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Quantum Cosmologies under Geometrical Unification of Gravity and Dark Energy

1,†, 2,*,† and 3,4,5,†
Facultad de Física, Pontificia Universidad Católica de Chile, Santiago 7820436, Chile
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago 7941169, Chile
Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Santiago 7941169, Chile
Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago 7800003, Chile
Centro de Recursos Educativos Avanzados, CREA, Santiago 7500018, Chile
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2019, 11(7), 860;
Received: 7 June 2019 / Revised: 26 June 2019 / Accepted: 26 June 2019 / Published: 2 July 2019
PDF [373 KB, uploaded 12 July 2019]


A Friedmann–Robertson–Walker Universe was studied with a dark energy component represented by a quintessence field. The Lagrangian for this system, hereafter called the Friedmann–Robertson–Walker–quintessence (FRWq) system, was presented. It was shown that the classical Lagrangian reproduces the usual two (second order) dynamical equations for the radius of the Universe and for the quintessence scalar field, as well as a (first order) constraint equation. Our approach naturally unified gravity and dark energy, as it was obtained that the Lagrangian and the equations of motion are those of a relativistic particle moving on a two-dimensional, conformally flat spacetime. The conformal metric factor was related to the dark energy scalar field potential. We proceeded to quantize the system in three different schemes. First, we assumed the Universe was a spinless particle (as it is common in literature), obtaining a quantum theory for a Universe described by the Klein–Gordon equation. Second, we pushed the quantization scheme further, assuming the Universe as a Dirac particle, and therefore constructing its corresponding Dirac and Majorana theories. With the different theories, we calculated the expected values for the scale factor of the Universe. They depend on the type of quantization scheme used. The differences between the Dirac and Majorana schemes are highlighted here. The implications of the different quantization procedures are discussed. Finally, the possible consequences for a multiverse theory of the Dirac and Majorana quantized Universe are briefly considered. View Full-Text
Keywords: quantum cosmology; quintessence; Dirac and Majorana quantization quantum cosmology; quintessence; Dirac and Majorana quantization
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Rubio, C.A.; Asenjo, F.A.; Hojman, S.A. Quantum Cosmologies under Geometrical Unification of Gravity and Dark Energy. Symmetry 2019, 11, 860.

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