Correspondence between classical and pseudoclassical descriptions of the spin degree of freedom of a relativistic particle in an external field

Abstract

A generalization of the Lagrangian description of a relativistic color charged spinning particle interacting with external non-Abelian gauge and fermion fields, previously introduced by the authors, is proposed which takes into account the time-varying spin degree of freedom of the particle. For the case under consideration, the degree of freedom is described by a classical commuting Dirac spinor ψα . This spinor has been mapped to new (anticommuting) variables: a pseudovector ξμ , a pseudoscalar ξ5, a pseudotensor *ζμν, a vector $ {{\overset{\scriptscriptstyle\frown}{\xi}}_{\mu}} $ ,and a scalar $ {{\overset{\scriptscriptstyle\frown}{\xi}}_5} $ . The first two of them are usually used in a pseudoclassical description of the spin degree of freedom of a massive spin 1/2 particle. The mapping $ \left( {{\psi_{\alpha}},\;{{\bar{\psi}}_{\alpha}}} \right)\Leftrightarrow \left( {{\xi_{\mu}},\;{\xi_5},\ldots } \right) $ has been examined for oneoneness. It has been shown that to attain one-oneness, it is necessary either to narrow the class of the initial spinor ψα to that of Majorana particles or to double the number of variables in the set of tensors (ξμ ξ5 *ζμν $ {{\overset{\scriptscriptstyle\frown}{\xi}}_{\mu}} $ $ {{\overset{\scriptscriptstyle\frown}{\xi}}_5} $ ).