Abstract
We study the dynamics of a massive pointlike particle coupled to gravity in four space–time dimensions. It has the same degrees of freedom as an ordinary particle: its coordinates with respect to a chosen origin (observer) and the canonically conjugate momenta. The effect of gravity is that such a particle is a black hole: its momentum becomes spacelike at a distances to the origin less than the Schwarzschild radius. This happens because the phase space of the particle has a nontrivial structure: the momentum space has curvature, and this curvature depends on the position in the coordinate space. The momentum space curvature in turn leads to the coordinate operator in quantum theory having a nontrivial spectrum. This spectrum is independent of the particle mass and determines the accessible points of space–time.
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References
C. Rovelli and L. Smolin, “Discreteness of area and volume in quantum gravity,” Nucl. Phys. B, 442, 593–619 (1995); Erratum, 456, 753–754 (1995); arXiv:gr-qc/9411005v1 (1994)
A. Ashtekar and J. Lewandowski, “Quantum theory of geometry. I. Area operators,” Class. Q. Grav., 14, A55–A81 (1997); arXiv:gr-qc/9602046v2 (1996).
J. Ambjørn, J. Jurkiewicz, and R. Loll, “Nonperturbative Lorentzian path integral for gravity,” Phys. Rev. Lett., 924–927 (2000); arXiv:hep-th/0002050v3 (2000); “Emergence of a 4D world from causal quantum gravity,” Phys. Rev. Lett., 93, 131301 (2004);arXiv:hep-th/0404156v4 (2004); “Reconstructing the universe,” Phys. Rev. D, 72, 064014 (2005); arXiv:hep-th/0505154v2 (2005).
M. Bronstein, “Quantentheorie schwacher Gravitationsfelder,” Phys. Z. Sowjetunion, 9, 140–157 (1936).
G.’t Hooft, “Canonical quantization of gravitating point particles in 2+1 dimensions,” Class. Q. Grav., 10, 1653–1664 (1993); “Quantization of point particles in (2+1)-dimensional gravity and spacetime discreteness,” Class. Q. Grav., 13, 1023–1039 (1996).
H.-J. Matschull and M. Welling, “Quantum mechanics of a point particle in 2+1 dimensional gravity,” Class. Q. Grav., 15, 2981–3030 (1998); arXiv:gr-qc/9708054v2 (1997).
A. N. Starodubtsev, “Phase space of a gravitating particle and dimensional reduction at the Planck scale,” Theor. Math. Phys., 185, 1527–1532 (2015).
S. W. MacDowell and F. Mansouri, “Unified geometric theory of gravity and supergravity,” Phys. Rev. Lett., 38, 739–742 (1977); Erratum, 38, 1376 (1977).
G. W. Moore and N. Seiberg, “Taming the conformal zoo,” Phys. Lett. B, 220, 422–430 (1989)
S. Elitzur, G. W. Moore, A. Schwimmer, and N. Seiberg, “Remarks on the canonical quantization of the Chern–Simons–Witten theory,” Nucl. Phys. B, 326, 108–134 (1989).
S. Carlip, “Statistical mechanics and black hole thermodynamics,” Nucl. Phys. B: Proc. Suppl., 57, 8–12 (1997)
E. Witten, “On holomorphic factorization of WZW and coset models,” Commun. Math. Phys., 144, 189–212 (1992).
G.’t Hooft, “Magnetic monopoles in unified gauge theories,” Nucl. Phys. B, 79, 276–284 (1974)
A. M. Polyakov, “Particle spectrum in quantum field theory,” JETP Lett., 20, 194–194 (1974).
A. Yu. Alekseev and A. Z. Malkin, “Symplectic structure of the moduli space of flat connection on a Riemann surface,” Commun. Math. Phys., 169, 99–119 (1995); arXiv:hep-th/9312004v1 (1993)
C. Meusburger and B. J. Schroers, “Phase space structure of Chern–Simons theory with a non-standard puncture,” Nucl. Phys. B, 738, 425–456 (2006); arXiv:hep-th/0505143v1 (2005).
T. T. Wu and C. N. Yang, “Concept of nonintegrable phase factors and global formulation of gauge fields,” Phys. Rev. D, 12, 3845–3857 (1975).
S. Alexandrov and D. Vassilevich, “Area spectrum in Lorentz covariant loop gravity,” Phys. Rev. D, 64, 044023 (2001);arXiv:gr-qc/0103105v3 (2001)
S. Alexandrov and Z. Kádár, “Timelike surfaces in Lorentz covariant loop gravity and spin foam models,” Class. Q. Grav., 22, 3491–3509 (2005); arXiv:gr-qc/0501093v2 (2005).
V. A. Berezin, A. M. Boyarsky, and A. Yu. Neronov, “Quantum geometrodynamics for black holes and wormholes,” Phys. Rev. D, 57, 1118–1128 (1998); arXiv:gr-qc/9708060v1 (1997)
V. Berezin, “Towards a theory of quantum black holes,” Internat. J. Modern Phys. A, 17, 979–988 (2002); arXiv:gr-qc/0112022v1 (2001).
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 190, No. 3, pp. 511–518, March, 2017.
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Starodubtsev, A.N. New approach to calculating the spectrum of a quantum space–time. Theor Math Phys 190, 439–445 (2017). https://doi.org/10.1134/S0040577917030138
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DOI: https://doi.org/10.1134/S0040577917030138