Strictly Canonical Quantization of a Massless Spinning Particle and a Quaternionic Extension of Pseudoclassical Mechanics

A pseudoclassical model, reproducing, upon quantization, the dynamics of the chiral sectors of the massless spin-1/2 field theory is proposed. The discrete symmetries of the action are studied in details. In order to reproduce the positive and negative chiral sectors of the particle and antiparticle, we promote the algebra of functions on the phase space to a bimodule over the complexified quaternions - biquaternions. The quantization is performed by means of strictly canonical methods (Dirac brackets formalism) reproducing the Dirac equation in the Foldy-Wouthuysen representation in the particle and antiparticle sectors and the Weyl equation in the chiral sectors.