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de Broglie–Bohm Formulation of Dirac Fields

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Abstract

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].

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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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Correspondence to Luca Fabbri.

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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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