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

TL;DR

This paper derives an exact solution for bilinear identities related to a relativistic spinning particle's tensor variables, and maps interaction terms with background fermionic fields, extending to Dirac spinors and linking with higher-order Lagrangians.

Contribution

It provides a novel exact solution for tensor identities of a spinning particle and constructs a mapping to background fermionic fields, extending previous work to Dirac spinors and higher-order Lagrangians.

Findings

Exact solution for bilinear tensor identities

Mapping of interaction terms with background fermions

Extension to Dirac spinors and higher-order Lagrangians

Abstract

The exact solution of a system of bilinear identities derived in the first part of our work [Nucl.Phys.A 938 (2015) 59] for the case of real Grassmann-odd tensor aggregate of the type $(S,V_{\mu},\!\,^{\ast}T_{\mu \nu},A_{\mu}, P)$ is obtained. The consistency of the solution with 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 considered. The alternative approach in solving of the algebraic system based on introducing complex tensor quantities is discussed. This solution is used in constructing the mapping of the interaction terms of spinning particle with a background (Majorana) fermion field $\Psi^{i}_{{\rm M}\hspace{0.02cm}\alpha}(x)$. A way of the extension of the obtained results for the case of the Dirac spinors $(\psi_{{\rm D}\hspace{0.03cm}\alpha},\theta_{{\rm D}\hspace{0.03cm}\alpha})$ and a background Dirac field $\Psi^{i}_{{\rm D}\hspace{0.03cm}\alpha}(x)$, is suggested. It is shown that for the construction of one-to-one correspondence between the most general spinors and the tensor variables, we need a four-fold increase of the number of the tensor ones. A connection with the higher-order derivative Lagrangians for a point particle and in particular, with the Lagrangian suggested by A.M. Polyakov, is proposed.