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Progress of Theoretical Physics

Observable of Particles of Spins 0 and 1

P. M. Mathews, A. Sankaranarayanan

Published Jul 1, 1964• 6 citations
Publisher

Abstract

The `Observable' position and other operators of particles of spins 0 and 1 (described by the Sakata-Taketani equations in Hamiltonian form) are investigated. It is shown that these are uniquely determined if it is required that the observable position must satisfy certain physically reasonable criteria, already applied by the authors to Dirac particles. The position operator coincides with that obtained by Case, who, however, assumed its form in a particular representation.

DOI
10.1143/ptp.32.159

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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Feb 15, 1970•27 citations
The `Observable' position and other operators of particles of spins 0 and 1 (described by the Sakata-Taketani equations in Hamiltonian form) are investigated. It is shown that these are uniquely determined if it is required that the observable position must satisfy certain physically reasonable criteria, already applied by the authors to Dirac particles. The position operator coincides with that obtained by Case, who, however, assumed its form in a particular representation.
An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
The problem of defining position and intrinsic spin operators in terms of the generators of the inhomogeneous Lorentz group is considered. These operators are taken to have the properties usually attributed to them in nonrelativistic quantum theory: their commutation relations must have the commonly accepted form. These commutation relations are then shown to define the intrinsic spin uniquely. The position operator is shown to be also essentially uniquely determined. Explicit forms of the spin and position operators for some special representations are exhibited. The relation of these operators to the spin and position operators of the spin-½ Dirac theory in the Foldy-Wouthuysen representation is considered. An Appendix gives a heuristic derivation of the spin operator.
The purpose of this paper is to find to what extent the notion of position in the relativistic quantum mechanics of free elementary systems of nonzero mass is defined (or precluded) by the requirement that the descriptions of a localized state as seen from two observers in different frames of reference should be physically consistent. Neither the existence of a time component of the position operator nor the compatibility of the space components of position is assumed. Hermiticity with respect to the quantum-relativistic scalar product in state-vector space is neither assumed nor rejected. There are three possibilities: (i) The position is represented as usual by a Hermitian operator; this is the point-type position. (ii) Non-Hermitian operators are accepted, and the kkk component of position can represent segments of the kkk axis with length of the order of 1m\frac{1}{m}m1​; when the operator is not only non-Hermitian, but also non-normal, this is the extended-type position. (iii) Non-Hermitian operators are accepted, but the segments are reduced to points; this is the limiting case of the extended-type position. Case (ii) is not studied in this paper. General conditions are found for any spin as regards cases (i) and (iii), and their consequences are obtained for spin 0 and spin \textonehalf{}. The results are the following: There is no position operator in case (i), but in case (iii) the position operator is unique for spin 0 and unique up to a parameter for spin \textonehalf{}. For both spins, in spite of the lack of relativistic Hermiticity, the eigenvalues of position are real, and the velocity is the expected one. The eigenstates of one component of the position are found. The components of the position are compatible with each other for spin 0, and incompatible for spin \textonehalf{}. In the former case, the simultaneous eigenstates of all components are Philip's class-I localized states. Related work is discussed.