Generalized Heisenberg Uncertainty Principle in Quantum Geometrodynamics and General Relativity
DOI:
https://doi.org/10.15407/ujpe62.08.0727Keywords:
generalized Heisenberg uncertainty principle, general relativity, Universe, cosmology, quantum geometrodynamicsAbstract
We focus on the energy flows in the Universe as a simple quantum system and are concentrating on the nonlinear Hamilton–Jacobi equation, which appears in the standard quantum formalism based on the Schr¨odinger equation. The cases of the domination of radiation, barotropic fluid, and the quantum matter-energy are considered too. As a result, the generalized Heisenberg uncertainty principle (GHUP) is formulated for a metric tensor. We also use the Kuzmichev–Kuzmichev geometrodynamics as a way to quantify the interrelationship between the GHUP for a metric tensor and conditions postulated as to a barotropic fluid, i.e. a dust for the early Universe conditions.
References
C. Will. Was Einstein right? A centenary assessment.In General Relativity and Gravitation, A Centennial Perspective, edited by A. Ashtekar, B. Berger, J. Isenberg, M. MacCallum (Cambridge Univ. Press, 2015).
https://doi.org/10.1017/CBO9781139583961.004
C. Will. The confrontation between general relativity and experiment. Living Rev. Relativity 17, 4 (2014).
https://doi.org/10.12942/lrr-2014-4
T.G. Downes, G.J. Milburn, C.M. Caves. Optimal quantum estimation for gravitation. General Relativity and Quantum Cosmology 1, 1 (2012); arXiv gr-qc/1108.5220.
V.E. Kuzmichev, V.V. Kuzmichev. Can quantum geometrodynamics complement general relativity? Ukr. J. Phys. 61, (5) 449 (2016).
https://doi.org/10.15407/ujpe61.05.0449
K.A. Olive et al. Review of particle physics. (Particle Data Group).Chin. Phys. C 38, 090001 (2014).
https://doi.org/10.1088/1674-1137/38/9/090001
J.D. Brown, D. Marolf. Relativistic material reference systems. Phys. Rev. D 53, 1835 (1996).
https://doi.org/10.1103/PhysRevD.53.1835
V.E. Kuzmichev, V.V. Kuzmichev. The big bang quantum cosmology: The matter-energy production epoch. Acta Phys. Pol. B 39, 979 (2008).
V.E. Kuzmichev, V.V. Kuzmichev. Quantum corrections to the dynamics of the expanding universe. Acta Phys. Pol. B 44, 2051 (2013).
https://doi.org/10.5506/APhysPolB.44.2051
V.E. Kuzmichev, V.V. Kuzmichev. Quantum geometrodynamical description of the dark sector of the matter-energy content of the universe. Ukr. J. Phys. 60, 664 (2015).
https://doi.org/10.15407/ujpe60.07.0664
P.A.M. Dirac. The theory of gravitation in Hamiltonian form. Proc. Roy. Soc. A 246, 333 (1958).
https://doi.org/10.1098/rspa.1958.0142
R. Arnowitt, S. Deser, C.M. Misner. The dynamics of general relativity. In Gravitation: An Introduction to Current Research. Edited by. L. Witten (Wiley, 1962) [arXiv:grqc/0405109].
C.M. Misner, K.S. Thorne, J.A. Wheeler. Gravitation (Freeman, 1973).
L.D. Landau, E.M. Lifshitz. The Classical Theory of Fields. Course of Theoretical Physics, Vol. 2 (Butterworth-Heinemann, 1975).
A.D. Linde. Elementary Particle Physics and Inflationary Cosmology (Harwood, 1990).
https://doi.org/10.1201/b16971
V.E. Kuzmichev, V.V. Kuzmichev. Properties of the quantum universe in quasistationary states and cosmological puzzles. Eur. Phys. J. C 23, 337 (2002).
https://doi.org/10.1007/s100520100850
W.G. Unruh. Why study quantum theory? Canad. J. Phys. 64, 128 (1986).
https://doi.org/10.1139/p86-019
M. Giovannini. A Primer on the Physics of the Cosmic Microwave Background (World Scientific, 2008).
E.W. Kolb, M.S. Turner. The Early Universe (AddisonWesley, 1990).
A. Beckwith. Gedanken experiment for refining the unruh metric tensor uncertainty principle via Schwarzschild geometry and Planckian space-time with initial nonzero entropy and applying the Riemannian-Penrose inequality and initial kinetic energy for a lower bound to graviton mass (massive gravity). J. High Energy Physics, Gravitation and Cosmology 2, 106 (2016).
https://doi.org/10.4236/jhepgc.2016.21012
C.S. Camara, M.R. de Garcia Maia, J.C. Carvalho, J.A.S. Lima. Nonsingular FRW cosmology and non linear dynamics. Phys. Rev. D 69, 123504 (2004).
https://doi.org/10.1103/PhysRevD.69.123504
J. Barbour. The nature of time. General Relativity and Quantum Cosmology, 2009; http://arxiv.org/pdf/0903.3489.pdf.
J. Barbour. Shape dynamics: An introduction. In Quantum Field Theory and Gravity, Conceptual and Mathematical Advances in the Search for a Unified Framework Edited by F. Finster, O. Muller, M. Nardmann, J. Tolksdorf, E. Zeidler (Birkh¨auser, 2010).
