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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Ahrens, S.-Y. Zhu, J. Jiang, Y. Sun, New J. Phys. 17, 113021 (2015)
F.A. An, E.J. Meier, B. Gadway, Sci. Adv. 3, e1602685 (2017)
J. Autschbach, J. Chem. Phys. 136, 150902 (2012)
V. Bargmann, L. Michel, V.L. Telegdi, Phys. Rev. Lett. 2, 435 (1959)
W. Baylis, Phys. Rev. A 45, 4293 (1992)
W. Baylis, Electrodynamics: a modern geometric approach (Birkhauser, 1999)
W. Baylis, R. Cabrera, J. Keselica, Adv. Appl. Clifford Al. 20, 517 (2010)
W. Baylis, Y. Yao, Phys. Rev. A 60, 785 (1999)
M.V. Berry, in Quantum Mechanics: Scientific Perspectives on Divine Action (Vatican Observatory CTNS Publications, 2001), pp. 41–54
I. Bialynicki-Birula, Acta Phys. Aust. Supl. XVIII, 111 (1977)
I. Bialynicki-Birula, EPJ Web Conf. 78, 01001 (2014)
I. Bialynicki-Birula, P. Górnicki, J. Rafelski, Phys. Rev. D 44, 1825 (1991)
R. Blatt, C. Roos, Nat. Phys. 8, 277 (2012)
O. Boada, A. Celi, J. Latorre, M. Lewenstein, New J. Phys. 13, 035002 (2011)
A. Bolivar, Quantum-classical correspondence: dynamical quantization and the classical limit (Springer-Verlag, 2004)
J. Bolte, R. Glaser, J. Phys. A 37, 6359 (2004)
D.I. Bondar, R. Cabrera, A. Campos, S. Mukamel, H.A. Rabitz, J. Phys. Chem. Lett. 7, 1632 (2016)
D.I. Bondar, R. Cabrera, R.R. Lompay, M.Y. Ivanov, H.A. Rabitz, Phys. Rev. Lett. 109, 190403 (2012)
D.I. Bondar, R. Cabrera, H.A. Rabitz, Phys. Rev. A 88, 012116 (2013)
D.I. Bondar, R. Cabrera, D.V. Zhdanov, H.A. Rabitz, Phys. Rev. A 88, 052108 (2013)
D.I. Bondar, A.G. Campos, R. Cabrera, H.A. Rabitz, Phys. Rev. E 93, 063304 (2016)
D.I. Bondar, R.R. Lompay, W.-K. Liu, Am. J. Phys. 79, 392 (2011)
R. Cabrera, D.I. Bondar, A.G. Campos, H.A. Rabitz, Phys. Rev. A. 94, 052111 (2016)
R. Cabrera, D.I. Bondar, K. Jacobs, H.A. Rabitz, Phys. Rev. A 92, 042122 (2015)
A.G. Campos, R. Cabrera, D.I. Bondar, H. Rabitz, https://doi.org/arXiv:1502.03025 (2015)
A.G. Campos, R. Cabrera, D.I. Bondar, H.A. Rabitz, Phys. Rev. A 90, 034102 (2014)
G. Coddens, From Spinors to Quantum Mechanics (World Scientific, 2015)
E. Deotto, E. Gozzi, D. Mauro, J. Math. Phys. 44, 5902 (2003)
E. Deotto, E. Gozzi, D. Mauro, J. Math. Phys. 44, 5937 (2003)
H. Elze, M. Gyulassy, D. Vasak, Phys. Lett. B 177, 402 (1986)
L.D. Faddeev, L. Khalfin, I. Komarov, in VA Fock-selected works: Quantum mechanics and quantum field theory (CRC Press, 2004), Sect. 29-2
J. Fanchi, Found. Phys. 23, 487 (1993)
C.M. Flores, Classical propagation in the quantum inverted oscillator, https://doi.org/arXiv:1612.01604 (2016)
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)
R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, C. Roos, Nature 463, 68 (2010)
