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

TL;DR

This paper develops a generalized Lagrangian for a classical color spinning particle interacting with non-Abelian gauge and fermion fields, incorporating pseudoclassical spin variables and exploring supersymmetric extensions.

Contribution

It introduces a novel mapping of Dirac spinors into pseudovector and pseudoscalar variables, extending the class of tensor quantities for a consistent description of spin in non-Abelian backgrounds.

Findings

Established a one-to-one correspondence between spinor and tensor variables.

Connected the new Lagrangian with Polyakov's approach and supersymmetric formulations.

Identified necessary conditions for the mapping, including variable doubling or Majorana spinor restriction.

Abstract

A generalization of the Lagrangian introduced earlier in [2011 {\it J. Phys. G} ${\bf 37}$ 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 $\xi_5$.