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Physics of Particles and Nuclei Letters

The Wigner-Newton state localization and the commutativity of the configuration space

R. M. Mir‐Kasimov

Published Oct 1, 2006• 5 citations
Publisher

Abstract

The commutativity requirement of the components of the position operator is implicitly included in the set of the Wigner-Newton postulates defining the localized state of the relativistic particle. By omitting this constraint, we obtain a more general concept of localization based on the representation of the Poincare group.
DOI
10.1134/s1547477106050025

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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.
The kind of operator algebra familiar in ordinary quantum mechanics is used to show formally that in an irreducible unitary representation of the Poincaré group for positive mass, the Newton–Wigner position operator is the only Hermitian operator with commuting components that transforms as a position operator should for translations, rotations, and time reversal and does not behave in a singular way that contradicts what can be learned from Lorentz transformations in the nonrelativistic limit.
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.
Representations of the proper Poincaré group on spaces of functions of the phase space variables q and p have been studied, by starting with the rotation subgroup SO(3), and carrying out an inducing procedure. A complete decomposition theory, for the physically interesting representations, have been developed. Explicit computations of the forms of the generators and certain systems of imprimitivity have been carried out. The role of the Newton–Wigner position and momentum operators, in the setting of phase space representations, has been discussed. The work is a continuation of an earlier paper on phase space representations, inspired ultimately by the possibility of developing a consistent relativistic quantum mechanics for single particles on phase space.
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.
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.