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Zitterbewegung in quantum mechanics. (English) Zbl 1193.81020
This interesting paper contains a research program investigating implications of the real Dirac equation for the interpretation and extension of quantum mechanics. The possibility that zitterbewegung opens a window to particle substructure in quantum mechanics is explored by constructing a particle model with some features inherent in the well known Dirac equation. The author discusses a self-contained dynamical model of the electron as a lightlike particle with helical zitterbewegung and electromagnetic interactions. This model admits periodic solutions with quantized energy, and the correct magnetic moment is generated by charge circulation. It attributes to the electron an electric dipole moment rotating with ultrahigh frequency. The possibility of observing this directly as a resonance in electron channeling is analyzed in detail. Correspondence with the Dirac equation is discussed. A modification of the Dirac equation is suggested to incorporate the rotating dipole moment. An interesting conclusion is that the relation of the zitterbewegung particle model to the Dirac equation can be considered from two different perspectives. On the one hand, it can be regarded as a “quasiclassical” approximation that embodies structural features of the Dirac equation in a convenient form. On the other hand, it can be regarded as formulating fundamental properties of the electron that are manifested in the Dirac equation in some kind of average form.

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
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[1] Hestenes, D.: Spacetime physics with geometric algebra. Am. J. Phys. 71, 691–704 (2003) · doi:10.1119/1.1571836
[2] Hestenes, D.: Mysteries and insights of Dirac theory. Ann. Fond. Louis Broglie 28, 390–408 (2003) · Zbl 1329.81221
[3] Hestenes, D.: Real Dirac theory. In: The Theory of the Electron. Advances in Applied Clifford Algebras, vol. 7, pp. 97–144. UNAM, Mexico (1997). Unfortunately, the published text is marred by annoying font substitutions
[4] Bender, D., et al.: Tests of QED at 29 GeV center-of-mass energy. Phys. Rev. D 30, 515 (1984) · doi:10.1103/PhysRevD.30.515
[5] Bohm, D., Hiley, B.: The Undivided Universe, an Ontological Interpretation of Quantum Theory, 2nd edn. Routledge, London (1993), p. 220 · Zbl 0990.81503
[6] Schroedinger, E.: Über die kräftfreie bewegung in der relativistischen quantenmechanik. Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. 24(418) (1930)
[7] Frenkel, J.: Die electrodynamik des rotierenden electrons. Z. Phys. 36, 243–262 (1926) · JFM 52.0960.06
[8] Thomas, L.H.: Kinematics of an electron with an axis. Philos. Mag. 3, 1–22 (1927) · JFM 53.0875.07
[9] Mathisson, M.: Neue mechanik materieller systeme. Acta Phys. Pol. 6, 163–200 (1937) · JFM 63.1262.01
[10] Weyssenhoff, J.: On two relativistic models of Dirac’s electron. Acta Phys. Pol. 9, 47–53 (1947)
[11] Corben, H.: Classical and Quantum Theory of Spinning Particles, 2nd edn. Holden-Day, San Francisco (1948) · Zbl 0035.13002
[12] Gürsey, F.: Relativistic kinematics of a classical point particle in spinor form. Nuovo Cimento 5, 785–809 (1957)
[13] Rivas, M.: Kinematical Theory of Spinning Particles. Kluwer, Dordrecht (2001) · Zbl 1053.81002
[14] Bargman, V., Michel, L., Telegdi, V.: Precession of the polarization of particles moving in a homogeneous electromagnetic field. Phys. Rev. Lett. 2, 435–437 (1959) · doi:10.1103/PhysRevLett.2.435
[15] Costa de Beauregard, O.: Noncollinearity of velocity and momentum of spinning particles. Found. Phys. 2, 111–126 (1972) · doi:10.1007/BF00708496
