Вісник Львівського університету. Серія фізична 58 (2021) с. 30-28
DOI: https://doi.org/10.30970/vph.58.2021.30

Parameters of the deformed algebra with minimal uncertainties in position and momentum and the weak equivalence principle

Kh. P. Gnatenko, N. A. Susulovska, V. M. Tkachuk

Deformed algebra [X,P]=i\hbar(1+\beta P2+\alpha X2) leading to the minimal uncertainties in position and momentum is considered. It is shown that the trajectory of motion of a free particle in the quantized space depends on its mass. Also the momentum of the particle is not proportional to mass. We conclude that the free motion does not depend on mass and the momentum is proportional to mass if the parameter of the deformed algebra \beta is inversely proportional to the squared mass and \alpha does not depend on the mass. It is also shown that the same conditions lead to preservation of the weak equivalence principle in the quantized space.

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Список посилань
  1. Gross D. J. String theory beyond the Planck scale / D. J. Gross, P. F. Mende // Nucl. Phys. B -- 1988. -- Vol. 303, No. 3. -- P. 407-454. doi:10.1016/0550-3213(88)90390-2.
  2. Maggiore M. A generalized uncertainty principle in quantum gravity / M. Maggiore // Phys. Lett. B -- 1993. -- Vol. 304, No. 1-2. -- P. 65-69. doi:10.1016/0370-2693(93)91401-8.
  3. Witten E. Reflections on the Fate of Spacetime / E. Witten // Physics Today -- 1996. -- Vol. 49, No. 4. -- P. 24-30. doi:10.1063/1.881493.
  4. Kempf A. Hilbert space representation of the minimal length uncertainty relation / A. Kempf, G. Mangano, R. B. Mann // Phys. Rev. D -- 1995. -- Vol. 52, No. 2. -- P. 1108-1118. doi:10.1103/PhysRevD.52.1108.
  5. Kempf A. Noncommutative geometric regularization / A. Kempf // Phys. Rev. D -- 1996. -- Vol. 54, No. 8. -- P. 5174-5178. doi:10.1103/PhysRevD.54.5174.
  6. Casadio R. Generalized Uncertainty Principle, Classical Mechanics, and General Relativity / R. Casadio, F. Scardigli // Phys. Lett. B -- 2020. -- Vol. 807. -- Art. 135558. -- 5 p. doi:10.1016/j.physletb.2020.135558.
  7. Luciano G. G. Generalized uncertainty principle and its implications on geometric phases in quantum mechanics / G. G. Luciano, L. Petruzziello // Eur. Phys. J. Plus -- 2021. -- Vol. 136. -- Art. 179. doi:10.1140/epjp/s13360-021-01161-0.
  8. Kempf A. Uncertainty relation in quantum mechanics with quantum group symmetry / A. Kempf // J. Math. Phys. -- 1994. -- Vol. 35, No. 9. -- P. 4483-4496. doi:10.1063/1.530798.
  9. Hinrichsen H. Maximal localization in the presence of minimal uncertainties in positions and in momenta / H. Hinrichsen, A. Kempf // J. Math. Phys. -- 1996. -- Vol. 37, No. 5. -- P. 2121-2137. doi:10.1063/1.531501
  10. Bagchi B. Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass / B Bagchi, A. Banerjee, C. Quesne, V. M. Tkachuk // J. Phys. A: Math. Gen. -- 2005. -- Vol. 38, No. 13. -- P. 2929-2945. doi:10.1088/0305-4470/38/13/008.
  11. Quesne C. Hamiltonians with Position-Dependent Mass, Deformations and Supersymmetry / C. Quesne, B. Bagchi, A. Banerjee, V. M. Tkachuk // Bulg. J. Phys. -- 2006. -- Vol. 33, No. 4. -- P. 308-318. \texttt{http://www.bjp-bg.com/papers/bjp2006/\_4/\_308-318.pdf}.
  12. Quesne C. Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem / C. Quesne, V. M. Tkachuk // J. Phys. A: Math. Gen. -- 2004. -- Vol. 37, No. 14. -- P. 4267-4281. doi:10.1088/0305-4470/37/14/006.
  13. Quesne C. Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework / C. Quesne, V. M. Tkachuk // J. Phys. A: Math. Gen. -- 2003. -- Vol. 36, No. 41. -- P. 10373-10389. doi:10.1088/0305-4470/36/41/009.
  14. Quesne C. Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics / C. Quesne, V. M. Tkachuk // SIGMA -- 2007. -- Vol. 3. -- Art. 016. -- 18 p. doi:10.3842/SIGMA.2007.016.
  15. Tkachuk V. M. Deformed Heisenberg algebra with minimal length and the equivalence principle / V. M. Tkachuk // Phys. Rev. A -- 2012. -- Vol. 86, No. 6. -- Art. 062112 -- 4 p. doi:10.1103/PhysRevA.86.062112.
  16. Quesne C. Composite system in deformed space with minimal length / C. Quesne, V.M. Tkachuk // Phys. Rev. A -- 2010. -- Vol. 81, No. 1. -- Art. 012106 -- 8p. doi:10.1103/PhysRevA.81.012106.
  17. Gnatenko Kh. P. Macroscopic body in the Snyder space and minimal length estimation / Kh. P. Gnatenko, V. M. Tkachuk // EPL (Europhysics Letters) -- 2019. -- Vol. 125, No. 5. -- Art. 50003. -- 5 p. doi:10.1209/0295-5075/125/50003.
  18. Snyder H. Quantized Space-Time / H. Snyder // Phys. Rev. -- 1947. -- Vol. 71, No. 1. -- P. 38-41. doi:10.1103/PhysRev.71.38.
  19. Romero J. The area quantum and Snyder space / J. Romero, A. Zamora // Phys. Lett. B -- 2008. -- Vol. 661, No. 1. -- P. 11-13. doi:10.1016/j.physletb.2008.02.001.
  20. Mignemi S. Classical and quantum mechanics of the nonrelativistic Snyder model / S. Mignemi // Phys. Rev. D -- 2011. -- Vol. 84, No. 2. -- Art. 025021. -- 11 p. doi:10.1103/PhysRevD.84.025021.
  21. Mignemi S. Snyder dynamics in a Schwarzschild spacetime / S. Mignemi, R. Strajn // Phys. Rev. D -- 2014. -- Vol. 90, No. 4. -- Art. 044019. -- 5 p. doi:10.1103/PhysRevD.90.044019.
  22. Lu L. Particle dynamics on Snyder space / L. Lu, A. Stern // Nucl. Phys. B -- 2012. -- Vol. 860, No. 1. -- P. 186-205. doi:10.1016/j.nuclphysb.2012.02.012.
  23. Gnatenko Kh. P. The Soccer-ball problem in quantum space / Kh. P. Gnatenko, V. M. Tkachuk // TASK Quarterly -- 2019. -- Vol. 23, No. 4. -- P. 361-450. doi:10.17466/tq2019/23.4/a.