Nonlocalisation of electrons in their wave
Dirac's theory has a very nice presentation when we use multivector functions, i.e., mappings \(\psi: M\to \text{Cl}_{1,3}\), where \(M\) is Minkowski spacetime, and \(T_x M\simeq\mathbb{R}^{1,3}\) for all \(x\in M, \mathbb{R}^{1,3}\) being the Minkowski vector space with signature \((1,3)\), and \(\text{Cl}_{1,3}\simeq H(2)\) [\(H(2)\) is the set of \(2\times 2\) quaternionic matrices] is the so-called real spacetime algebra. In this formalism Dirac-spinor fields are represented by some equivalence classes of even multivector functions [cf. \textit{W. A. Rodrigues jun.}, \textit{Q. A. G. Souza} and \textit{J. Vaz jun.}, Spinors fields and superfields as equivalence classes of exterior algebras fields, in R. Ablanowicz and P. Lounesto (eds.), ``Clifford algebras and spinor structures'', pp. 177- 198, Kluwer Acad. Publ., Dordrecht (1995)]. The Dirac equation is represented in this formalism by the so-called Dirac-Hestenes equation (DHE) [\textit{D. Hestenes}, Space-time algebra, New York (1966; Zbl 0183.289)]. The DHE suggests by itself a realistic interpretation of the quantum mechanical formalism [see, e.g., \textit{D. Hestenes} and \textit{A. Weingartenhofer}, ``The Electron'', Kluwer Acad. Publ., Dordrecht (1991)], much in accord with the Bohm and Vigier ideas. Now, the author pretends to have shown that the electron described by the DHE cannot be localized in its wave function. This conclusion is drawn after the analysis of a particular example where one electron is submitted to a constant confined cylindrical magnetic field (i.e., the field is zero outside a cylindrical domain). From the exact solution of the DHE we get the probability density. This, according to the realistic point of view, must be interpreted as a dispersion of reassemblage of the possible trajectories for the electron. It is a fact that, at first sight, the author's example seems to imply that it is very hard to maintain the semiclassical probability interpretation. But according to the reviewers the example is not a realistic one, since there are no physical magnetic fields under the author's condition. So, we don't think that the example implies in the discredit of the hidden variables the theory of Bohm and Vigier. The author's example shows indirectly that in a realistic context, where the contradiction wave/particle can be avoided, the initial conditions for stationary states can only be infered from a quantum theory of ensembles. But, this is a too long issue to be discussed here.
scientific article; zbMATH DE number 778902