Generalized Heisenberg Uncertainty Principle in Quantum Geometrodynamics and General Relativity

  • A. W. Beckwith Physics Department, Chongqing University, College of Physics, Chongqing University Huxi Campus
  • S. S. Moskaliuk Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
Keywords: generalized Heisenberg uncertainty principle, general relativity, Universe, cosmology, quantum geometrodynamics

Abstract

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).

https://doi.org/10.1142/6730

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).

https://doi.org/10.1142/7873

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).

https://doi.org/10.1017/CBO9780511807787

Published
2018-12-13
How to Cite
Beckwith, A., & Moskaliuk, S. (2018). Generalized Heisenberg Uncertainty Principle in Quantum Geometrodynamics and General Relativity. Ukrainian Journal of Physics, 62(8), 727. https://doi.org/10.15407/ujpe62.08.0727
Section
Astrophysics and cosmology