Newton-Wigner Operator Research Articles

Localizability, gauge symmetry and Newton–Wigner operator for massless particles

The notion of position operator for massless spinning particles is discussed in some detail. The noncommutativity of coordinates is related to the gauge symmetry following from the freedom in definition of standard state in Wigner’s construction of induced representations of Poincare group. The counterpart of Newton–Wigner operator is discussed. It is explained why the Newton–Wigner construction works only for helicity |λ|≤1∕2.

Localizability in de Sitter space

An analogue of the Newton–Wigner position operator is defined for a massive neutral scalar field in de Sitter space. The one-particle subspace of the theory, consisting of positive-energy solutions of the Klein–Gordon equation selected by the Hadamard condition, is identified with an irreducible representation of the de Sitter group. Postulates of localizability analogous to those written by Wightman for fields in Minkowski space are formulated on it, and a unique solution is shown to exist. Representations in both the principal and the complementary series are considered. A simple expression for the time evolution of the Newton–Wigner operator is presented.

A PHYSICAL REALIZATION OF THE GENERALIZED PT-, C-, AND CPT-SYMMETRIES AND THE POSITION OPERATOR FOR KLEIN–GORDON FIELDS

Generalized parity [Formula: see text], time-reversal [Formula: see text], and charge-conjugation [Formula: see text] operators were initially defined in the study of the pseudo-Hermitian Hamiltonians. We construct a concrete realization of these operators for Klein–Gordon fields and show that in this realization [Formula: see text] and [Formula: see text] operators respectively correspond to the ordinary time-reversal and charge-grading operations. Furthermore, we present a complete description of the quantum mechanics of Klein–Gordon fields that is based on the construction of a Hilbert space with a relativistically invariant, positive-definite, and conserved inner product. In particular we offer a natural construction of a position operator and the corresponding localized and coherent states. The restriction of this position operator to the positive-frequency fields coincides with the Newton–Wigner operator. Our approach does not rely on the conventional restriction to positive-frequency fields. Yet it provides a consistent quantum mechanical description of Klein–Gordon fields with a genuine probabilistic interpretation.

Doubly special quantum and statistical mechanics from quantum κ-Poincaré algebra

Recently Amelino-Camelia proposed a “Doubly Special Relativity” theory with two observer independent scales (of speed and mass) that could replace the standard Special Relativity at energies close to the Planck scale. Such a theory might be a starting point in construction of a quantum theory of spacetime. In this Letter we investigate the quantum and statistical mechanical consequences of such a proposal. We construct the generalized Newton–Wigner operator and find relations between energy/momentum and frequency/wavevector for position eigenstates of this operator. These relations indicate the existence of a minimum length scale. Next we analyze the statistical mechanics of the corresponding systems. We find that depending on the value of a parameter defining the canonical commutational algebra one has to do either with a system with maximal possible temperature or with the one, which in the high temperature limit becomes discrete.

Quantum mechanics in Riemannian spacetime. II. Operators of observables

The formulation of the generally covariant analog of standard (nonrelativistic) quantum mechanics in a general Riemannian spacetime begun in earlier studies of the author is continued with the introduction of asymptotic (with respect toc−2) operators of the spatial position of a spinless particle and of the projection of its momentum onto an arbitrary spacetime direction. The connection between the position operator and the generalization of theV1,3 Newton—Wigner operator is established. It is shown that the projection of the momentum onto the 4-velocity of the frame of reference (the energy operator) is unitarily equivalent to the Hamiltonian in the Schrodinger equation.

Rediscovering the Newton-Wigner operator from a space-time description of quantum field theory

The Newton-Wigner position operator is rediscovered directly from a space-time description of quantum field theory which is based on the actual physical situation where particles travel between emitters and detectors. It avoids methods used in nonrelativistic quantum physics and so-called wave equations.

The point form of quantum dynamics and a 4-vector coordinate operator for a spinless particle

We construct the analog in the quantum mechanics of a free spinless particle, of Dirac’s formula for the generators of space–time translations in his point form of classical dynamics, where one takes as fundamental variables the generators of homogeneous Lorentz transformations and the coordinate 4-vector of the point where the world line of the particle meets one sheet of a two-sheeted hyperboloid in space–time. A 4-vector coordinate operator is determined for such a particle, with commuting Hermitian components. The corresponding observable is the analog of the coordinate 4-vector of the point on the hyperboloid. This operator bears the same relation to such a surface as the Newton–Wigner operator does to an instant.

The four-dimensional position operator in theories with broken conformal symmetry

The four-dimensional position operator obtained previously for conformally invariant field theory can be also used when the conformal symmetry is broken. If one uses the symmetry-breaking scheme based on the notion of complex dimensionalities it is shown that for the scalar free field with mass our position operator coincides with the Newton-Wigner operator. The generalized free fields with the Breit-Wigner mass spectrum are considered from the point of view of localization in time.