E. Gozzi, C. Pagani, Phys. Rev. Lett. 105, 150604 (2010)
W. Greiner, Classical electrodynamics (Springer-Verlag, 1998)
W. Greiner, Relativistic quantum mechanics: wave equations (Springer-Verlag, 2000)
R. Hakim, Introduction to relativistic statistical mechanics (World Scientific, 2011)
R. Hakim, J. Heyvaerts, Phys. Rev. A 18, 1250 (1978)
R. Hakim, H. Sivak, Ann. Phys. 139, 230 (1982)
M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)
D. Hestenes, J. Math. Phys. 15, 1768 (1974)
D. Hestenes, New foundations for classical mechanics (Springer, 1999)
K. Jacobs, Quantum measurement theory and its applications (Cambridge University Press, 2014)
M. Katsnelson, K. Novoselov, A. Geim, Nat. Phys. 2, 620 (2006)
S.-I. Koda, J. Chem. Phys. 144, 154108 (2016)
B.O. Koopman, Proc. Nat. Acad. Sci. 17, 315 (1931)
K. Kowalski, J. Rembieliński, Ann. Phys. 375, 1 (2016)
H. Krüger, Found. Phys. 23, 1265 (1993)
P. Kustaanheimo, E. Stiefel, J. Math. Bd 218, 27 (1965)
W. Liu, Phys. Chem. Chem. Phys. 14, 35 (2012)
P. Lounesto, in Clifford algebras and spinors (Cambridge University Press, 2001), Vol. 286
S. Mane, Y.M. Shatunov, K. Yokoya, Rep. Prog. Phys. 68, 1997 (2005)
D. Mauro, Topics in Koopman-von Neumann Theory, PhD thesis, Università degli Studi di Trieste, 2002
K. Novoselov, A.K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, A. Firsov, Nature 438, 197 (2005)
J. Otterbach, R. Unanyan, M. Fleischhauer, Phys. Rev. Lett. 102, 063602 (2009)
J. Pedernales, R. Di Candia, D. Ballester, E. Solano, New J. Phys. 15, 055008 (2013)
M. Radonjić, D. Popović, S. Prvanović, N. Burić, Phys. Rev. A 89, 024104 (2014)
J. Schwinger, B.-G. Englert, Quantum Mechanics: Symbolism of Atomic Measurements (Springer Science & Business Media, 2001)
B. Shanahan, A. Chenu, N. Margolus, A. Del Campo, Phys. Rev. Lett. 120, 070401 (2018)
G. Shin, I. Bialynicki-Birula, J. Rafelski, Phys. Rev. A 46, 645 (1992)
G. Shin, J. Rafelski, Phys. Rev. A 48, 1869 (1993)
H. Spohn, Ann. Phys. 282, 420 (2000)
J. Vaishnav, C.W. Clark, Phys. Rev. Lett. 100, 153002 (2008)
S. Varró, J. Javanainen, J. Opt. B: Quant. Semiclassical Opt. 5, S402 (2003)
D. Vasak, M. Gyulassy, H. Elze, Ann. Phys. 173, 462 (1987)
J. von Neumann, Ann. Math. 33, 587 (1932)
J. von Neumann, Ann. Math. 33, 789 (1932)
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)
M. Wen, H. Bauke, C.H. Keitel, Sci. Rep. 6, 31624 (2016)
D.V. Zhdanov, D.I. Bondar, T. Seideman, Quantum friction: environment engineering perspectives, https://doi.org/arXiv:1612.00573 (2016)
D.V. Zhdanov, D.I. Bondar, T. Seideman, Phys. Rev. Lett. 119, 170402 (2017)
D.V. Zhdanov, T. Seideman, Phys. Rev. A 92, 012129 (2015)
W.H. Zurek, Phys. Today 44, 36 (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2018-800075-7