Stochastic quantization of a Fermi field: Fermions as solitons

Abstract

It is shown that the stochastic quantization of a fermion introducing an anisotropy in the internal space so that this gives rise to two internal helicities corresponding to particle and antiparticle leads us to describe a fermion as a Skyrme soliton. The Skyrme term appears here as a consequence of this internal anisotropy and can be treated as a quantum effect. Some topological properties of this fermionization are then discussed.

Spin transport in tilted electron vortex beams

“…Bandyopadhyay and Hajra in some of their papers [6,7] demonstrated that the quantization of a fermion can be achieved when we introduce an internal variable that appears as a direction vector at each and every space time points in Nelson's stochastic quantization procedure (Nelson 1966(Nelson , 1967 [8,9]. This direction vector essentially corresponds to the spin degree of freedom.…”

The geometric phase and the spin-statistics relation

“…In earlier papers (Bandyopadhyay & Hajra 1987;Hajra & Bandyopadhyay 1991), it was pointed out that, in the framework of Nelson's (1966Nelson's ( , 1967 stochastic mechanics, the quantization of a fermion can be achieved when we introduce an internal variable that appears as a direction vector. Indeed, the relativistic generalization of Nelson's stochastic quantization procedure is attained when we introduce Brownian-motion processes both in the internal space as well as in the external observable space.…”

“…In an earlier paper (Bandyopadhyay & Hajra 1987;Hajra & Bandyopadhyay 1991), it has been pointed out that the relativistic generalization of Nelson's stochastic mechanics, as well as the quantization of a fermion, can be achieved by introducing an internal variable that acts as a direction vector. In fact, this direction vector depicts the internal degrees of freedom representing spin.…”

Electron vortex beams in a magnetic field and spin filter

“…The quantization of a fermion can be achieved [6,7] in the framework of Nelson's stochastic quantization procedure [8,9] when an internal variable is introduced to represent a direction vector, giving rise to the spin degrees of freedom. This effectively gives rise to the SL(2,C) gauge theory and demanding Hermiticity the gauge field belongs to the SU(2) group.…”

“…It has been shown [6,7] earlier that a fermion can be quantized in the framework of Nelson's stochastic quantization procedure [8,9] when an internal variable is introduced to represent a direction vector (vortex line) attached to a space-time point. The direction vector (vortex line), which is topologically equivalent to a magnetic flux line gives rise to spin degree of freedom and essentially represents a spin vortex.…”