Skip to main content
SpringerLink
Log in

Operational dynamical modeling of spin 1/2 relativistic particles

The Dirac equation and its classical limit

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

The formalism of Operational Dynamical Modeling [Bondar et al., Phys. Rev. Lett. 109, 190403 (2012)] is employed to analyze dynamics of spin half relativistic particles. We arrive at the Dirac equation from specially constructed relativistic Ehrenfest theorems by assuming that the coordinates and momenta do not commute. Forbidding creation of antiparticles and requiring the commutativity of the coordinates and momenta lead to classical Spohn’s equation [Spohn, Ann. Phys. 282, 420 (2000)]. Moreover, Spohn’s equation turns out to be the classical Koopman-von Neumann theory underlying the Dirac equation.

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. S. Ahrens, S.-Y. Zhu, J. Jiang, Y. Sun, New J. Phys. 17, 113021 (2015)

    Article  ADS  Google Scholar 

  2. F.A. An, E.J. Meier, B. Gadway, Sci. Adv. 3, e1602685 (2017)

    Article  ADS  Google Scholar 

  3. J. Autschbach, J. Chem. Phys. 136, 150902 (2012)

    Article  ADS  Google Scholar 

  4. V. Bargmann, L. Michel, V.L. Telegdi, Phys. Rev. Lett. 2, 435 (1959)

    Article  ADS  Google Scholar 

  5. W. Baylis, Phys. Rev. A 45, 4293 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  6. W. Baylis, Electrodynamics: a modern geometric approach (Birkhauser, 1999)

  7. W. Baylis, R. Cabrera, J. Keselica, Adv. Appl. Clifford Al. 20, 517 (2010)

    Article  Google Scholar 

  8. W. Baylis, Y. Yao, Phys. Rev. A 60, 785 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  9. M.V. Berry, in Quantum Mechanics: Scientific Perspectives on Divine Action (Vatican Observatory CTNS Publications, 2001), pp. 41–54

  10. I. Bialynicki-Birula, Acta Phys. Aust. Supl. XVIII, 111 (1977)

    Google Scholar 

  11. I. Bialynicki-Birula, EPJ Web Conf. 78, 01001 (2014)

    Article  Google Scholar 

  12. I. Bialynicki-Birula, P. Górnicki, J. Rafelski, Phys. Rev. D 44, 1825 (1991)

    Article  ADS  Google Scholar 

  13. R. Blatt, C. Roos, Nat. Phys. 8, 277 (2012)

    Article  Google Scholar 

  14. O. Boada, A. Celi, J. Latorre, M. Lewenstein, New J. Phys. 13, 035002 (2011)

    Article  ADS  Google Scholar 

  15. A. Bolivar, Quantum-classical correspondence: dynamical quantization and the classical limit (Springer-Verlag, 2004)

  16. J. Bolte, R. Glaser, J. Phys. A 37, 6359 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  17. D.I. Bondar, R. Cabrera, A. Campos, S. Mukamel, H.A. Rabitz, J. Phys. Chem. Lett. 7, 1632 (2016)

