Mapping between the classical and pseudoclassical models of a relativistic spinning particle in external bosonic and fermionic fields. I

The problem on mapping between two Lagrangian descriptions (using a commuting $c$-number spinor $\psi_{\alpha}$ or anticommuting pseudovector $\xi_{\mu}$ and pseudoscalar $\xi_5$ variables) of the spin degrees of freedom of a color spinning massive particle interacting with background non-Abelian gauge field, is considered. A general analysis of the mapping between a pair of {\it Majorana} spinors $(\psi_{\alpha},\theta_{\alpha})$ ($\theta_{\alpha}$ is some auxiliary anticommuting spinor) and a real anticommuting tensor aggregate $(S,V_{\mu},\!\,^{\ast}T_{\mu \nu},A_{\mu},P)$, is presented. A complete system of bilinear relations between the tensor quantities, is obtained. The analysis we have given is used for the above problem of the equivalence of two different ways of describing the spin degrees of freedom of the relativistic particle. The mapping of the kinetic term $(i\hbar/2)(\bar{\theta}\theta) (\dot{\bar{\psi}}\psi-\bar{\psi}\dot{\psi})$, the term $(1/e)(\bar{\theta}\theta)\dot{x}_{\mu}(\bar{\psi}\gamma^{\mu}\psi)$ that provides a couple of the spinning variable $\psi$ and the particle velocity $\dot{x}_{\mu}$, and the interaction term $\hbar(\bar{\theta}\theta)Q^{a}F_{\mu\nu}^a (\bar{\psi}\sigma^{\mu\nu}\psi)$ with an external non-Abelian gauge field, are considered in detail. In the former case a corresponding system of bilinear identities including both the tensor variables and their derivatives $(\dot{S},\dot{V}_{\mu},\!\,^{\ast}\dot{T}_{\mu\nu},\dot{A}_{\mu}, \dot{P})$, is defined. A detailed analysis of the local bosonic symmetry of the Lagrangian with the commuting spinor $\psi_{\alpha}$, is carried out. A connection of this symmetry with the local SUSY transformation of the Lagrangian containing anticommuting pseudovector and pseudoscalar variables, is considered.