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Physical Review

Position Operators in Relativistic Single-Particle Theories

A. Sankaranarayanan, Robert H. Good

Published Oct 25, 1965• 31 citations
Publisher

Abstract

This paper gives formulas for position operators that apply uniformly for all spins of particle. The three-vector X, of the Newton-Wigner and Foldy type, is treated first. Then it is shown that, although X has complicated Lorentz transformation properties, it is linearly related to a certain four-vector ${Y}_{\ensuremath{\mu}}$ which is built up from the Poincar\'e group generators. The four-vector is the generalization of the classical notion of the component of the position four-vector in the direction perpendicular to the world line of the particle.

DOI
10.1103/physrev.140.b509

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Physical Review•Nov 15, 1963•77 citations
This paper gives formulas for position operators that apply uniformly for all spins of particle. The three-vector X, of the Newton-Wigner and Foldy type, is treated first. Then it is shown that, although X has complicated Lorentz transformation properties, it is linearly related to a certain four-vector Y\ensuremathμ{Y}_{\ensuremath{\mu}}Y\ensuremathμ​ which is built up from the Poincar'e group generators. The four-vector is the generalization of the classical notion of the component of the position four-vector in the direction perpendicular to the world line of the particle.
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 manifestly covariant formalism is developed suitable for a discussion of position operators for relativistic systems. It is shown that the rest-frame center of mass and the frame-dependent center of mass, which were studied previously by Pryce and M\o{}ller, and the local Newton-Wigner operator all transform like the spatial part of a four-vector when appropriately interpreted. The formalism allows an equivalent treatment of the space and time components of a position four-vector operator, and manifestly covariant canonical commutation rules are obtained for the three operators mentioned above. Finally it is suggested that the Newton-Wigner operator be interpreted as a center of spin.
Some general assumptions based on the physical properties of elementary and of composite quantized systems lead in a natural way to general expressions for the position and spin operators of massive particles. All previously proposed position and spin operators, defined in the enveloping algebra of the Poincaré group, appear as special cases. By algebraic arguments a general local position operator is obtained which is proved to coincide under special conditions with the Pryce-Newton-Wigner position operator.
Certain definitions and derivations for the relativistic center of mass are considered and are related to the position operator corresponding to the representation proposed in a previous paper. It is shown that the classical definition of Pryce has indeed a more direct correspondence with our representation than the Foldy-Wouthuysen representation, in spite of the fact that Pryce's particular method of symmetrization leads to the latter. In the process of derivation, the points of view of Synge, Pryce, and Shirokov are compared and the expressions for the velocity and the intrinsic angular momentum corresponding to the different definitions are given. In the Appendix, our operator is compared with that of Newton and Wigner for the case of spin ½.
We consider a massive particle of arbitrary spin and the basis vectors that carry the unitary, irreducible representations of the Poincaré group. From the complex coefficients in normalizable superpositions of these basis vectors, we identify momentum/spin-component probability amplitudes with the same interpretation as in the nonrelativistic theory. We find the relativistic transformations of these amplitudes, which are unitary in that they preserve the modulus-squared of scalar products from frame to frame. Space inversion and time reversal are also treated. We reconsider the Newton- Wigner construction of eigenvectors of position and the position operator. Position/spin-component probability amplitudes are also identified and their relativistic, unitary, transformations derived. Again, space inversion and time reversal are considered. For reference, we show how to construct positive energy solutions of the Klein-Gordon and Dirac equations in terms of probability amplitudes. We find the boost transformation of the position operator in the spinless case and present some results on the relativity of position measurements. We consider issues surrounding the classical concept of causality as it applies in quantum mechanics. We briefly examine the relevance of the results presented here for theories of interaction.
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.