Skip to main content
SpringerLink
Log in

Deformation of the Poisson Structure of a Point Particle Due to Gravitational Back Reaction

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

The dynamics of a massive particle in a frame of a test particle in 3 + 1 spacetime dimensions is considered with gravitational interaction taken into account. The total action (gravity+particles) collapses to a boundary separating the massive particle and the test particle, and is further reduced to a finite dimensional action depending only on relative particle coordinates and momenta. It turns out that the momentum space is a coadjoint orbit of the Lorentz group. The momentum space is thus curved and its curvature falls off with the particle relative distance according to the Newton law. This defines the modified form of the Poisson brackets. At the quantum level, this results in non-commutativity and partial discreteness in coordinate space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Bronstein, Phys. Zeitschr. Sow, 9 (1936).

  2. E. Witten, “2 + 1 dimensional gravity as an exactly soluble system,” Nucl. Phys. B, 311, No. 1, 1–19 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  3. G. ’t Hooft, “Canonical quantization of gravitating point particles in 2 + 1 dimensions,” Class. Quantum Grav., 10, 1653 (1993).

  4. G. ’t Hooft, “Quantization of point particles in (2+1)-dimensional gravity and spacetime discreteness,” Class. Quantum Grav., 13, 1023 (1996).

  5. H. J. Matschull and M. Welling, “Quantum mechanics of a point particle in 2+1 dimensional gravity,” Class. Quant. Grav. 15, 2981 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Ashtekar and P. Singh, “Loop quantum cosmology: a status report,” Class. Quant. Grav., 28 (2011).

  7. A. A. Andrianov, O. Novikov, and Chen Lan, “Quantum cosmology of multifield scalar matter: Some exact solutions,” Theor. Math. Phys., 184, 1224–1233 (2015).

  8. W. Israel, “Singular hypersurfaces and thin shells in general relativity,” Il Nuovo Cimento, B, 44 (1967).

  9. V. K. Kuchař, “Geometrodynamics of Schwarzschild black holes,” Phys. Rev., D50 (1994).

  10. P. Hájíček, “Spherically symmetric gravitating shell as a reparametrization invariant system,” Phys. Rev., D57 (1998).

  11. J. Louko, B. F. Whiting, and J. L. Friedman, “Hamiltonian space-time dynamics with a spherical null dust shell,” Phys. Rev., D57 (1998).

  12. P. Hájíček and C. Kiefer, “Embedding variables in the canonical theory of gravitating shells,” Nucl. Phys., B603 (2001).

  13. V. A. Berezin, A. M. Boyarsky, and A. Yu. Neronov, Phys. Rev. D 57 1118 (1998), e-print archive gr-qc/9708060.

  14. V. Berezin, Published Int. J. Mod. Phys. A, 17, 979–988 (2002).

    Article  Google Scholar 

  15. J. Wess and B. Zumino, “Consequences of anomalous ward identities,” Phys. Lett. B. 37, 95 (1971).

    Article  MathSciNet  Google Scholar 

  16. E. Witten, “Global aspects of current algebra,” Nucl. Phys. B., 223, No. 2, 422–432 (1983).

    Article  MathSciNet  Google Scholar 

  17. A. Y. Alekseev and A. Z. Malkin, “Symplectic structure of the moduli space of at connection on a Riemann surface,” Comm. Math. Phys., 169, No. 99 (1995).

  18. C. Meusburger and B. J. Schroers, “Phase space structure of Chern-Simons theory with a nonstandard puncture,” Nucl. Phys. B, 738, No. 425 (2006).

  19. I. M. Gel’fand and M. A. Naimark, “Unitary representations of the classical groups,” Trudy Mat. Inst. Steklov Acad. Sci. USSR, 36, 3–288 (1950).

  20. A. A. Andrianov, Y. Elmahalawy, and A. Starodubtsev, “Quantum analysis of BTZ black hole formation due to the collapse of a dust shell universe,” 6, No. 11, 201 (2020).

    Google Scholar 

  21. A. A. Andrianov, Y. Elmahalawy, and A. Starodubtsev, “(2 + 1)-dimensional gravity coupled to a dust shell: quantization in terms of global phase space variables Published,” Theor. Math. Phys., 200, No. 3, 1269–1281 (2019).

    Article  MATH  Google Scholar 

  22. A. N. Starodubtsev, “New approach to calculating the spectrum of a quantum space-time,” Theor. Math. Phys., 190, No. 3, 439–445 (2017).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St.Petersburg State University, St.Petersburg, Russia

    D. A. Lyozin & A. N. Starodubtsev

  2. Leonard Euler St.Petersburg International Mathematical Institute, St.Petersburg, Russia

    A. N. Starodubtsev

Authors
  1. D. A. Lyozin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. N. Starodubtsev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. A. Lyozin.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 509, 2021, pp. 153–175.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lyozin, D.A., Starodubtsev, A.N. Deformation of the Poisson Structure of a Point Particle Due to Gravitational Back Reaction. J Math Sci 275, 326–341 (2023). https://doi.org/10.1007/s10958-023-06684-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06684-8

Search

Navigation