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de Broglie–Bohm Formulation of Dirac Fields

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We present the theory of Dirac spinors in the formulation given by Bohm on the idea of de Broglie: the quantum relativistic matter field is equivalently re-written as a special type of classical fluid and in this formulation it is shown how a relativistic environment can host the non-local aspects of the above-mentioned hidden-variables theory. Sketches for extensions are given at last.

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Notes

  1. As a matter of fact, this is an instance in which, like in many other cases, the chronological order does not follow the logical order: in fact the de Broglie-Bohm theory was set before the results of Bell, with Bell proving his theorem on the guess that the dBB non-locality could be a general feature of quantum mechanics.

  2. This is usually denoted as a gamma with index five, but it has no sense in the space-time and so we use a notation with no index.

  3. Notice that such a polar decomposition is always possible so long as \(\Theta\) and \(\Phi\) are not identically zero as it generally happens. In the specific circumstance in which \(\Theta ^{2}\!+\!\Phi ^{2}\!\equiv \!0\) we would still have a polar decomposition [31]. However, in this case the fields would be pure Goldstone states [32]. Therefore, we are not going to consider this singular case in the following.

  4. We assume the reader familiar with the Pauli matrices \(\vec {\varvec{\sigma }}\) above.

  5. The non-quantum limit would be implemented by \(\hbar \!\rightarrow \!0\) which is hidden in our presentation with natural units. If we were not to assume them, \(\hbar \!\rightarrow \!0\) would clearly give the spinless condition.

  6. In fact, [27] even pre-dates [3].

References

  1. Pusey, M.F., Barrett, J., Rudolph, T.: On the reality of the quantum state. Nat. Phys. 8, 476 (2012)

    Article  Google Scholar 

  2. Einstein, A., Podolsky, B., Rosen, N.: Can quantum: mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)

    Article  ADS  MATH  Google Scholar 

  3. John Stewart Bell: On the Einstein Podolsky Rosen paradox. Phys. Phys. Fizika 1, 195 (1964)

    Article  MathSciNet  Google Scholar 

  4. Bohm, D.: A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. Phys. Rev. 85, 166 (1952)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Dürr, D., Munch-Berndl, K.: A hypersurface Bohm–Dirac theory. Phys. Rev. A 60, 2729 (1999)

    Article  ADS  Google Scholar 

  6. Dürr, D., Goldstein, S., Norsen, T., Struyve, W., Zanghì, N.: Can Bohmian mechanics be made relativistic? Proc. R. Soc. Lond. A 470, 20130699 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  7. Bohm, D., Lochak, G., Vigier, J. P.: “Interprétation de l’équation de Dirac comme approximation linéaire de l’équation d’une onde se propageant dans un fluide tourbillonnaire en agitation chaotique du type éther de Dirac”. Séminaire L. de Broglie 25, n. 8 (1955)

  8. Bohm, D., Albwachs, F. H., Lochak, G., Vigier, J. P.: “Interprétation de l’équation de Dirac comme approximation linéaire de l’équation d’une onde se propageant dans un fluide tourbillonnaire en agitation chaotique du type éther de Dirac”, Séminaire L. de Broglie 25, n. 15 (1955)

  9. Takabayasi, T.: Variational principle in the hydrodynamical formulation of the Dirac field. Phys. Rev. 102, 297 (1955)

    Article  ADS  MathSciNet  Google Scholar 

  10. Takabayasi, T.: “Relativistic hydrodynamics of the Dirac matter”, Séminaire L. de Broglie 26, n. 3 (1956)

  11. Takabayasi, T.: Relativistic hydrodynamics equivalent to the dirac equation. Prog. Theor. Phys. 13, 222 (1955)

    Article  ADS  Google Scholar 

  12. Takabayasi, T.: Hydrodynamical description of the dirac equation. Nuovo Cimento 3, 233 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  13. Takabayasi, Takehiko: On the formulation of quantum mechanics associated with classical pictures. Prog. Theor. Phys. 8, 143 (1952)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Bohm, D.: Comments on an article of Takabayasi concerning the formulation of quantum mechanics with classical pictures. Prog. Theor. Phys. 9, 273 (1953)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. de Broglie, L.: Sur l’introduction des idées d’onde-pilote et de double solution dans la théorie de l’électron de Dirac. Comp. Rend. Acad. Sci. 235, 557 (1952)

    MATH  Google Scholar 

  16. Vigier, J.P.: Forces s’exerçant sur les lignes de courant usuelles des particules de spin \(0\), \(1/2\) et \(1\) en théorie de l’onde-pilote. Comp. Rend. Acad. Sci. 235, 1107 (1952)

    MATH  Google Scholar 

  17. Bohm, D., Schiller, R., Tiomno, J.: A causal interpretation of the Pauli equation (A and B). Nuovo Cimento 1, 48–67 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  18. Holland, P.R.: The Dirac equation in the de Broglie–Bohm theory of motion. Found. Phys. 22, 1287 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  19. Gasperini, M.: Theory of Gravitational Interactions. Springer Nature, Berlin (2017)

    Book  MATH  Google Scholar 

  20. Fabbri, Luca: Spinors in polar form. Eur. Phys. J. Plus 136, 354 (2021)

    Article  Google Scholar 

  21. Jakobi, G., Lochak, G.: Introduction des parametres relativistes de Cayley–Klein dans la representation hydrodynamique de l’equation de Dirac. Comp. Rend. Acad. Sci. 243, 234 (1956)

    Google Scholar 

  22. Fabbri, Luca: Covariant inertial forces for spinors. Eur. Phys. J. C 78, 783 (2018)

    Article  ADS  Google Scholar 

  23. Fabbri, Luca: Polar solutions with tensorial connection of the spinor equation. Eur. Phys. J. C 79, 188 (2019)

    Article  ADS  Google Scholar 

  24. Yvon, J.: Équations de Dirac–Madelung. J. Phys. Radium 1, 18 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fabbri, L.: Polar analysis of the Dirac equation through dimensions. Eur. Phys. J. Plus 134, 185 (2019)

    Article  Google Scholar 

  26. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden variable theories. Phys. Rev. Lett. 23, 880 (1969)

    Article  ADS  MATH  Google Scholar 

  27. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447 (1966)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Kochen, S., Specker, E.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59 (1968)

    MathSciNet  MATH  Google Scholar 

  29. Conway, J., Kochen, S.: The free will theorem. Found. Phys. 36, 1441 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Fabbri, L.: Goldstone states as non-local hidden variables. Universe 8, 277 (2022)

    Article  ADS  Google Scholar 

  31. Fabbri, Luca, Rogerio, Rodolfo José Bueno.: Polar form of spinor fields from regular to singular: the flag-dipoles. Eur. Phys. J. C 80, 880 (2020)

    Article  ADS  Google Scholar 

  32. Fabbri, Luca: Weyl and Majorana spinors as pure goldstone bosons. Adv. Appl. Clifford Algebras 32, 3 (2022)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I wish to thank Dr. Marie-Hélène Genest for the useful discussions that we have had on this subject.

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  1. DIME, Sez. Metodi e Modelli Matematici, Università di Genova, Via all’Opera Pia 15, 16145, Genova, Italy

    Luca Fabbri

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LF did all calculations, wrote the manuscript text and reviewed the final version

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Fabbri, L. de Broglie–Bohm Formulation of Dirac Fields. Found Phys 52, 116 (2022). https://doi.org/10.1007/s10701-022-00641-2

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