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Physical Review A

Lorentz boost of the Newton-Wigner-Pryce position operator

Adam H. Monahan, Malcolm McMillan

Published Oct 1, 1997• 7 citations
Publisher

Abstract

The effect of a canonical Lorentz boost on the Newton-Wigner-Pryce position operator of a spin zero system is calculated. The resulting equations differ significantly from those corresponding to the Lorentz boost of a position coordinate in classical special relativity. The components of the boosted Newton-Wigner-Pryce position operator are interpreted in terms of a classical analog. It is seen that the transformed operator corresponds not to the coordinates of the same event in two reference frames, but to the coordinates of the intersection of the worldline of the system's center of momentum with hyperplanes of the same coordinate time in the two frames.

DOI
10.1103/physreva.56.2563

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The effect of a canonical Lorentz boost on the Newton-Wigner-Pryce position operator of a spin zero system is calculated. The resulting equations differ significantly from those corresponding to the Lorentz boost of a position coordinate in classical special relativity. The components of the boosted Newton-Wigner-Pryce position operator are interpreted in terms of a classical analog. It is seen that the transformed operator corresponds not to the coordinates of the same event in two reference frames, but to the coordinates of the intersection of the worldline of the system's center of momentum with hyperplanes of the same coordinate time in the two frames.
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.
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.
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.
A relativistic spin operator is to be the difference between the total and orbital angular momentum. As the unique position operator for a localized state, the remarkable Newton-Wigner position operator, which has all desirable commutation relations as a position operator, can give a proper spin operator. Historically important three spin operators respectively proposed by Bogolubov et al., Pryce, and Foldy-Woutheysen are investigated to manifest a corresponding spin operator to the Newton-Wigner position operator. We clarify a unique spin operator in relativistic quantum mechanics described by the Dirac Hamiltonian.
Abstract Quantum reference frame as in the example of observing a quantum particle from another has been a topic of much recent interest. Quantum spatial translations, quantum rotations, and quantum Lorentz boosts in the sense have been studied to some extent. The article aims at using a consistent formulation of all that to give a full picture of what would be the symmetry of Special Quantum Relativity as special relativity as seen from quantum particles. We first look at a direct formulation of the mathematical quantum analog of any classical one-parameter Lie group of such transformations suggested by our Heisenberg picture thinking of quantum mechanics. A quantum rotation and a quantum Lorentz boost are presented. Analysis of the compositions of two such transformations shows unnatural results with various problems, especially with their commutators. From a physical point of view, we argue for a modified formulation. A quantum reference frame transformation, as the description of firstly the spacetime position of an object, idealized as a particle, as seen from another has to involve a full set of four-coordinate quantum quantities instead of one in a single coordinate only. Unlike the classical case, we illustrate that the analog of a change by the zero classical coordinate values has nontrivial effects in the transformation. A consistent full picture of the symmetry is then obtained. Results that could be interpreted essentially as the quantum analogs of the translation of a single coordinate observable, including angular ones for a rotation or a boost, can be retrieved from that but only as approximations. Under the traditional perspective of relativity symmetry being about spacetime reference frames, our task is then completed. We further look at the quantum version of Heisenberg–Weyl symmetry and perspectives on its relation with the subject matter and argue for the necessity of our background noncommutative geometric perspective for a conceptual consistent story.
A relativistic spin operator is the difference between the total and the orbital angular momentum. As the unique position operator for a localized state, the remarkable Newton-Wigner position operator, which has all the desirable commutation relations of a position operator, can give a proper spin operator. Historically, the three important spin operators proposed by Bogolubov et al. , Pryce, and Foldy-Woutheysen, respectively were investigated to manifest a spin operator corresponding to the Newton-Wigner position operator. We clarify a unique spin operator in relativistic quantum mechanics, which can be described by using the Dirac Hamiltonian.