Equations of motion for a classical color particle in background non-Abelian fermionic and bosonic fields: Inclusion of pseudoclassical spin

Abstract

A generalization of the Lagrangian introduced earlier in [2011 {\it J. Phys. G} $37$ 105001] for a classical color spinning particle interacting with background non-Abelian gauge and fermion fields for purpose of considering a change in time of the spin particle degree of freedom, is suggested. In the case under consideration the spin degree of freedom is described by a commuting $c$-number Dirac spinor ${\psi }_{\alpha }$. A mapping of this spinor into new variables: anticommuting pseudovector ${\xi }_{\mu }$ and pseudoscalar ${\xi }_5$ commonly used in a description of the spin degree of freedom of a massive spin-1/2 particle, is constructed. An analysis of one-to-one correspondence of this mapping is given. It is shown that for the one-to-one correspondence it is necessary to extend a class of real tensor quantities including besides ${\xi }_{\mu }$ and ${\xi }_5$, also odd vector ${\hat{\xi }}_{\mu }$, scalar ${\hat{\xi }}_5$, and (dual) pseudotensor ${}^{\ast }{\zeta }_{\mu \nu }$. In addition, it is shown that it is necessary either to restrict a class of the initial spinor ${\psi }_{\alpha }$ to Majorana one or to double the number of variables in the tensor aggregate $({\xi }_{\mu },{\xi }_5,{,}^{\ast }{\zeta }_{\mu \nu },{\hat{\xi }}_{\mu },{\hat{\xi }}_5)$. Various special cases of the desired mapping are considered. In particular, a connection with the Lagrangian suggested by A.M. Polyakov, is studied. It is also offered the way of obtaining the supersymmetric Lagrangian in terms of the even ${\psi }_{\alpha }$ and odd ${\theta }_{\alpha }$ spinors. The map of the Lagrandian leads to local SUSY Lagrangian in terms of ${\xi }_{\mu }$ and ${\xi }_5$.

Comments: 41 pages