17, L133. [Google Scholar] [CrossRef] - Dirac, P.A.M. Lectures on Quantum Mechanics; Dover Publications, Inc.: Mineola, NY, USA, 2001. [Google Scholar]
- Henneaux, M.; Teitelboim, C. Quantization of Gauge Systems; Princeton University: Princeton, NJ, USA, 1992. [Google Scholar]
- Mukhanov, V.F.; Winitzki, S. Introduction to Quantum Effects in Gravity; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Strange, P. Relativistic Quantums Mechanic, with Applications in Condensed Matter and Atomic Physics; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Pedram, P.; Mirzaei, M.; Gousheh, S.S. Using spectral method as an approximation for solving hyperbolic PDEs. Comput. Phys. Commun.
**2007**, 176, 581. [Google Scholar] [CrossRef] - Pedram, P.; Mirzaei, M.; Jalalzadeh, S.; Gousheh, S.S. Perfect fluid quantum Universe in the presence of negative cosmological constant. Gen. Relativ. Gravit.
**2008**, 40, 1663. [Google Scholar] [CrossRef] - Sánchez-Monroy, J.A.; Quimbay, C.J. Dirac equation in low dimensions: The factorization method. Ann. Phys.
**2014**, 350, 69. [Google Scholar] [CrossRef] - Yeşiltaş, Ö. $\mathcal{P}\mathcal{T}$ symmetric Hamiltonian model and Dirac equation in 1+1 dimensions. J. Phys. A Math. Theor.
**2013**, 46, 015302. [Google Scholar] [CrossRef] - Eleuch, H.; Alhaidari, A.D.; Bahlouli, H. Analytical solutions to the Dirac equation in 1+1 Space-Time Dimension. Appl. Math. Inf. Sci.
**2012**, 6, 149. [Google Scholar] - Cooper, F.; Khare, A.; Musto, R.; Wipf, A. Supersymmetry and the Dirac equation. Ann. Phys.
**1988**, 187, 1. [Google Scholar] [CrossRef] - De Crombrugghe, M.; Rittenberg, V. Supersymmetric quantum mechanics. Ann. Phys.
**1983**, 151, 99. [Google Scholar] [CrossRef] - Cooper, F.; Khare, A.; Sukhatme, U. Supersymmetry in Quantum Mechanics; World Scientific: Singapore, 2001. [Google Scholar]
- Savasta, S.; Di Stefano, O. Classical origin of the spin of relativistic pointlike particles and geometric interpretation of Dirac solutions. arXiv
**2008**, arXiv:0803.4013v1. [Google Scholar] - Savasta, S.; Di Stefano, O.; Maragò, O.M. Quantum-classical correspondence of the Dirac matrices: The Dirac Lagrangian as a Total Derivative. arXiv
**2009**, arXiv:0905.4741v1. [Google Scholar] - Barceló, C.; Visser, M.; Liberati, S. Einstein gravity as an emergent phenomenon? Int. J. Mod. Phys. D
**2001**, 10, 799. [Google Scholar] [CrossRef] - Padmanabhan, T. Emergent perspective of Gravity and Dark Energy. Res. Astron. Astrophys.
**2012**, 12, 891. [Google Scholar] [CrossRef] - Horiguchi, T. Quantum potential interpretation of the Wheeler-DeWitt equation. Mod. Phys. Lett. A
**1994**, 9, 1429. [Google Scholar] [CrossRef] - Vink, J.C. Quantum potential interpretation of the wave function of the universe. Nucl. Phys. B
**1992**, 369, 707. [Google Scholar] [CrossRef] - Halliwell, J.J. Derivation of the Wheeler-DeWitt equation from a path integral for minisuperspace models. Phys. Rev. D
**1988**, 38, 2468. [Google Scholar] [CrossRef] - Barvinsky, A.O.; Ponomariov, V.N. Quantum geometrodynamics: The path integral and the initial value problem for the wave function of the universe. Phys. Lett. B
