Abstract
In this paper we provide using the Clifford and spin-Clifford formalism and some few results of the extensor calculus a derivation of the conservation laws that follow directly from the Dirac–Hestenes equation (DHE) describing a Dirac–Hestenes spinor field (DHSF) in interaction with an external electromagnetic field without using the Lagrangian formalism. In particular, we show that the energy-momentum and total angular momentum extensors of a DHSF is not conserved in spacetime regions permitting the existence of a null electromagnetic field F but a non null electromagnetic potential \(A \). These results have been used together with some others recently obtained (e.g., that the classical relativistic Hamilton–Jacobi equation is equivalent to a DHE satisfied by a particular class of DHSF) to obtain the correct relativistic quantum potential when the Dirac theory is interpreted as a de Broglie–Bohm theory. Some results appearing in the literature on this issue are criticized and the origin of some misconceptions is detailed with a rigorous mathematical analysis.
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References
Bohm, D., Hiley, B.J.: The Undivided Universe—An Ontological Interpretation of Quantum Theory. Routledge, London (1993)
Chen, P., Kleinert, H.: Bohm Trajectories as Approximations to Properly Fluctuating Quantum Trajectories. arXiv:1308.5021v1 [quant-ph]
Chen, P., Kleinert, H.: Deficiencies of Bohmian Trajectories in View of Basic Quantum Principles. Electron. J. Theor. Phys. 13, 1–12 (2016). http://www.ejtp.com/articles/ejtpv13i35.pdf
de Broglie, L.: Non-Linear Wave Mechanics. A Causal Interpretation. Elsevier, Amsterdam (1960)
Englert, B.-G., Scully, M.O., Süssmann, G., Walther, H.: Surrealistic Bohm trajectories. Z. Naturforsch. 47A, 1175 (1992)
Figueiredo, V.L., Rodrigues Jr., W.A., Capelas de Oliveira, E.: Covariant, algebraic and operator spinors (with V. L. Figueiredo and E. C. de Oliveira). Int. J. Theor. Phys. 29, 371–396 (1990)
Geroch, R.: Spinor structure of space-times in general relativity I. J. Math. Phys. 9, 1739–1744 (1968)
Geroch, R.: Spinor structure of space-times in general relativity. II. J. Math. Phys. 11, 343–348 (1970)
Gurtler, R., Hestenes, D.: Consistency in the formulation of the Dirac, Pauli, and Schrödinger theories. J. Math. Phys. 16, 573–584 (1975)
Hestenes, D.: Local observables in Dirac theory. J. Math. Phys. 14, 893–905 (1973)
Hestenes, D.: Observables, operators and complex numbers in classical and quantum physics. J. Math. Phys. 16, 556–572 (1975)
Hestenes, D.: Real Dirac theory. In: Keller, J., Oziewicz, Z. (eds.) Proceedings of the International Conference on Theory of the Electron (Mexico City, September 24–27,1995), Adv. Applied Clifford Algbras, vol. 7(S), pp. 97–144 (1997)
Hiley, B.J., Callaghan, R.E.: The Clifford Algebra Approach to Quantum Mechanics B: The Dirac particle and its Relation to Bohm Approach. arXiv:1011.4033v1 [math-ph]
Hiley, B.J., Callaghan, R.E.: Clifford algebras and the Dirac–Bohm quantum Hamilton–Jacobi equation. Found. Phys. 42, 192–208 (2012)
Holland, P.R.: The Quantum Theory of Motion. University Press, London (1995)
Jones, E., Bach, R., Batelaan, H.: Path integrals, matter waves, and the double slit. Eur. J. Phys. 36, 065048 (2015). http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1001&context=physicsbatelaan
Lasenby, A.N., Doran, C.J.L., Gull, S.F.: A multivector derivative approach to Lagrangian field theory. Found. Phys. 23, 1295–1327 (1993)
Lounesto, P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (1997)
Mahler, D.H., Rozema, L., Fisher, K., Vermeyden, L., Resch, K.J., Wiseman, H.M., Steinberg ,A.: Experimental nonlocal and surreal Bohmian trajectories. Sci. Adv. 2, e1501466 (2016). http://advances.sciencemag.org/content/advances/2/2/e1501466.full.pdf
Moya, A.M.: Lagrangian formalism for multivector fields in spacetime, Ph.D. thesis (in Portuguese), IMECC-UNICAMP (1993)
Rodrigues Jr., W.A., Souza, Q.A.G., Vaz Jr., J.: Lagrangian formulation in the Clifford bundle of Dirac–Hestenes equation on a Riemann–Cartan spacetime. In: Letelier, P., Rodrigues Jr., W.A. (eds.) Gravitation: The Space Time Structure. Proceedings of SILARG VIII (Águas de Lindóia, Brazil, July 25–39, 1993), pp. 534–543. World Sci. Publ, Singapore (1994)
Fernández, V.V., Moya, A.M., Rodrigues Jr., W.A.: Special Supplement of AACA on extensor calculus. Adv. Appl. Clifford Algebras 11(Suppl 3), 1–103 (2001)
Mosna, R.A., Rodrigues Jr., W.A.: The bundles of algebraic and Dirac–Hestenes spinor fields. J. Math. Phys. 45, 2945–2966 (2004)
Rodrigues Jr., W.A.: Algebraic and Dirac–Hestenes spinors and spinor fields. J. Math. Phys. 45, 2908–2994 (2004)
Rodrigues Jr., W.A., de Oliveira, Capelas, E.: The Many Faces of Maxwell Dirac and Einstein Equations. A Clifford Bundle Approach, Lecture Notes in Physics 922 (second edition revised and enlarged). Springer, Heidelberg (first published as. Lecture Notes in Physics 722, 2007) (2016)
Rodrigues Jr., W.A., Wainer, S.A.: The Relativistic Hamilton–Jacobi Equation for a Massive, Charged and Spinning Particle, its Equivalent Dirac Equation and the de Broglie–Bohm Theory. Adv. Applied Clifford Algebras (2017) (to appear). arXiv:1610.03310v1 [math-ph]
Sawant, R., Samuel, J., Sinha, A., Sinha, S., Sinha, U.: Non-Classical Paths in Interference Experiments. arXiv:1308.2022v2 [quant-ph]
Vaz Jr., J., da Rocha, R.: An Introduction to Clifford Algebras and Spinors. Oxford University Press, Oxford (2016)
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Communicated by Rafał Abłamowicz
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Moya, A.M., Rodrigues, W.A. & Wainer, S.A. The Dirac–Hestenes Equation and its Relation with the Relativistic de Broglie–Bohm Theory. Adv. Appl. Clifford Algebras 27, 2639–2657 (2017). https://doi.org/10.1007/s00006-017-0779-x
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DOI: https://doi.org/10.1007/s00006-017-0779-x