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Annalen der Physik

On the Newton – Wigner Coordinate for a Scalar Particle

M. I. Shirokov

Published Jan 1, 1962• 5 citations
Publisher

Abstract

Abstract In 1949 Newton and Wigner have obtained a non‐covariant coordinate q of the scalar particle on the basis of some postulates. It is shown that the ordinary coordinate x satisfies all physical and mathematical requirements underlying their postulates if the definition (3) (see below) of the probability density in the x‐space is used. Therefore from the point of view of these requirements the coordinate x is in no way worse than the coordinate q. Only the requirement that the position operator must be hermitian is crucial for rejecting the x‐coordinate.

DOI
10.1002/andp.19624650105

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Abstract In 1949 Newton and Wigner have obtained a non‐covariant coordinate q of the scalar particle on the basis of some postulates. It is shown that the ordinary coordinate x satisfies all physical and mathematical requirements underlying their postulates if the definition (3) (see below) of the probability density in the x‐space is used. Therefore from the point of view of these requirements the coordinate x is in no way worse than the coordinate q. Only the requirement that the position operator must be hermitian is crucial for rejecting the x‐coordinate.
We put forward an interpretation of scalar quantum field theory as relativistic quantum mechanics by curing well known problems related to locality. A probabilistic interpretation of quantum field theory similar to quantum mechanics is difficult if particle localization is defined using the Newton-Wigner position operator as it is non-local and non-covariant. An alternative bilinear covariant position operator is discussed which incorporates a time operator that can be exponentiated to a unitary operator. Moreover, it satisfies an algebra that unifies special relativity and quantum mechanics and has the same form for particles with spin. Higher power position operators are derived which yield Heisenberg's uncertainty relations. Our ideas are illustrated with a relativistic wave function whose probability density can be perfectly localized.
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.
A description of scalar charged particles, based on the Feshbach-Villars formalism, is proposed. Particles are described by an object that is a Wigner function in usual coordinates and momenta and a density matrix in the charge variable. It is possible to introduce the usual Wigner function for a large class of dynamical variables. Such an approach explicitly contains a measuring device frame. From our point of view it corresponds to the Copenhagen interpretation of quantum mechanics. It is shown how physical properties of such particles depend on the definition of the coordinate operator. The evolution equation for the Wigner function of a single-charge state in the classical limit coincides with the Liouville equation. Localization peculiarities manifest themselves in specific constraints on possible initial conditions.