    Article  Google Scholar 

  18. D.I. Bondar, R. Cabrera, R.R. Lompay, M.Y. Ivanov, H.A. Rabitz, Phys. Rev. Lett. 109, 190403 (2012)

    Article  ADS  Google Scholar 

  19. D.I. Bondar, R. Cabrera, H.A. Rabitz, Phys. Rev. A 88, 012116 (2013)

    Article  ADS  Google Scholar 

  20. D.I. Bondar, R. Cabrera, D.V. Zhdanov, H.A. Rabitz, Phys. Rev. A 88, 052108 (2013)

    Article  ADS  Google Scholar 

  21. D.I. Bondar, A.G. Campos, R. Cabrera, H.A. Rabitz, Phys. Rev. E 93, 063304 (2016)

    Article  ADS  Google Scholar 

  22. D.I. Bondar, R.R. Lompay, W.-K. Liu, Am. J. Phys. 79, 392 (2011)

    Article  ADS  Google Scholar 

  23. R. Cabrera, D.I. Bondar, A.G. Campos, H.A. Rabitz, Phys. Rev. A. 94, 052111 (2016)

    Article  ADS  Google Scholar 

  24. R. Cabrera, D.I. Bondar, K. Jacobs, H.A. Rabitz, Phys. Rev. A 92, 042122 (2015)

    Article  ADS  Google Scholar 

  25. A.G. Campos, R. Cabrera, D.I. Bondar, H. Rabitz, https://doi.org/arXiv:1502.03025 (2015)

  26. A.G. Campos, R. Cabrera, D.I. Bondar, H.A. Rabitz, Phys. Rev. A 90, 034102 (2014)

    Article  ADS  Google Scholar 

  27. G. Coddens, From Spinors to Quantum Mechanics (World Scientific, 2015)

  28. E. Deotto, E. Gozzi, D. Mauro, J. Math. Phys. 44, 5902 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  29. E. Deotto, E. Gozzi, D. Mauro, J. Math. Phys. 44, 5937 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  30. H. Elze, M. Gyulassy, D. Vasak, Phys. Lett. B 177, 402 (1986)

    Article  ADS  Google Scholar 

  31. L.D. Faddeev, L. Khalfin, I. Komarov, in VA Fock-selected works: Quantum mechanics and quantum field theory (CRC Press, 2004), Sect. 29-2

  32. J. Fanchi, Found. Phys. 23, 487 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  33. C.M. Flores, Classical propagation in the quantum inverted oscillator, https://doi.org/arXiv:1612.01604 (2016)

  34. F. Gay-Balmaz, C. Tronci, The hamiltonian setting of Koopman-von Neumann theory and the dynamics of hybrid classical-quantum systems, https://doi.org/arXiv:1802.04787 (2018)

  35. R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, C. Roos, Nature 463, 68 (2010)

    Article  ADS  Google Scholar 

  36. E. Gozzi, C. Pagani, Phys. Rev. Lett. 105, 150604 (2010)

    Article  ADS  Google Scholar 

  37. W. Greiner, Classical electrodynamics (Springer-Verlag, 1998)

  38. W. Greiner, Relativistic quantum mechanics: wave equations (Springer-Verlag, 2000)

  39. R. Hakim, Introduction to relativistic statistical mechanics (World Scientific, 2011)

  40. R. Hakim, J. Heyvaerts, Phys. Rev. A 18, 1250 (1978)

    Article  ADS  Google Scholar 

  41. R. Hakim, H. Sivak, Ann. Phys. 139, 230 (1982)

    Article  ADS  Google Scholar 

  42. M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)

    Article  ADS  Google Scholar 

  43. D. Hestenes, J. Math. Phys. 15, 1768 (1974)

    Article  ADS  Google Scholar 

  44. D. Hestenes, New foundations for classical mechanics (Springer, 1999)

  45. K. Jacobs, Quantum measurement theory and its applications (Cambridge University Press, 2014)

  46. M. Katsnelson, K. Novoselov, A. Geim, Nat. Phys. 2, 620 (2006)

    Article  Google Scholar 

  47. S.-I. Koda, J. Chem. Phys. 144, 154108 (2016)

    Article  ADS  Google Scholar 

  48. B.O. Koopman, Proc. Nat. Acad. Sci. 17, 315 (1931)

    Article  ADS  Google Scholar 

  49. K. Kowalski, J. Rembieliński, Ann. Phys. 375, 1 (2016)

    Article  ADS  Google Scholar 

  50. H. Krüger, Found. Phys. 23, 1265 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  51. P. Kustaanheimo, E. Stiefel, J. Math. Bd 218, 27 (1965)