**1986**, 167, 289. [Google Scholar] [CrossRef] - Arisue, H.; Fujiwara, T.; Kato, M.; Ogawa, K. Path-integral and operator formalism in quantum gravity. Phys. Rev. D
**1987**, 35, 2309. [Google Scholar] [CrossRef] - Faizal, M. Deformation of the Wheeler–DeWitt equation. Int. J. Mod. Phys. A
**2014**, 29, 1450106. [Google Scholar] [CrossRef] - Garattini, R.; Faizal, M. Cosmological constant from a deformation of the Wheeler–DeWitt equation. Nucl. Phys. B
**2016**, 905, 313. [Google Scholar] [CrossRef] - Pedram, P. Generalized uncertainty principle and the conformally coupled scalar field quantum cosmology. Phys. Rev. D
**2015**, 91, 063517. [Google Scholar] [CrossRef] - Everett, H. Theory of the Universal Wavefunction. Ph.D. Thesis, Princeton University, Princeton, NJ, USA, 1956. [Google Scholar]
- Everett, H. “Relative State” formulation of quantum mechanics. Rev. Mod. Phys.
**1957**, 29, 454. [Google Scholar] [CrossRef] - Weinberg, S. Living in the Multiverse, Universe or Multiverse? Carr, B., Ed.; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Tegmark, M. The interpretation of quantum mechanics: Many worlds or many words? Fortschr. Phys.
**1998**, 46, 855–862. [Google Scholar] [CrossRef] - Tegmark, M.; Aguirre, A.; Rees, M.J.; Wilczek, F. Dimensionless constants, cosmology, and other dark matters. Phys. Rev. D
**2006**, 73, 023505. [Google Scholar] [CrossRef][Green Version] - Tegmark, M. Parallel Universes, Science and Ultimate Reality; Barrow, J.D., Davies, P.C.W., Harper, C.L., Eds.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Feeney, S.M.; Johnson, M.C.; Mortlock, D.J.; Peiris, H.V. First observational tests of eternal inflation. Phys. Rev. Lett.
**2011**, 107, 071301. [Google Scholar] [CrossRef] - Hall, M.J.W.; Deckert, D.-A.; Wiseman, H.M. Quantum phenomena modeled by interactions between many classical worlds. Phys. Rev. X
**2014**, 4, 041013. [Google Scholar] [CrossRef] - Antonov, A.A. Hidden Multiverse: Explanation of Dark Matter and Dark Energy phenomena. Int. J. Phys.
**2015**, 3, 84. [Google Scholar] - Vilekin, A. A quantum measure of the multiverse. arXiv
**2013**, arXiv:1312.0682. [Google Scholar] - Caroll, S.M. Is our Universe natural? Nature
**2006**, 440, 1132. [Google Scholar] [CrossRef] - Carroll, S.M.; Sebens, C.T. Many Worlds, the Born Rule, and Self-Locating uncertainty. In Quantum Theory: A Two-Time Success Story; Struppa, D.C., Tollaksen, J.M., Eds.; Springer: Milan, Italy, 2014. [Google Scholar]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rubio, C.A.; Asenjo, F.A.; Hojman, S.A. Quantum Cosmologies under Geometrical Unification of Gravity and Dark Energy. *Symmetry* **2019**, *11*, 860.
https://doi.org/10.3390/sym11070860

**AMA Style**

Rubio CA, Asenjo FA, Hojman SA. Quantum Cosmologies under Geometrical Unification of Gravity and Dark Energy. *Symmetry*. 2019; 11(7):860.
https://doi.org/10.3390/sym11070860

**Chicago/Turabian Style**

Rubio, Carlos A., Felipe A. Asenjo, and Sergio A. Hojman. 2019. "Quantum Cosmologies under Geometrical Unification of Gravity and Dark Energy" *Symmetry* 11, no. 7: 860.
https://doi.org/10.3390/sym11070860