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Deformations of Fluid Dynamics and Friedmann Equation on Noncommutative Phase Space

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Abstract

Noncommutative phase space is one of the widely studied extensions of ordinary phase space, and has profound implications in cosmological physics. In this paper we study the dynamics of perfect fluid on noncommutative phase space, as well as deformations of the Friedmann equation. The Lagrangian formalism is used to take into account of the phase space noncommutativities. Then a map from canonical Lagrangian variables to Eulerian variables is employed to derive the equations of motion of the mass and current densities. We find that both these two equations receive noncommutative corrections that are linear in the noncommutative parameters. However, we also find that in the approximation of vanishing comoving velocity the leading order noncommutative correction due to momentum noncommutativity on the Friedmann equation is zero.

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References

  1. Snyder, H.S.: Quantized space-time. Phys. Rev. 71, 38-41 (1947)

    Article  ADS  MathSciNet  Google Scholar 

  2. Peierls, R.: Zur theorie des diamagnetismus von leitungselektronen. Zeitschrift für Physik 80, 763791 (1933)

    Article  Google Scholar 

  3. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 09, 032 (1999). arXiv:hep-th/9908142 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  4. Freidel, L., Livine, E.R.: Effective 3-D quantum gravity and non-commutative quantum field theory, 4th International Symposium on Quantum Theory and Symmetries and 6th International Workshop on Lie Theory and Its Applications in Physics (QTS-4) (LT-6) Varna, Bulgaria, August 15-21, 2005. Phys. Rev. Lett. 96, 221301 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  5. Moffat, J.W.: Noncommutative quantum gravity. Phys. Lett. B 491, 345-352 (2000). arXiv:hep-th/0007181 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  6. Moffat, J.W.: Perturbative noncommutative quantum gravity. Phys. Lett. B 493, 142-148 (2000). arXiv:hep-th/0008089 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  7. Faizal, M.: Noncommutative quantum gravity. Mod. Phys. Lett. A 28, 1350034 (2013). arXiv:1302.5156 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  8. Douglas, M.R., Nekrasov, N.A.: Noncommutative field theory. Rev. Mod. Phys. 73, 977-1029 (2001). arXiv:hep-th/0106048 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  9. Szabo, R.J.: Quantum field theory on noncommutative spaces, Frontiers of Mathematical Physics: Summer Workshop on Particles, Fields and Strings Burnaby, Canada, July 16-27, 2001. Phys. Rept. 378, 207-299 (2003). arXiv:hep-th/0109162 [hep-th]

    Article  ADS  Google Scholar 

  10. Chaichian, M., Sheikh-Jabbari, M.M., Tureanu, A.: Hydrogen atom spectrum and the Lamb shift in noncommutative QED. Phys. Rev. Lett. 86, 2716 (2001). arXiv:hep-th/0010175 [hep-th]

    Article  ADS  Google Scholar 

  11. Zhang, J.-z.: Testing spatial noncommutativity via Rydberg atoms. Phys. Rev. Lett. 93, 043002 (2004). arXiv:hep-ph/0405143 [hep-ph]

    Article  ADS  Google Scholar 

  12. Zhang, P.M., Horvathy, P.A.: Kohn condition and exotic Newton-Hooke symmetry in the non-commutative Landau problem. Phys. Lett. B 706, 442-446 (2012). arXiv:1111.1595 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  13. Zhang, P.-M., Horvathy, P.A.: Chiral decomposition in the non-commutative landau problem. Annals Phys. 327, 1730-1743 (2012). arXiv:1112.0409 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  14. Ramrez, W.G., Deriglazov, A.A.: Relativistic effects due to gravimagnetic moment of a rotating body. Phys. Rev. D 96, 124013 (2017). arXiv:1709.06894 [gr-qc]

    Article  ADS  Google Scholar 

  15. Chaichian, M., Demichev, A., Presnajder, P., Sheikh-Jabbari, M.M., Tureanu, A.: Aharonov-Bohm effect in noncommutative spaces. Phys. Lett. B 527, 149-154 (2002). arXiv:hep-th/0012175 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  16. Chaichian, M., Demichev, A., Presnajder, P., Sheikh-Jabbari, M.M., Tureanu, A.: Quantum theories on noncommutative spaces with nontrivial topology: Aharonov-Bohm and Casimir effects. Nucl. Phys. B 611, 383-402 (2001). arXiv:hep-th/0101209 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  17. Ma, K., Wang, J.-H., Yang, H.-X.: Time-dependent AharonovBohm effect on the noncommutative space. Phys. Lett. B 759, 306-312 (2016). arXiv:1604.02110 [hep-th]