[16] Doran, C., Lasenby, A., Gull, S., Somaroo, S., Challinor, A.: Spacetime algebra and electron physics. Adv. Imaging Electron Phys. 95, 271 (1996) · doi:10.1016/S1076-5670(08)70158-7
[17] Rivas, M.: Is there a classical spin contribution to the tunnel effect? Phys. Lett. A 248, 279 (1998) · doi:10.1016/S0375-9601(98)00703-8
[18] Weyssenhoff, J.: Relativistically invariant homogeneous canonical formalism with higher derivatives. Acta Phys. Pol. 11, 49–70 (1951) · Zbl 0044.43804
[19] Krüger, H.: The electron as a self-interacting lightlike point charge: Classification of lightlike curves in spacetime under the group of SO(1,3) motions. In: The Theory of the Electron. Advances in Applied Clifford Algebras, vol. 7, pp. 145–162. UNAM, Mexico (1997) · Zbl 1221.81091
[20] Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003) · Zbl 1078.53001
[21] Proca, A.: Mechanique du point. J. Phys. Radium 15, 15–72 (1954) · Zbl 0055.22008 · doi:10.1051/jphysrad:0195400150206500
[22] Barut, A.O., Zanghi, N.: Classical model of the Dirac electron. Phys. Rev. Lett. 52, 2009–2012 (1984) · doi:10.1103/PhysRevLett.52.2009
[23] Gull, S.F.: Charged particles at potential steps. In: Hestenes, D., Weingartshafer, A. (eds.) The Electron, pp. 37–48. Kluwer, Dordrecht (1991)
[24] Lasenby, A., Doran, C., Gull, S.: A multivector derivative approach to Lagrangian field theory. Found. Phys. 23, 1295–1327 (1993) · doi:10.1007/BF01883781
[25] Schwinberg, P., Van Dyck, R. Jr., Dehmelt, H.: Electron magnetic moment from geonium spectra: Early experiments and background concepts. Phys. Rev. D 34, 722–736 (1986) · doi:10.1103/PhysRevD.34.722
[26] Hestenes, D.: Spin and uncertainty in the interpretation of quantum mechanics. Am. J. Phys. 47, 399–415 (1979) · doi:10.1119/1.11806
[27] Yoshioka, D.: The Quantum Hall Effect. Springer, Berlin (2002) · Zbl 1140.81024
[28] Bjorken, J., Drell, S.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1964) · Zbl 0184.54201
[29] Gouanère, M., Spighel, M., Cue, N., Gaillard, M.J., Genre, R., Kirsh, R.G., Poizat, J.C., Remillieux, J., Catillon, P., Roussel, L.: Experimental observation compatible with the particle internal clock. Ann. Fond. Louis Broglie 30, 109–115 (2005)
[30] Gemmell, D.: Channeling and related effects in the motion of charged particles through crystals. Rev. Mod. Phys. 46, 129–227 (1974) · doi:10.1103/RevModPhys.46.129
[31] Lindhard, J.: Influence of crystal lattice on motion of energetic charged particles. Mat. Fys. Medd. Dan. Vid. Selsk. 34(14), 1–64 (1974)
[32] Morse, P., Feshbach, H.: Methods of Theoretical Physics, vol. I. McGraw-Hill, New York (1953) · Zbl 0051.40603
[33] Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1972) · Zbl 0543.33001
[34] Landau, L., Lifshitz, E.: Mechanics. Pergamon, Oxford (1969) (p. 80)
[35] Hestenes, D.: Gauge gravity and electroweak theory. In: Jantzen, R., Kleinert, H., Ruffini, R. (eds.) Proceedings of the Eleventh Marcel Grossmann Meeting. World Scientific, Singapore (2007) · Zbl 1161.83006
[36] Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. Lond. A 117, 610 (1928) · JFM 54.0973.01 · doi:10.1098/rspa.1928.0023
[37] Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, Oxford (1957), pp. 261–267
[38] Greiner, G.: Relativistic Quantum Mechanics, 4th edn. Springer, Berlin (1990), pp. 91–93, 233–236
[39] Recami, E., Salesi, G.: Kinematics and hydrodynamics of spinning particles. Phys. Rev. A 57, 98–105 (1998) · Zbl 1221.81102 · doi:10.1103/PhysRevA.57.98
[40] de Broglie, L.: Ondes et quanta. C. R. Math. 177, 507–510 (1923)
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