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The Newton-Wigner Position Operator and the Domain of Validity of One Particle Relativistic Theory

Zurab K. Silagadze

Published Jan 1, 1993• 3 citations
OpenAlex

Abstract

Itis suggested that the minimal localization of the relativistic system in terms of the Newton-Wigner mean size can be used as the end criteria for one-particle interpretation of relativistic wave equations.

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Although the concept of localization in non-relativistic quantum mechanics is mathematically well-defined there is no obvious and unambiguous way to generalize it to relativistic quantum theory. After a brief review of localization in quantum mechanics the Newton-Wigner localization scheme is introduced and it is shown by the example of a massive spinless system how it leads to a position operator with seemingly unphysical properties. This motivated the development of several theorems which claim to rule out the existence of localizable particles. A particularly important one of them is presented and its validity discussed. In the last section however it is shown that if localization is considered with respect to spacelike hyperplanes, the properties of the Newton-Wigner position operators are not necessarily unphysical and localizable particles can exist.

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The connection between the (orthodox) single-particle Newton-Wigner position and the Bohm/de Broglie hidden-particle position in the relativistic regime is investigated. The effect of non-positive excursions of the ensemble density for extreme cases of positive-energy waves is easily computed using an integral of the equations of motion developed here and is interpreted in terms of virtual-like pair creation and annihilation beneath the Compton wavelength. A Bohm-theoretic description of the acausal explosion of Newton-Wigner-localized states is presented in detail for spin-0 particles in attempt to relate Bohm/de Broglie and orthodox positions. Spin-1/2 particles are considered as well with the assumption that the convective current component describes ensemble dynamics as with the spin-0 theory. For spin-1/2 particles, a specific second-rank tensor is defined in terms of Dirac spinors for generalizing from simply a quantum potential to a stress tensor required to account for the force of pilot wave on particle. A simple dependence of the stress tensor on a two-component spin pseudovector field is determined for approaching a non-relativistic limit. Consistency is found with an earlier non-relativistic theory of objects with spin.

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Definition of Position and Spin Operators for Massive Elementary Relativistic Systems of Arbitrary Spin

J. A. de Azcárraga, Luis Oliver, J. Pascual

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.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 15, 1973•6 citations
Although the concept of localization in non-relativistic quantum mechanics is mathematically well-defined there is no obvious and unambiguous way to generalize it to relativistic quantum theory. After a brief review of localization in quantum mechanics the Newton-Wigner localization scheme is introduced and it is shown by the example of a massive spinless system how it leads to a position operator with seemingly unphysical properties. This motivated the development of several theorems which claim to rule out the existence of localizable particles. A particularly important one of them is presented and its validity discussed. In the last section however it is shown that if localization is considered with respect to spacelike hyperplanes, the properties of the Newton-Wigner position operators are not necessarily unphysical and localizable particles can exist.
The connection between the (orthodox) single-particle Newton-Wigner position and the Bohm/de Broglie hidden-particle position in the relativistic regime is investigated. The effect of non-positive excursions of the ensemble density for extreme cases of positive-energy waves is easily computed using an integral of the equations of motion developed here and is interpreted in terms of virtual-like pair creation and annihilation beneath the Compton wavelength. A Bohm-theoretic description of the acausal explosion of Newton-Wigner-localized states is presented in detail for spin-0 particles in attempt to relate Bohm/de Broglie and orthodox positions. Spin-1/2 particles are considered as well with the assumption that the convective current component describes ensemble dynamics as with the spin-0 theory. For spin-1/2 particles, a specific second-rank tensor is defined in terms of Dirac spinors for generalizing from simply a quantum potential to a stress tensor required to account for the force of pilot wave on particle. A simple dependence of the stress tensor on a two-component spin pseudovector field is determined for approaching a non-relativistic limit. Consistency is found with an earlier non-relativistic theory of objects with spin.
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.