A. Goldhaber, M. Nieto. Photon and graviton mass limits. Rev. Mod. Phys. 82, 939 (2010).
https://doi.org/10.1103/RevModPhys.82.939
W.J. Handley, S.D. Brechet, A.N. Lasenby, M.P. Hobson. Kinetic initial conditions for inflation. Phys. Rev. D 89, 063505 (2014).
https://doi.org/10.1103/PhysRevD.89.063505
M. Roos. Introduction to Cosmology (Wiley, 2003) [ISBN: 0-470-84909-6].
A.Z. Petrov. Einstein Spaces (Pergamon Press, 1969).
https://doi.org/10.1016/B978-0-08-012315-8.50007-0
D. Gorbunov, V. Rubakov. Introduction to the Theory of the Early Universe, Cosmological Perturbations and Inflationary Theory (World Scientific, 2011).
S.A. Fulling. Aspects of Quantum Field Theory in Curved Space-Time (Cambridge Univ. Press, 1991).
H. Gutfreund, J. Renn. The Road to Relativity, the History and Meaning of Einstein's "The Foundation of General Relativity" (Princeton Univ. Press, 2015).
J. Griffiths, J. Podolsky. Exact Space-Times in Einstein's General Relativity (Cambridge Univ. Press, 2009).
https://doi.org/10.1017/CBO9780511635397
R.M. Wald. Quantum Field theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, 1994).
K. Fredenhagen, K. Rejzner. Local covariance and background independence. In Quantum Field Theory and Gravity, Conceptual and Mathematical Advances in the Search for a Unified Framework, Edited by F. Finster, O. Muller, M. Nardmann, J. Tolksdorf, E. Zeidler (Birkh¨auser, 2010).
C. Corda. Primordial gravity's breath. EJTP 9 26 1 (2012).
S. Gilen, D. Oriti. Discrete and Continuum Third Quantization of Gravity. In Quantum Field Theory and Gravity, Conceptual and Mathematical Advances in the Search for a Unified Framework, Edited by F. Finster, O. Muller, M. Nardmann, J. Tolksdorf, E. Zeidler (Birkh¨auser, 2010).
N.D. Birrell, P.C.W. Davies. Quantum Fields in Curved Space (Cambridge Univ. Press, 1982).
https://doi.org/10.1017/CBO9780511622632
A. Beckwith. Gedanken experiment for fluctuation of mass of a graviton, based on the trace of GR stress energy tensorpre Planckian conditions that lead to gaining of graviton mass, and Planckian conditions that lead to graviton mass shrinking to 10–62 grams. J. of High Energy Physics, Gravitation and Cosmology, 2, 19 (2016).
https://doi.org/10.4236/jhepgc.2016.21002
A. Meszhlumian. Towards the Theory of Stationary Universe. In Texas/PASCOS 92: Relativistic astrophysics and particle cosmology. Edited by C. Akerlof, M. Srednicki (Annals of the New York Academy of Sciences, 1992), V. 688.
A. Beckwith. Gedanken experiment examining how kinetic energy would dominate potential energy, in pre-Planckian space-time physics, and allow us to avoid the BICEP 2 mistake. J. of High Energy Physics, Gravitation and Cosmology 2, 75 (2016).
https://doi.org/10.4236/jhepgc.2016.21008
Y. Jack Ng. Holographic foam, dark energy and infinite statistics. Phys. Lett. B 657, 10 (2007).
https://doi.org/10.1016/j.physletb.2007.09.052
Y. Jack Ng. Spacetime foam: From entropy and holography to infinite statistics and nonlocality. Entropy 10, 441 (2008).
https://doi.org/10.3390/e10040441
I. Ciufolini and J. Wheeler. Gravitation and Inertia (Princeton Univ. Press, 1995).
T. Padmanabhan. Cosmological constant – the weight of the vacuum. Phys. Rept. 380, 235 (2003).
https://doi.org/10.1016/S0370-1573(03)00120-0
C. Egan, C.H. Lineweaver. A larger estimate of the entropy of the universe. Astrophys. J. 710, 1825 (2010).
https://doi.org/10.1088/0004-637X/710/2/1825
A.F. Ali, S. Das. Cosmology from quantum potential. Phys. Lett. B 741, 276 (2015).
https://doi.org/10.1016/j.physletb.2014.12.057
I. Haranas, I. Gkigkitzis. The mass of graviton and its relation to the number of information according to the holographic principle. Int. Scholarly Research Notices 2014, 1 (2014).
https://doi.org/10.1155/2014/718251
C. Rovelli, F. Vidotto. Covariant Loop Quantum Gravity, an Elementary Introduction to Quantum Gravity and Spinfoam Theory (Cambridge Univ. Press, 2015).
G. Galloway, P. Miao, R. Schoen. Initial data and the Einstein constraint equations. In General Relativity and Gravitation, A centennial Perspective. Edited by A. Ashtekar, B. Berger, J. Isenberg, M. MacCallum (Cambridge Univ. Press, 2015).
https://doi.org/10.1017/CBO9781139583961.012
H. Wen, F.Y. Li, Z.Y. Fang. Very high frequency gravitational waves from magnetars and gamma-ray burst. arXiv:1608.03186 (2016).
M. Bojowald. Can the Arrow of Time be understood from Quantum Cosmology? In The Arrows of Time, a Debate in Cosmology. Edited by L. Mersini, R. Vaas (Springer, 2012).
T. Padmanabhan. Gravitation, Foundations and Frontiers (Cambridge Univ. Press, 2010).
Downloads
Published
Issue
Section
License
Copyright Agreement
License to Publish the Paper
Kyiv, Ukraine
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4. Duration.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5. Loyalty.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.
.png){kind=small}