    Article  ADS  Google Scholar 

  18. Ma, K., Wang, J.-h., Yang, H.-X.: SeibergWitten map and quantum phase effects for neutral Dirac particle on noncommutative plane. Phys. Lett. B 756, 221-227 (2016). arXiv:1410.6363 [hep-th]

    Article  ADS  Google Scholar 

  19. Rodriguez, M.E.: Quantum effects of Aharonov-Bohm type and noncommutative quantum mechanics. arXiv:1710.10357 [quant-ph] (2017)

  20. Ma, K., Dulat, S.: Spin Hall effect on a noncommutative space. Phys. Rev. A 84, 012104 (2011). arXiv:1104.4955 [hep-th]

    Article  ADS  Google Scholar 

  21. Deriglazov, A.A., Pupasov-Maksimov, A.M.: Relativistic corrections to the algebra of position variables and spin-orbital interaction. Phys. Lett. B 761, 207-212 (2016). arXiv:1609.00043 [quant-ph]

    Article  ADS  Google Scholar 

  22. Wang, J.-h., Ma, K., Li, K.: Influences of a topological defect on the spin Hall effect. Phys. Rev. A 87, 032107 (2013). arXiv:1301.4363 [cond-mat.mes-hall]

    Article  ADS  Google Scholar 

  23. Wang, J., Ma, K., Li, K., Fan, H.: Deformations of the spin currents by topological screw dislocation and cosmic dispiration. Annals Phys. 362, 327-335 (2015). arXiv:1510.07741 [cond-mat.mes-hall]

    Article  ADS  MathSciNet  Google Scholar 

  24. Ren, Y.-J., Ma, K.: Influences of the coordinate dependent noncommutative space on charged and spin currents. arXiv:1802.10452 [physics.gen-ph] (2018)

  25. Wang, K., Zhang, Y.-F., Wang, Q., Long, Z.-W., Jing, J.: Quantum speed limit for a relativistic electron in the noncommutative phase space. Int. J. Mod. Phys. A 32, 1750143 (2017). arXiv:1702.03167 [hep-th]

    Article  ADS  Google Scholar 

  26. Wang, K., Zhang, Y.F., Wang, Q., Long, Z.W., Jing, J.: Quantum speed limit for relativistic spin-0 and spin-1 bosons on commutative and noncommutative planes. Adv. High Energy Phys. 2017, 4739596 (2017). arXiv:1703.01063 [hep-th]

    MathSciNet  MATH  Google Scholar 

  27. Deriglazov, A.A., Ramrez, W.G.: Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations in ultra-relativistic regime and gravimagnetic moment. Int. J. Mod. Phys. D 26, 1750047 (2016). arXiv:1509.05357 [gr-qc]

    Article  ADS  Google Scholar 

  28. Deriglazov, A.A., Ramrez, W.G.: Ultrarelativistic spinning particle and a rotating body in external fields. Adv. High Energy Phys. 2016, 1376016 (2016). arXiv:1511.00645 [gr-qc]

    Article  Google Scholar 

  29. Chang, L.N., Minic, D., Okamura, N., Takeuchi, T.: The Effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem. Phys. Rev. D 65, 125028 (2002). arXiv:hep-th/0201017 [hep-th]

    Article  ADS  Google Scholar 

  30. Ma, K., Ren, Y.-J., Wang, K.-Q.: Probing Noncommutativities of phase space by using persistent charged current and its asymmetry. arXiv:1703.10923 [hep-th] (2017)

  31. Ma, K., Wang, J.-H., Yang, H.-X.: Probing the noncommutative effects of phase space in the time-dependent AharonovBohm effect. Annals Phys. 383, 120-129 (2017). arXiv:1605.03902 [hep-ph]

    Article  ADS  Google Scholar 

  32. Fateme, H., Ma, K., Hassan, H.: Motion of a Nonrelativistic Quantum Particle in Non-commutative Phase Space. Chin. Phys. Lett. 32, 100302 (2015)

