# Quantum and Classical Cosmology in the Brans–Dicke Theory

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## Abstract

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## 1. Introduction

## 2. A Short Review of the Brans–Dicke Theory

## 3. Bouncing Solutions and the Energy Conditions

## 4. Canonical Quantization of the BD Theory

## 5. Analysis of the Solution via the de Broglie–Bohm Approach

#### 5.1. The Scalar Field Is Absent

#### 5.2. A Single Scalar Mode

#### 5.3. Multiple Scalar Modes

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Brans, C.; Dicke, R.H. Mach’s Principle and a Relativistic Theory of Gravitation. Phys. Rev.
**1961**, 124, 925–935. [Google Scholar] [CrossRef] - DeWitt, B.S. Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev.
**1967**, 160, 1113. [Google Scholar] [CrossRef] [Green Version] - Kiefer, C. Quantum Gravity; Oxford University Press: Oxford, UK, 2007. [Google Scholar]
- Kuchar, K. Time and interpretation of quantum gravity. Int. J. Mod. Phys. D
**2011**, 20, 3–86. [Google Scholar] [CrossRef] - Isham, C.J. Canonical quantum gravity and the problem of time. Sci. Ser. C
**1993**, 409, 157–287. [Google Scholar] - Schutz, B.F. Perfect Fluids in General Relativity: Velocity Potentials and a Variational Principle. Phys. Rev. D
**1970**, 2, 2762. [Google Scholar] [CrossRef] [Green Version] - Everett, H. Relative state formulation of quantum mechanics. Rev. Mod. Phys.
**1957**, 29, 454–462. [Google Scholar] [CrossRef] [Green Version] - Omnès, R. The Interpretation of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1994. [Google Scholar]
- Bohm, D.; Hiley, B.J. The Undivided Universe: An Ontological Interpretation of Quantum Theory; Routledge: London, UK, 1993. [Google Scholar]
- Holland, P.R. The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Pinto-Neto, N.; Fabris, J.C. Quantum cosmology from the de Broglie–Bohm perspective. Class. Quantum Gravity
**2013**, 30, 143001. [Google Scholar] [CrossRef] [Green Version] - Dirac, P.A.M. The Cosmological Constants. Nature
**1937**, 139, 323. [Google Scholar] [CrossRef] - Jordan, P. The present state of Dirac’s cosmological hypothesis. Z. Phys.
**1959**, 157, 112. [Google Scholar] [CrossRef] - Will, C.M. Theory and Experiment in Gravitational Physics; Cambridge University Press: Cambridge, UK, 2018. [Google Scholar]
- Weinberg, S. Gravitation and Cosmology; John Wiley and Sons: Hoboken, NJ, USA, 1972. [Google Scholar]
- Banerjee, N.; Sen, S. Does Brans–Dicke theory always yield general relativity in the infinite limit? Phys. Rev. D
**1997**, 56, 1334. [Google Scholar] [CrossRef] - Faraoni, V. Illusions of general relativity in Brans–Dicke gravity. Phys. Rev. D
**1999**, 59, 084021. [Google Scholar] [CrossRef] [Green Version] - Chauvineau, B. On the limit of Brans–Dicke theory when ω→∞. Class. Quantum Gravity
**2003**, 20, 2617. [Google Scholar] [CrossRef] - Brando, G.; Fabris, J.C.; Falciano, F.T.; Galkina, O. Stiff matter solution in Brans–Dicke theory and the general relativity limit. Int. J. Mod. Phys. D
**2019**, 28, 1950156. [Google Scholar] [CrossRef] [Green Version] - Will, C.M. The Confrontation between General Relativity and Experiment. Living Rev. Relativ.
**2014**, 17, 4. [Google Scholar] [CrossRef] [Green Version] - Gasperini, M. From Pre- to Post-Big Bang: An (almost) self-dual cosmological history. arXiv
**2021**, arXiv:2106.12865. [Google Scholar] - Khoury, J.; Ovrut, B.A.; Steinhardt, P.J.; Turok, N. Ekpyrotic universe: Colliding branes and the origin of the hot big bang. Phys. Rev. D
**2001**, 64, 123522. [Google Scholar] [CrossRef] [Green Version] - La, D.; Steinhardt, P.J. Extended Inflationary Cosmology. Phys. Rev. Lett.
**1989**, 62, 376. [Google Scholar] [CrossRef] - Bailin, D.; Love, A. Kaluza-Klein theories. Rep. Prog. Phys.
**1987**, 50, 1087. [Google Scholar] [CrossRef] - Lidsey, J.E.; Wands, D.; Copeland, E.J. Superstring cosmology. Phys. Rep.
