Abstract
The \((k=-1)\)-Robertson–Walker spacetime is under investigation. With the derived Hamilton operator, we are solving the Wheeler–De Witt Equation and its Schrödinger-like extension, for physically important forms of the effective potential. The closed form solutions, expressed in terms of Heun’s functions, allow us to comment on the occurrence of Universe from highly probable quantum states.
Similar content being viewed by others
References
Gupta, S.N.: Gravitation and Electromagnetism. Phys. Rev. 96, 1683–1685 (1954)
Fang, J., Fronsdal, C.: Deformations of gauge groups. Gravitation. J. Math. Phys 20, 2264–2270 (1979)
Smolin, L.: Linking topological quantum field theory and nonperturbative quantum gravity. J. Math. Phys. 36, 6417–6455 (1995)
Ashtekar, A., Singh, P.: Loop Quantum Cosmology: A Status Report. Class. Quant. Grav. 28, 213001 (2011)
Chiou, Dah-Wei: Loop Quantum Gravity. Int. J. Mod. Phys. D 24, 1530005 (2014)
Gonzalo, J.: Olmo, Rubiera-Garcia, D.: Brane-world and loop cosmology from a gravity-matter coupling perspective. Phys. Lett. B 740, 73–79 (2015)
Gambini, R., Pullin, J.: Emergence of stringlike physics from Lorentz invariance in loop quantum gravity. Int. J. Mod. Phys. D 23, 1442023 (2014)
Fang, J., et al.: A generalized consistency condition for massless fields. Letters in Mathematical Physics 38, 213–216 (1996)
Muller, D., et al.: Casimir energy in a small volume multiply connected static hyperbolic preinflationary universe. Phys. Rev. D 63, 123508 (2001)
Muller, D., Fagundes, H.V.: Casimir energy density in closed hyperbolic universes. Int. J. Mod. Phys. A 17, 4385–4392 (2002)
Aurich, R., Steiner, F.: Dark energy in a hyperbolic universe. Mon. Not. Roy. Astron. Soc. 334, 735–742 (2002)
Wheeler, J.A.: Superspace. In: Gilbert, R.D., Newton, R. (eds.) Analytic Methods in Mathematical Physics, pp. 335–378. Gordon and Breach, New York (1970)
De Witt, B.: Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev. 160, 1113–1148 (1967)
Miritzis, J.: Isotropic cosmologies in Weyl geometry. Class. Quant. Grav. 21, 3043–3056 (2004)
Romero, C., et al.: General Relativity and Weyl Geometry. Class. Quant. Grav. 29, 155015 (2012)
Anderson, E.: The Problem of Time in Quantum Gravity (2010). arXiv:1009.2157 [gr-qc]
Kuchař, K. V.: Time and interpretations of quantum gravity. Int. J. Mod. Phys. Proc. Suppl. D 20, 3–86 (2011)
Isham, C.J.: Canonical quantum gravity and the problem of time. In: Ibort, L.A., Rodriguez, M.A. (eds.) Integrable Systems, Quantum Groups and Quantum Field Theories, pp. 157–288. Kluwer, Dordrecht (1993)
Ashtekar, A., et al.: Quantum Nature of the Big Bang: Improved dynamics. Phys. Rev. D 74, 084003 (2006)
Vakili, B.: Scalar field quantum cosmology: a Schrödinger picture. Phys. Lett. B 718, 34–42 (2012)
Dariescu, M. A., Dariescu, C.: From a Five Dimensional Warped Friedmann-Robertson-Walker Universe to the Weyl Integrable Spacetime. Int. J. Theor. Phys. (2015). doi:10.1007/s10773-014-2469-y
Canuto, V., et al.: Scale Covariant Theory of Gravitation and Astrophysical Applications. Phys. Rev. D 16, 1643–1663 (1977)
Gaztanaga, E., et al.: Clustering of Luminous Red Galaxies IV: Baryon Acoustic Peak in the Line-of-Sight Direction and a Direct Measurement of H(z). Mon. Not. Roy. Astron. Soc. 399, 1663–1680 (2009)
Zhai, Zhong-Xu, et al.: Reconstruction and constraining of the jerk parameter from OHD and SNe Ia observations. Phys. Lett. B 727, 8–20 (2013)
Busca, N.G., et al.: Baryon Acoustic Oscillations in the Ly-\(\alpha \) forest of BOSS quasars. Astronomy and Astrophysics 552, A96 (2013)
Almeida, T.S., et al.: From Brans-Dicke gravity to a geometrical scalar-tensor theory. Phys. Rev. D 89, 064047 (2014)
Arscott, F. M.: Part A. Heun’s Equation. In: Ronveaux, A. (ed.) Heun’s Differential Equations, pp. 3–86. Oxford University Press, Oxford, UK (1995)
Fiziev, P.P., Staicova, D.R.: Application of the confluent Heun functions for finding the quasinormal modes of nonrotating black holes. Phys. Rev. D 84, 127502 (2011)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 4th edn. Academic, New York (1965)
Liu, Xiao, et al.: Dynamical behaviors of FRW Universe containing a positive/negative potential scalar field in loop quantum cosmology. Gen. Rel. Grav. 45, 1021–1031 (2013)
Acknowledgments
The authors are most grateful to the anonymous referees for insightful remarks and suggestions which have been very helpful in improving the original form of our paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dariescu, C., Dariescu, MA. Quantum Analysis of \(k=-1\) Robertson–Walker Universe. Found Phys 45, 1495–1513 (2015). https://doi.org/10.1007/s10701-015-9922-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-015-9922-5