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Scientific Herald of Uzhhorod University Series Physics

On the lagrange approach and dynamical variables for the spinor field in Foldy – Wouthuysen canonical representation

І. Ю. Кривський, В. М. Симулик, R.V. Tymchyk

Published Dec 24, 2009• 0 citations
Publisher

Abstract

The relativistic invariant Lagrange approach for the spinor field in Foldy – Wouthuysen representation is constructed. The analogy with classical mechanics for the system with arbitrary number degrees of freedom is used for this construction. The dynamical variables for the spinor field in this representation are found. The conservation laws in terms of quantum mechanical particle-antiparticle momentum-spin amplitudes are presented. In this way the direct physical sense of the conserved quantities is demonstrated. 12 important additional conserved quantities are obtained together with 10 well-known conservation laws for the spinor field, which are the consequences of the Poincare symmetry of the theory. The additional dynamical variables are the consequences of the fact that both orbital and spin parts of the total 4-angular momentum of the spinor field are conserved independently.

DOI
10.24144/2415-8038.2009.25.175-186

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The relativistic invariant Lagrange approach for the spinor field in Foldy – Wouthuysen representation is constructed. The analogy with classical mechanics for the system with arbitrary number degrees of freedom is used for this construction. The dynamical variables for the spinor field in this representation are found. The conservation laws in terms of quantum mechanical particle-antiparticle momentum-spin amplitudes are presented. In this way the direct physical sense of the conserved quantities is demonstrated. 12 important additional conserved quantities are obtained together with 10 well-known conservation laws for the spinor field, which are the consequences of the Poincare symmetry of the theory. The additional dynamical variables are the consequences of the fact that both orbital and spin parts of the total 4-angular momentum of the spinor field are conserved independently.
The problem of the position and spin in relativistic quantum mechanics is analyzed in detail. It is definitively shown that the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one) are quantum-mechanical counterparts of the classical position and spin variables. The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions. The relativistic spin operators are discussed. The spin-orbit interaction does not exist for a free particle if the conventional operators of the orbital angular momentum and the rest-frame spin are used. Alternative definitions of the orbital angular momentum and the spin are based on noncommutative geometry, do not satisfy standard commutation relations, and can allow the spin-orbit interaction.
The detailed consideration of the relativistic canonical quantum-mechanical model of an arbitrary spin-multiplet is given. The group-theoretical analysis of the algebra of experimentally observables physical quantities for the spin 1/2 doublet is presented. It is shown that both the Foldy-Wouthuysen equation for the fermionic spin 1/2 doublet and the Dirac equation in its local representation are the consequences of the relativistic canonical quantum mechanics of the corresponding doublet. The mathematically well-defined consideration on the level of modern axiomatic approaches to the field theory is provided. The hypothesis of the antiparticle negative mass is discussed briefly in the section conclusions.
Covariant four-vector position operators ${X}^{\ensuremath{\mu}}$ are proposed, which form a natural operator generalization of the four-position in relativistic classical mechanics. These ${X}^{\ensuremath{\mu}}$ are defined by specifying commutation relations of the ${X}^{\ensuremath{\mu}}$ with the Poincar\'e generators ${P}^{\ensuremath{\mu}}$ and ${M}^{\ensuremath{\mu}\ensuremath{\nu}}$, and thereby extending the Poincar\'e algebra to a larger algebra whose representations are subsequently found. The ${X}^{\ensuremath{\mu}}$ are shown to be acceptable relativistic position operators within a proper-time dynamical framework. A single Hamiltonian is used for all spins, with a covariant proper-time description. The dynamics is capable of describing the time evolution of states which are not mass eigenstates. An automatic Foldy-Wouthuysen-type diagonalization is achieved for all spin representations, with spin and orbital angular momentum being separately conserved. The connection with the standard theory is made via the specific field equations. In making this connection to the standard theory of half-integral spins, the origin of Zitterbewegung and the nonseparate conservation of spin and angular momentum are clarified. The connection to Maxwell's equations provides an interesting statement of those equations and of gauge invariance. The unphysical representations of negative and imaginary mass and continuous spin are not present in this formalism. Other features are discussed.
A relativistic spin operator cannot be uniquely defined within relativistic quantum mechanics. Previously, different proper relativistic spin operators have been proposed, such as spin operators of the Foldy-Wouthuysen and Pryce type, that both commute with the free-particle Dirac Hamiltonian and represent constants of motion. Here we consider the dynamics of a relativistic electron spin in an external electromagnetic field. We use two different Hamiltonians to derive the corresponding spin dynamics. These two are: (a) the Dirac Hamiltonian in the presence of an external field, and (b) the semirelativistic expansion of the same. Considering the Foldy-Wouthuysen and Pryce spin operators we show that these lead to different spin dynamics in an external electromagnetic field, which offers possibilities to distinguish their action. We find that the dynamics of both spin operators involve spin-dependent and spin-independent terms, however, the Foldy-Wouthuysen spin dynamics additionally accounts for the relativistic particle-antiparticle coupling. We conclude that the Pryce spin operator provides a suitable description of the relativistic spin dynamics in a weak-to-intermediate external field, whereas the Foldy-Wouthuysen spin operator is more suitable in the strong field regime.
The relationship between Foldy–Wouthuysen and Lorentz transformations has been clarified throughout this paper. We propose a generalized FW transformation connecting two particular realizations of the (m,j) representation of the Poincaré group: the covariant realization and a canonical realization acting on relativistic probability amplitudes. Fermions and bosons must be considered separately because the intrinsic parity of the particle–antiparticle systems is (−1)2j. Thus for fermions we can directly take the 2(2j+1) - dimensional Joos–Weinberg covariant realization, while for bosons we must double it to get a reducible 4(2j+1) - dimensional realization where particles and antiparticles lie in orthogonal subspaces. In short, in momentum space the FW transformation is the matrix representing a Lorentz boost times the factor (m/p0)1/2, while in configuration space the FW transformation does not belong to the Poincaré group. The last part of the paper is devoted to getting quantum-mechanical representations of the Galileo group as a contraction of Poincaré group representations by using mathematical methods earlier developed by Mickelsson and Niederle. The relevance of our generalized FW transformation for getting a smooth, well defined, nonrelativistic limit is a remarkable result.
The classical dynamics for a charged spin particle is governed by the Lorentz force equation for orbital motion and by the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation for spin precession. In static and homogeneous electromagnetic fields, it has been shown that the Foldy-Wouthuysen (FW) transform of the Dirac-Pauli Hamiltonian, which describes the relativistic quantum theory for a spin-1/21/21/2 particle, is consistent with the classical Hamiltonian (with both the orbital and spin parts) up to the order of 1/m141/{m}^{14}1/m14 (mmm is the particle's mass) in the low-energy and weak-field limit. In this paper, we extend this correspondence to the case of inhomogeneous fields. Regardless of the field gradient (e.g. Stern-Gerlach) force, the T-BMT equation is unaltered and thus the classical Hamiltonian remains the same, but subtleties arise and need to be clarified. For the relativistic quantum theory, we apply Eriksen's method to obtain the exact FW transformations for the two special cases, which in conjunction strongly suggest that, in the weak-field limit, the FW transformed Dirac-Pauli Hamiltonian (except for the Darwin term) is in agreement with the classical Hamiltonian in a manner that classical variables correspond to quantum operators via a specific Weyl ordering. Meanwhile, the Darwin term is shown to have no classical correspondence.