    Google Scholar 

  52. W. Liu, Phys. Chem. Chem. Phys. 14, 35 (2012)

    Article  Google Scholar 

  53. P. Lounesto, in Clifford algebras and spinors (Cambridge University Press, 2001), Vol. 286

  54. S. Mane, Y.M. Shatunov, K. Yokoya, Rep. Prog. Phys. 68, 1997 (2005)

    Article  ADS  Google Scholar 

  55. D. Mauro, Topics in Koopman-von Neumann Theory, PhD thesis, Università degli Studi di Trieste, 2002

  56. K. Novoselov, A.K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, A. Firsov, Nature 438, 197 (2005)

    Article  ADS  Google Scholar 

  57. J. Otterbach, R. Unanyan, M. Fleischhauer, Phys. Rev. Lett. 102, 063602 (2009)

    Article  ADS  Google Scholar 

  58. J. Pedernales, R. Di Candia, D. Ballester, E. Solano, New J. Phys. 15, 055008 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  59. M. Radonjić, D. Popović, S. Prvanović, N. Burić, Phys. Rev. A 89, 024104 (2014)

    Article  ADS  Google Scholar 

  60. J. Schwinger, B.-G. Englert, Quantum Mechanics: Symbolism of Atomic Measurements (Springer Science & Business Media, 2001)

  61. B. Shanahan, A. Chenu, N. Margolus, A. Del Campo, Phys. Rev. Lett. 120, 070401 (2018)

    Article  ADS  Google Scholar 

  62. G. Shin, I. Bialynicki-Birula, J. Rafelski, Phys. Rev. A 46, 645 (1992)

    Article  ADS  Google Scholar 

  63. G. Shin, J. Rafelski, Phys. Rev. A 48, 1869 (1993)

    Article  ADS  Google Scholar 

  64. H. Spohn, Ann. Phys. 282, 420 (2000)

    Article  ADS  Google Scholar 

  65. J. Vaishnav, C.W. Clark, Phys. Rev. Lett. 100, 153002 (2008)

    Article  ADS  Google Scholar 

  66. S. Varró, J. Javanainen, J. Opt. B: Quant. Semiclassical Opt. 5, S402 (2003)

    Article  ADS  Google Scholar 

  67. D. Vasak, M. Gyulassy, H. Elze, Ann. Phys. 173, 462 (1987)

    Article  ADS  Google Scholar 

  68. J. von Neumann, Ann. Math. 33, 587 (1932)

    Article  MathSciNet  Google Scholar 

  69. J. von Neumann, Ann. Math. 33, 789 (1932)

    Article  MathSciNet  Google Scholar 

  70. S.L. Vuglar, D.V. Zhdanov, R. Cabrera, T. Seideman, C. Jarzynski, D.I. Bondar, Quantum statistical forces via reservoir engineering, https://doi.org/arXiv:1611.02736 (2016)

  71. M. Wen, H. Bauke, C.H. Keitel, Sci. Rep. 6, 31624 (2016)

    Article  ADS  Google Scholar 

  72. D.V. Zhdanov, D.I. Bondar, T. Seideman, Quantum friction: environment engineering perspectives, https://doi.org/arXiv:1612.00573 (2016)

  73. D.V. Zhdanov, D.I. Bondar, T. Seideman, Phys. Rev. Lett. 119, 170402 (2017)

    Article  ADS  Google Scholar 

  74. D.V. Zhdanov, T. Seideman, Phys. Rev. A 92, 012129 (2015)

    Article  ADS  Google Scholar 

  75. W.H. Zurek, Phys. Today 44, 36 (1991)

    Article  Google Scholar 

Download references

Author information

Author notes

  1. Renan Cabrera

    Present address: Current affiliation: Arctan, Inc., Arlington, VA, 22201, USA

Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08544, USA

    Renan Cabrera, Andre G. Campos & Herschel A. Rabitz

  2. Max Planck Institute for Nuclear Physics, 69117, Heidelberg, Germany

    Andre G. Campos

  3. Tulane University, New Orleans, LA, 70118, USA

    Denys I. Bondar

Authors
  1. Renan Cabrera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andre G. Campos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Herschel A. Rabitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Denys I. Bondar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Denys I. Bondar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cabrera, R., Campos, A.G., Rabitz, H.A. et al. Operational dynamical modeling of spin 1/2 relativistic particles. Eur. Phys. J. Spec. Top. 227, 2195–2207 (2019). https://doi.org/10.1140/epjst/e2018-800075-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2018-800075-7

Search

Navigation