Representation of the spin in the Dirac equation for the electron

Maurizio Dapor

V.A.Induchoodan Menon, Manoj Kumar Rajendran, Vadakkedathu P. N. Nampoori

The authors after showing that a confined helical wave possesses rest mass, electric charge and half spin [1][2] now proves that such a wave is a solution of the Dirac equation. Using the confined helical wave representation of electron, the authors explain in a simple manner why the eigen value for Dirac’s velocity operator is ± c while the average value of the velocity of the wave packet formed by its positive energy solutions is the particle velocity. Further, the confined helical wave representation is seen to be consistent with van der Waerden equation and the zig-zag picture of electron proposed by Pen Rose.

Joe McEvoy, Oscar Sergio Zárate

Loris D’Alessi

In this work, the action of the relativistic electron is derived from the hydrodynamic formulation of the Dirac equation. In particular, in the hydrodynamic scenario, the four-velocity of the electron is regarded as an Eulerian field and the electron spin interacts with the corresponding vorticity field. In the second part of the work, it is shown how the second order Dirac equation can be derived from a principle of Minimum Fisher Information.

Hisaiti Yamasaki

This paper aims, in the theory of Dirac electron, at the deliberate use of the hyperbolic metric of velocity, the effectiveness of which Smorodinskii has recently emphasized in relativity. The metric of velocity is fully used together with a proper spin direction vector, to represent explicitly spinor ψ and vector, etc., of ψ*(matrix)ψ, of free Dirac electron. The 3-vector polarization operator is confirmed to correspond to the proper spin vectro. For the electron in a square well potential, ψ is a product of hyperbolic function of the metric with trigonometric or the Bessel function, etc., and the metric turns out to be imaginary or complex in the case of the bound state or free negative energy state. A classical model of Dirac electron is established concretely by obtaining simple time derivatives of the metric and spin vector, from the covariant equations of a spinning particle under general fields, suggested by Frenkel, Kramers and others. The so-called Thomas factor 1/2 is brought inevitably in the equation of the spin vector itself, regardless of the time average motion. These explicit representations may facilitate the visualization of Dirac electron.

Joe McEvoy, Oscar Zarate