Generalized Uncertainty Principle in Quantum Cosmology for the Maximally Symmetric Space

Authors

  • V. E. Kuzmichev Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
  • V. V. Kuzmichev Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine https://orcid.org/0000-0002-2589-8887

DOI:

https://doi.org/10.15407/ujpe64.2.100

Keywords:

quantum gravity, quantum geometrodynamics, cosmology, uncertainty principle

Abstract

The new uncertainty relation is derived in the context of the canonical quantum theory with gravity in the case of the maximally symmetric space. This relation establishes a connection between fluctuations of the quantities, which determine the intrinsic and extrinsic curvatures of the spacelike hypersurface in spacetime and introduces the uncertainty principle for quantum gravitational systems. The generalized time-energy uncertainty relation taking gravity into account gravity is proposed. It is shown that known Unruh’s uncertainty relation follows, as a particular case, from the new uncertainty relation. As an example, the sizes of fluctuations of the scale factor and its conjugate momentum are calculated within an exactly solvable model. All known modifications of the uncertainty principle deduced previously from different approaches in the theory of gravity and the string theory are obtained as particular cases of the proposed general expression.

Author Biography

V. V. Kuzmichev, Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine

Senior Researcher (Department for Astrophysics and Elementary Particles)

References

L. de Broglie. Les Incertitudes d'Heisenberg et l'Interpr?etation Probabiliste de la M?ecanique Ondulatoire (Gauthier-Villars, 1982) [ISBN: 978-2040154110].

M. Bronstein. Quantentheorie schwacher Gravitations-felder. Phys. Z. Sowjetunion 9, 140 (1936).

M. Bronstein. Quantization of gravitational waves. ZhETF 6, 195 (1936).

E.P.Wigner. Relativistic invariance and quantum phenomena. Rev. Mod. Phys. 29, 255 (1957). https://doi.org/10.1103/RevModPhys.29.255

J.L. Anderson. Quantization of general relativity. In: Gravitation and Relativity, ed. by H-Y. Chiu, W.F. Hoffmann, (Benjamin, 1964).

D. Amati, M. Ciafaloni, G. Veneziano. Can spacetime be probed below the string size? Phys. Lett. B 216, 41 (1989). https://doi.org/10.1016/0370-2693(89)91366-X

K. Konishi, G. Paffuti, P. Provero. Minimum physical length and the generalized uncertainty principle in string theory. Phys. Lett. B 234, 276 (1990). https://doi.org/10.1016/0370-2693(90)91927-4

E. Witten. Reflections on the fate of spacetime. Physics Today 49, 24 (1996). https://doi.org/10.1063/1.881493

M. Maggiore. A generalized uncertainty principle in quantum gravity. Phys. Lett. B 304, 65 (1993). https://doi.org/10.1016/0370-2693(93)91401-8

L.G. Garay. Quantum gravity and minimum length. Int. J. Mod. Phys. A 10, 145 (1995). https://doi.org/10.1142/S0217751X95000085

S. Capozziello, G. Lambiase, G. Scarpetta. Generalized uncertainty principle from quantum geometry. Int. J. Theor. Phys. 39, 15 (2000). https://doi.org/10.1023/A:1003634814685

A. Kempf, G. Mangano, R.B. Mann. Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52, 1108 (1995). https://doi.org/10.1103/PhysRevD.52.1108

F. Brau. Minimal length uncertainty relation and hydrogen atom. J. Phys. A 32, 7691 (1999). https://doi.org/10.1088/0305-4470/32/44/308

S. Das, E.C. Vagenas. Phenomenological implications of the generalized uncertainty principle. Can. J. Phys. 87, 233 (2009). https://doi.org/10.1139/P08-105

S. Hossenfelder. Minimal length scale scenarios for quantum gravity. Living Rev. Rel. 16, 2 (2013). https://doi.org/10.12942/lrr-2013-2

A. Tawfik, A. Diab. Generalized uncertainty principle: approaches and applications. Int. J. Mod. Phys. D 23, 1430025 (2014). https://doi.org/10.1142/S0218271814300250

A. Tawfik, A. Diab. A review of the generalized uncertainty principle. Rep. Prog. Phys. 78, 126001 (2015). https://doi.org/10.1088/0034-4885/78/12/126001

R.C.S. Bernardo, J.P.H. Esguerra. Euclidean path integral formalism in deformed space with minimum measurable length. J. Math. Phys. 58, 042103 (2017). https://doi.org/10.1063/1.4979797

C. Bambi, F.R. Urban. Natural extension of the generalised uncertainty principle. Class. Quant. Grav. 25, 095006 (2008). https://doi.org/10.1088/0264-9381/25/9/095006

R. Arnowitt, S. Deser, C.M. Misner. The dynamics of general relativity. In: Gravitation: An Introduction to Current Research, ed. by L. Witten, (Wiley, 1962) [ISBN: 978-1114291669].

K.V. Kucha?r, C.G. Torre. Gaussian reference fluid and interpretation of quantum geometrodynamics. Phys. Rev. D 43, 419 (1991). https://doi.org/10.1103/PhysRevD.43.419

J.A. Wheeler. Superspace and the nature of quantum geometrodynamics. In: Battelle Rencontres, eds. by C. De-Witt, J.A. Wheeler, (Benjamin, 1968).

J.D. Brown, D. Marolf. On 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 evolution of the very early universe. Ukr. J. Phys. 53, 837 (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. Can quantum geometrodynamics complement general relativity? Ukr. J. Phys. 61, 449 (2016). https://doi.org/10.15407/ujpe61.05.0449

V.E. Kuzmichev, V.V. Kuzmichev. The matter-energy intensity distribution in a quantum gravitational system. Quantum Stud.: Math. Found. 5(2), 245 (2018). https://doi.org/10.1007/s40509-017-0115-0

P.A.M. Dirac. The theory of gravitation in Hamiltonian form. Proc. Roy. Soc. A 246, 333 (1958).

F. Karolyhazy. Gravitation and quantum mechanics of macroscopic object. Nuovo Cimento A 42, 390 (1966). https://doi.org/10.1007/BF02717926

M. Maziashvili. Space–time in light of K?arolyh?azy uncertainty relation. Int. J. Mod. Phys. D 16, 1531 (2007). https://doi.org/10.1142/S0218271807010870

W.G. Unruh. Why study quantum theory? Can. J. Phys. 64, 128 (1986). https://doi.org/10.1139/p86-019

G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin. The principle of relative locality. Phys. Rev. D 84, 084010 (2011). https://doi.org/10.1103/PhysRevD.84.084010

Lay Nam Chang, Z. Lewis, D. Minic, T. Takeuchi. On the minimal length uncertainty relation and the foundations of string theory. Advances in High Energy Physics 2011, 493514 (2011).

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Published

2019-02-21

How to Cite

Kuzmichev, V. E., & Kuzmichev, V. V. (2019). Generalized Uncertainty Principle in Quantum Cosmology for the Maximally Symmetric Space. Ukrainian Journal of Physics, 64(2), 100. https://doi.org/10.15407/ujpe64.2.100

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Section

Fields and elementary particles