In parts II and III of the present series of papers, the localization problem was studied for the cases of relativistic nonzero mass (spin 0, $\frac{1}{2}$ ) and zero mass (spin 0, $\frac{1}{2}$ , and 1). With a set of postulates, of which the basic one is the imposition of Lorentz invariance of localization (LIL) (which is a self-consistency requirement), it was possible to define the position operator uniquely (except for a constant in the spin- $\frac{1}{2}$ , nonzero-mass case). For the determination of the position operators it was enough to consider only very special cases of homogeneous Lorentz transformations. One of the purposes of the present paper is to consider the effect of arbitrary inhomogeneous continuous Lorentz transformations in order to obtain general Poincar\'e-invariant localization (GPIL). Our main result will be that localization is LIL if and only if it is GPIL. In II and III, LIL was obtained in spite of the fact that the position operators were not 4-vector operators but only 3-vector operators. In this paper, following Fleming, we construct formally covariant position operators from the 3-vector operators. We prove that their eigenfunctions are just the localized states defined in the GPIL sense.In parts II and III of the present series of papers, the localization problem was studied for the cases of relativistic nonzero mass (spin 0, $\frac{1}{2}$) and zero mass (spin 0, $\frac{1}{2}$, and 1). With a set of postulates, of which the basic one is the imposition of Lorentz invariance of localization (LIL) (which is a self-consistency requirement), it was possible to define the position operator uniquely (except for a constant in the spin-$\frac{1}{2}$, nonzero-mass case). For the determination of the position operators it was enough to consider only very special cases of homogeneous Lorentz transformations. One of the purposes of the present paper is to consider the effect of arbitrary inhomogeneous continuous Lorentz transformations in order to obtain general Poincar\'e-invariant localization (GPIL). Our main result will be that localization is LIL if and only if it is GPIL. In II and III, LIL was obtained in spite of the fact that the position operators were not 4-vector operators but only 3-vector operators. In this paper, following Fleming, we construct formally covariant position operators from the 3-vector operators. We prove that their eigenfunctions are just the localized states defined in the GPIL sense.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsMar 15, 197311 citations

Integral representation of covariant observables

U. Cattaneo

Springer eBooksApr 7, 20080 citations

Manifestly covariant canonical quantization of the scalar field and particle localization

Matej Pavšič

Particle localization within quantum field theory is revisited. Canonical quantization of a free scalar field theory is performed in a manifestly Lorentz covariant way with respect to an arbitrary 3-surface [Formula: see text], which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to [Formula: see text]. Position on [Formula: see text] is determined by a 4-vector [Formula: see text]. The corresponding quantum position operator, formed in terms of the operators [Formula: see text], [Formula: see text], that create/annihilate particles on [Formula: see text], has thus well-behaved properties under Lorentz transformations. A generic state is a superposition of the states, created with [Formula: see text], the superposition coefficients forming multiparticle wave packet profiles — wave functions, including a single-particle wave function that satisfies the covariant generalization of the Foldy equation. The covariant center-of-mass operator is introduced and its expectation values in a generic multiparticle state calculated.Particle localization within quantum field theory is revisited. Canonical quantization of a free scalar field theory is performed in a manifestly Lorentz covariant way with respect to an arbitrary 3-surface [Formula: see text], which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to [Formula: see text]. Position on [Formula: see text] is determined by a 4-vector [Formula: see text]. The corresponding quantum position operator, formed in terms of the operators [Formula: see text], [Formula: see text], that create/annihilate particles on [Formula: see text], has thus well-behaved properties under Lorentz transformations. A generic state is a superposition of the states, created with [Formula: see text], the superposition coefficients forming multiparticle wave packet profiles — wave functions, including a single-particle wave function that satisfies the covariant generalization of the Foldy equation. The covariant center-of-mass operator is introduced and its expectation values in a generic multiparticle state calculated.

arXiv:1804.03404Modern Physics Letters AJun 27, 20187 citations

Quantum Field Theory and Local Observables

Bert Schroer

arXiv:hep-th/9707230Jul 28, 19971 citation

In parts II and III of the present series of papers, the localization problem was studied for the cases of relativistic nonzero mass (spin 0,

\frac{1}{2}

) and zero mass (spin 0,

\frac{1}{2}

, and 1). With a set of postulates, of which the basic one is the imposition of Lorentz invariance of localization (LIL) (which is a self-consistency requirement), it was possible to define the position operator uniquely (except for a constant in the spin-

\frac{1}{2}

, nonzero-mass case). For the determination of the position operators it was enough to consider only very special cases of homogeneous Lorentz transformations. One of the purposes of the present paper is to consider the effect of arbitrary inhomogeneous continuous Lorentz transformations in order to obtain general Poincar\'e-invariant localization (GPIL). Our main result will be that localization is LIL if and only if it is GPIL. In II and III, LIL was obtained in spite of the fact that the position operators were not 4-vector operators but only 3-vector operators. In this paper, following Fleming, we construct formally covariant position operators from the 3-vector operators. We prove that their eigenfunctions are just the localized states defined in the GPIL sense.