**2000**, 337, 343. [Google Scholar] [CrossRef] [Green Version] - Colistete, R., Jr.; Fabris, J.C.; Pinto-Neto, N. Singularities and classical limit in quantum cosmology with scalar fields. Phys. Rev. D
**1998**, 57, 4707. [Google Scholar] [CrossRef] [Green Version] - Colistete, R., Jr.; Fabris, J.C.; Pinto-Neto, N. Gaussian superpositions in scalar tensor quantum cosmological models. Phys. Rev. D
**2000**, 62, 083507. [Google Scholar] [CrossRef] [Green Version] - Almeida, C.R.; Batista, A.B.; Fabris, J.C.; Moniz, P.R.V. Quantum cosmology with scalar fields: Self-adjointness and cosmological scenarios. Gravit. Cosmol.
**2015**, 21, 191. [Google Scholar] [CrossRef] [Green Version] - Galkina, O.; Fabris, J.C.; Falciano, F.T.; Pinto-Neto, N. Regular bouncing solutions, energy conditions, and the Brans–Dicke theory. JETP Lett.
**2019**, 110, 523–528. [Google Scholar] [CrossRef] [Green Version] - Gurevich, L.E.; Finkelstein, A.M.; Ruban, V.A. On the problem of the initial state in the isotropic scalar-tensor cosmology of Brans–Dicke. Astrophys. Space Sci.
**1973**, 22, 231. [Google Scholar] [CrossRef] - Battefeld, D.; Peter, P. A Critical Review of Classical Bouncing Cosmologies. Phys. Rep.
**2015**, 571, 1–66. [Google Scholar] [CrossRef] [Green Version] - Novello, M.; Bergliaffa, S.E.P. Bouncing Cosmologies. Phys. Rep.
**2008**, 463, 127–213. [Google Scholar] [CrossRef] - Ijjas, A.; Steinhardt, P.J. Bouncing Cosmology made simple. Class. Quantum Gravity
**2018**, 35, 135004. [Google Scholar] [CrossRef] [Green Version] - Peter, P.; Pinto-Neto, N. Primordial perturbations in a non singular bouncing universe model. Phys. Rev. D
**2002**, 65, 023513. [Google Scholar] [CrossRef] [Green Version] - Bronnikov, K.A. Scalar-tensor gravity and conformal continuations. J. Math. Phys.
**2002**, 43, 6096–6115. [Google Scholar] [CrossRef] [Green Version] - Pinto-Neto, N. Hamiltonian Formulation of General Relativity and Application; PPGCosmo series; Livraria da Física: São Paulo, Brazil, 2020. [Google Scholar]
- Almeida, C.R.; Batista, A.B.; Fabris, J.C.; Moniz, P.V. Quantum Cosmology os Scalar-tensor Theories and Self-adjointness. J. Math. Phys.
**2017**, 58, 042301. [Google Scholar] [CrossRef] [Green Version] - Almeida, C.R.; Batista, A.B.; Fabris, J.C.; Pinto-Neto, N. Quantum Cosmological Scenarios of Brans–Dicke Gravity in Einstein and Jordan Frames. Gravit. Cosmol.
**2018**, 24, 245–253. [Google Scholar] [CrossRef] - Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products; Academic Press: Cambridge, MA, USA, 2007. [Google Scholar]
- de Broglie, L. An Introduction to the Study of Wave Machanics; E.P. Dutton and Company: New York, NY, USA, 1930. [Google Scholar]
- Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables I. Phys. Rev.
**1952**, 85, 166–179. [Google Scholar] [CrossRef] - Brown, H.R.; Wallace, D. Solving the Measurement Problem: De Broglie–Bohm Loses Out to Everett. Found. Phys.
**2005**, 35, 517–540. [Google Scholar] [CrossRef] [Green Version] - Holland, P. What’s Wrong with Einstein’s 1927 Hidden-Variable Interpretation of Quantum Mechanics? Found. Phys.
**2005**, 35, 177–196. [Google Scholar] [CrossRef] [Green Version] - Pinto-Neto, N. The Bohm Interpretation of Quantum Cosmology. Found. Phys.
**2005**, 35, 577–603. [Google Scholar] [CrossRef] [Green Version] - Marto, J.; Moniz, P.V. de Broglie–Bohm FRW universes in quantum string cosmology. Phys. Rev.
**2001**, 65, 023516. [Google Scholar] [CrossRef] - Delgado, P.C.M.; Pinto-Neto, N. Cosmological models with asymmetric quantum bounces. Class. Quantum Gravity
**2020**, 37, 125002. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Almeida, C.R.; Galkina, O.; Fabris, J.C.
Quantum and Classical Cosmology in the Brans–Dicke Theory. *Universe* **2021**, *7*, 286.
https://doi.org/10.3390/universe7080286

**AMA Style**

Almeida CR, Galkina O, Fabris JC.
Quantum and Classical Cosmology in the Brans–Dicke Theory. *Universe*. 2021; 7(8):286.
https://doi.org/10.3390/universe7080286

**Chicago/Turabian Style**

Almeida, Carla R., Olesya Galkina, and Julio César Fabris.
2021. "Quantum and Classical Cosmology in the Brans–Dicke Theory" *Universe* 7, no. 8: 286.
https://doi.org/10.3390/universe7080286