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Journal of Mathematical Physics

Observables for Massive Relativistic Particles of Arbitrary Spin

J. A. de Azcárraga, Luis Oliver

Published Oct 1, 1969• 3 citations
Publisher

Abstract

The expressions for the dynamical operators corresponding to a Bargmann-Wigner particle of mass κ and arbitrary spin s are given, making use of the Foldy-Wouthuysen and Chakrabarti transformations for the Bargmann-Wigner equations given by Pursey, and Azcárraga and Boya, respectively. The operators selecting the positive and negative energy states are also given in the two representations. The operators for arbitrary spin are studied and a condition for an operator be constant of motion is given. Finally, the Newton-Wigner position operator and those obtained in the Foldy-Wouthuysen and Chakrabarti generalized representations are compared.

DOI
10.1063/1.1664772

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The expressions for the dynamical operators corresponding to a Bargmann-Wigner particle of mass κ and arbitrary spin s are given, making use of the Foldy-Wouthuysen and Chakrabarti transformations for the Bargmann-Wigner equations given by Pursey, and Azcárraga and Boya, respectively. The operators selecting the positive and negative energy states are also given in the two representations. The operators for arbitrary spin are studied and a condition for an operator be constant of motion is given. Finally, the Newton-Wigner position operator and those obtained in the Foldy-Wouthuysen and Chakrabarti generalized representations are compared.
Since the important work of Newton and Wigner on the position operator and the localized states, several methods have been developed in the literature to deal with the problem of finding a position operator for relativistic systems. One of the most relevant has been the method based on the use of "canonical" transformations such as the Foldy-Wouthuysen transformation. In this paper, we strictly consider the Foldy-Wouthuysen transformation as a procedure which allows one to write the Bargmann-Wigner equations in a form which in the nonrelativistic limit leads to the Galilei-invariant Schr\"odinger equations for arbitrary spin. As a result of this interpretation, we derive for the position operator and the localized states of elementary systems the same expressions as Newton and Wigner; the spin operator is also obtained. Finally, the Chakrabarti transformation is considered in the same spirit.
Zitterbewegung of massless particles with an arbitrary spin is analyzed in various representations. Dynamics of the group velocity of a massless particle as a whole and of the corresponding radius vector is determined. This radius vector defines any fixed point of the envelope of the moving wave packet characterizing the particle and its group velocity differs from the group velocities of any points of the wavefront. We consider free massless scalar and Dirac particles, the photon, and massive and massless particles with an arbitrary spin and describe them in different representations. For particles with an arbitrary spin, the generalized Feshbach-Villars representation and the Foldy-Wouthuysen one are used. Zitterbewegung takes place in any representation except for the Foldy-Wouthuysen one. Formulas describing the motion of a ``trembling'' free particle are the same in any representation. In the Foldy-Wouthuysen representation, the operators of the velocity and momentum of a free particle are proportional and Zitterbewegung does not take place. Since the radius vector (position) and velocity operators are the quantum-mechanical counterparts of the classical position and velocity just in the Foldy-Wouthuysen representation, Zitterbewegung is not observable. The same conclusion has been previously made for free massive particles. For relativistic massive particles with spins 0, 1/2, 1 and massless particles with spins 0, 1/2 in arbitrarily strong electromagnetic fields, independent of the external fields Zitterbewegung does not appear in the Foldy-Wouthuysen representation either. This conclusion is made for leading terms in the Hamiltonian proportional to the zero and first powers of the Planck constant and for such terms proportional to ${\ensuremath{\hbar}}^{2}$ which describe contact interactions.
In a recent paper a relativistically covariant Schr\"odinger equation was derived for particles of arbitrary spin sss, locally covariant wave functions without redundant components being used to describe states of a particle. Here we determine the invariant scalar product with respect to which the representation of Poincar\'e transformations on these wave functions is unitary. It is shown that the conventional position and spin operators, not being Hermitian with respect to this scalar product, cannot be observables. New operators which can represent these observables are constructed with the aid of a generalized Foldy-Wouthuysen transformation which is determined explicitly for arbitrary spin.
We find all spin operators for a Dirac particle satisfying the following very general conditions: (i) spin does not convert positive (negative) energy states into negative (positive) energy states, (ii) spin is a pseudovector, and (iii) eigenvalues of the projection of a spin operator in an arbitrary direction are independent of this direction (isotropy condition). We show that there are four such operators and all of them fulfill the standard su(2) Lie algebra commutation relations. Nevertheless, only one of them has a proper nonrelativistic limit and acts in the same way on negative and positive energy states. We show also that this operator is equivalent to the Newton-Wigner spin operator and Foldy-Wouthuysen mean-spin operator. We also discuss another operator proposed in the literature.
Abstract Zitterbewegung of massive and massless scalar bosons and a massive Proca (spin-1) boson is analyzed. The equations describing the evolution of the velocity and position of the scalar boson in the generalized Feshbach–Villars representation and the corresponding equations for the massive Proca particle in the Sakata–Taketani representation are equivalent to each other and to the well-known equations for the Dirac particle. However, Zitterbewegung does not appear in the Foldy–Wouthuysen representation. Since the position and velocity operators in the Foldy–Wouthuysen representation and their transforms to other representations are the quantum-mechanical counterparts of the corresponding classical variables, Zitterbewegung is not observable.