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

Abstract

Abstract. The problem on mapping between two Lagrangian descriptions (using a commuting c-number spinor ψ α or anticommuting pseudovector ξ µ and pseudoscalar ξ 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 Majorana spinors (ψ α , θ α ) (θ α is some auxiliary anticommuting spinor) and a real anticommuting tensor aggregate (S, V µ , * T µν , A µ , 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 /2)(θθ)(ψψ −ψψ), the term (1/e)(θθ)ẋ µ (ψγ µ ψ) that provides a couple of the spinning variable ψ and the particle velocitẏ x µ , and the interaction term (θθ)Q a F a µν (ψσ µν ψ) 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 (Ṡ,V µ , * Ṫ µν ,Ȧ µ ,Ṗ ), is defined. A detailed analysis of the local bosonic symmetry of the Lagrangian with the commuting spinor ψ α , is carried out. A connection of this symmetry with the local SUSY transformation of the Lagrangian containing anticommuting pseudovector and pseudoscalar variables, is considered. The approach of obtaining a supersymmetric Lagrangian in terms of the even ψ α and odd θ α spinors, is offered. *

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

“…Synge [13]. In what follows we need only two vectors h (1) µ and h (2) µ . Let us define an antisymmetric tensor ω µν in the following form:…”

“…The reason of such preferability of the first solution (3.2) -(3.4) is unclear for us. We note also that another problem here, is the determination of an explicit form of the 4-vectors h (1) µ and h (2) µ . It is easy to construct vectors satisfying (5.4) from the analogous spacelike vectors (A.1) and (A.2).…”

“…The orthogonal triad (a (1) , a (2) , a (3) ) satisfying the orthonormality relation As is known the components of these vectors constitute the rotation matrix (a ij ) ≡ (a (j) i ) in the three-dimensional Euclidean space [30]. The other parameterizations of the triad are also possible: in terms of the angles θ, φ which describe the direction of rotation axes and the rotation angle ω or in terms of the Cayley-Klein parameters [30,31].…”

“….) by means of the general relation (I.2.1) (references to formulas in [1] are prefixed by the roman number I), if we preliminarily contract the latter with the auxiliary spinor θ β 1/2 (θθ)ψ α = 1 4…”

“…

The exact solution of a system of bilinear identities derived in the first part of our work [1] for the case of real Grassmann-odd tensor aggregate of the type (S, V µ , * T µν , A µ , P ) is obtained. The consistency of the solution with a corresponding system of bilinear identities including both the tensor variables and their derivatives ( Ṡ, Vµ , * Ṫµν , Ȧµ , Ṗ )

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