Fermions and Topology

Abstract

It has been shown in an earlier paper [Bandyopadhyay and Hajra (1987)] that Nelson’s stochastic quantization procedure can be generalized to have a relativistic framework and the quantization of a Fermi field can be achieved when we take into account Brownian motion processes in the internal space also apart from that in the external space. For the quantization of a Fermi field we have to intorduce an anisotropy in the internal space so that the internal variable appears as a “direction vector”. The opposite orientation of the “direction vector” corresponds to particle and antiparticle. To be equivalent to the Feynman path integral we have to take into accuont compexified space-time when the coordinate is given by z μ = x μ + iξ μ where ξ μ corresponds to the “direction vector” attached to the space-time point x μ [Hajra and Bandyopadhyay (1991)]. Since for quantization we have to introduce Brownian motion process both in the external and internal space, after quantization, for an observational procedure, we can think of the mean position of the particle in the external observable space with a stochastic extension as determined by the internal stochastic variable.