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Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields

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

Edward B. Manoukian

Published Oct 1, 1990• 2 citations
Publisher
DOI
10.1007/bf02820577

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Asymptotic Locality: Rectifying Newton--Wigner and Foldy--Wouthuysen Localization

E. J. Thompson

Relativistic quantum theory supports two competing notions of particle locality. Newton--Wigner localization provides a sharp position operator on the one-particle Hilbert space but is not Lorentz covariant and leads to instantaneous spreading for positive-energy states. Foldy--Wouthuysen localization, widely used in precision experiments, yields a robust quasi-classical particle description but is representation-dependent and does not coincide with strict field-theoretic locality. We show that this tension is not a paradox in UV-complete nonlocal quantum field theories whose ultraviolet physics is encoded by an entire-function regulator. In such theories the physically accessible observables form a quasi-local algebra generated by regulated smearings with an intrinsic nonlocality length. Sharp Newton--Wigner projectors are not operationally realistic, and Newton--Wigner and Foldy--Wouthuysen localization become equivalent non-singular notions of locality in the same measurement algebra.

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Time of arrival and localization of relativistic particles

Charis Anastopoulos, Ntina Savvidou

Constructing observables that describe the localization of relativistic particles is an important foundational problem in relativistic quantum field theory (QFT). The description of localization in terms of single-time observables leads to conflict with the requirement of causality. In this paper, we describe particle localization in terms of time-of-arrival observables, defined in terms of the interaction between a quantum field and a measuring apparatus. The resulting probabilities are linear functionals of QFT correlation functions. Specializing to the case of a scalar field, we identify several time-of-arrival observables differing on the way that the apparatus localizes particle-detection records. Maximum localization is obtained for a unique observable that is related to the Newton-Wigner position operator. Finally, we define a measure of localizability for relativistic particles in terms of a novel time-energy uncertainty relation for the variance of the time of arrival.

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From Quantum Field Theory to Quantum Mechanics

We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.

Tsinghua report and review in physics•Feb 1, 2014•0 citations

ON PHASE SPACE REPRESENTATIONS IN QUANTUM FIELD THEORY

I.V. Polubarinov

Submitted to•Jan 1, 1982•0 citations
Relativistic quantum theory supports two competing notions of particle locality. Newton--Wigner localization provides a sharp position operator on the one-particle Hilbert space but is not Lorentz covariant and leads to instantaneous spreading for positive-energy states. Foldy--Wouthuysen localization, widely used in precision experiments, yields a robust quasi-classical particle description but is representation-dependent and does not coincide with strict field-theoretic locality. We show that this tension is not a paradox in UV-complete nonlocal quantum field theories whose ultraviolet physics is encoded by an entire-function regulator. In such theories the physically accessible observables form a quasi-local algebra generated by regulated smearings with an intrinsic nonlocality length. Sharp Newton--Wigner projectors are not operationally realistic, and Newton--Wigner and Foldy--Wouthuysen localization become equivalent non-singular notions of locality in the same measurement algebra.
Constructing observables that describe the localization of relativistic particles is an important foundational problem in relativistic quantum field theory (QFT). The description of localization in terms of single-time observables leads to conflict with the requirement of causality. In this paper, we describe particle localization in terms of time-of-arrival observables, defined in terms of the interaction between a quantum field and a measuring apparatus. The resulting probabilities are linear functionals of QFT correlation functions. Specializing to the case of a scalar field, we identify several time-of-arrival observables differing on the way that the apparatus localizes particle-detection records. Maximum localization is obtained for a unique observable that is related to the Newton-Wigner position operator. Finally, we define a measure of localizability for relativistic particles in terms of a novel time-energy uncertainty relation for the variance of the time of arrival.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.