    Article  Google Scholar 

  33. Bertolami, O., Rosa, J.G., de Aragao, C.M.L., Castorina, P., Zappala, D.: Noncommutative gravitational quantum well. Phys. Rev. D 72, 025010 (2005). arXiv:hep-th/0505064 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  34. Ma, K.: Constrains of Charge-to-Mass Ratios on Noncommutative Phase Space. Adv. High Energy Phys. 2017, 1945156 (2017). arXiv:1705.05789 [hep-ph]

    MATH  Google Scholar 

  35. Pramanik, S., Ghosh, S.: GUP-based and snyder non-commutative algebras, relativistic particle models and deformed symmetries and interaction: a unified approach. Int. J. Mod. Phys. A 28, 1350131 (2013). arXiv:1301.4042 [hep-th]

    Article  ADS  Google Scholar 

  36. Pramanik, S., Ghosh, S., Pal, P.: Electrodynamics of a generalized charged particle in doubly special relativity framework. Annals Phys. 346, 113–128 (2014). arXiv:1212.6881 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  37. Ghosh, S.: A Lagrangian for DSR Particle and the Role of Noncommutativity. Phys. Rev. D 74, 084019 (2006). arXiv:hep-th/0608206 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  38. Deriglazov, A.A., Ramrez, W.G.: Lagrangian formulation for Mathisson- Papapetrou-Tulczyjew-Dixon (MPTD) equations. Phys. Rev. D 92, 124017 (2015). arXiv:1509.04926 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  39. Das, P., Ghosh, S.: Noncommutative geometry and fluid dynamics. Eur. Phys. J. C 76, 627 (2016). [Erratum: Eur. Phys. J.C77,no.2,64(2017)], arXiv:1601.01430 [hep-th]

    Article  ADS  Google Scholar 

  40. Holender, L., Santos, M.A., Orlando, M.T.D., Vancea, I.V.: Noncommutative fluid dynamics in the Kaehler parametrization. Phys. Rev. D 84, 105024 (2011)

    Article  ADS  Google Scholar 

  41. Abdalla, M.C.B., Holender, L., Santos, M.A., Vancea, I.V.: Noncommutative fluid dynamics in the Snyder space-time. Phys. Rev. D 86, 045019 (2012). arXiv:1206.3982 [hep-th]

    Article  ADS  Google Scholar 

  42. Alavi, S.A.: General noncommuting curvilinear coordinates and fluid Mechanics. Chin. Phys. Lett. 23, 10 (2006). arXiv:hep-th/0608106 [hep-th]

    Article  Google Scholar 

  43. Ma, K.: Noncommutative effects on the fluid dynamics and modifications of the Friedmann equation. arXiv:1801.02533 [hep-th] (2018)

  44. Gangopadhyay, S., Saha, A., Halder, A.: On the Landau system in noncommutative phase-space. Phys. Lett. A 379, 2956-2961 (2015). arXiv:1412.3581 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  45. Banerjee, R., Chakraborty, B., Ghosh, S., Mukherjee, P., Samanta, S.: Topics in Noncommutative Geometry Inspired Physics. Found. Phys. 39, 1297-1345 (2009). arXiv:0909.1000 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  46. Marcolli, M., Pierpaoli, E.: Early Universe models from Noncommutative Geometry. Adv. Theor. Math. Phys. 14, 1373-1432 (2010). arXiv:0908.3683 [hep-th]

    Article  MathSciNet  Google Scholar 

  47. Chamseddine, A.H., Connes, A., Mukhanov, V.: Quanta of Geometry: Noncommutative Aspects. Phys. Rev. Lett. 114, 091302 (2015). arXiv:1409.2471 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  48. Chamseddine, A.H., Connes, A.: Quantum gravity boundary terms from spectral action. Phys. Rev. Lett. 99, 071302 (2007). arXiv:0705.1786 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  49. Chamseddine, A.H., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991-1089 (2007). arXiv:hep-th/0610241 [hep-th]

    Article  MathSciNet  Google Scholar 

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Acknowledgements

K. M. is supported by the National Natural Science Foundation of China under Grant No. 11705113, and partially supported by the Natural Science Basic Research Plan in Shaanxi Province of China under Grant No. 2017JM1032.

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  1. School of Physics Science, Shaanxi University of Technology, Hanzhong, 723000, Shaanxi, People’s Republic of China

    Ya-Jie Ren & Kai Ma

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Ren, YJ., Ma, K. Deformations of Fluid Dynamics and Friedmann Equation on Noncommutative Phase Space. Int J Theor Phys 57, 3373–3380 (2018). https://doi.org/10.1007/s10773-018-3850-z

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