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arXiv:1510.05106

Localization properties and causality aspects of massless and massive scalar particles

Andreas Aste, Maik Frensel

Published Oct 17, 2015• 0 citations
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Abstract

Localization properties of scalar single particle states are analyzed by explicit calculational examples with a focus on the massless case. Problems arising from the non-existence of relativistic particle position operators respecting the causal structure of Minkowski spacetime are illustrated by exploring the conflicts arising from localization and causal properties commonly imposed on single particle states. These topics necessitate the introduction of quantum field theoretical localization concepts and are scarcely discussed and often misinterpreted in the literature.

DOI
10.48550/arxiv.1510.05106

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Localization properties of scalar single particle states are analyzed by explicit calculational examples with a focus on the massless case. Problems arising from the non-existence of relativistic particle position operators respecting the causal structure of Minkowski spacetime are illustrated by exploring the conflicts arising from localization and causal properties commonly imposed on single particle states. These topics necessitate the introduction of quantum field theoretical localization concepts and are scarcely discussed and often misinterpreted in the literature.
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Abstract We employ reduced density matrices to investigate the localization of relativistic particles within the framework of (1+1)(1+1)(1+1) ( 1 + 1 ) -dimensional quantum field theory. In the case of bosons, we prove that scalar particles cannot be localized within any compact region, although they can be made arbitrarily close to localized states. For massless fermions, by exploiting properties of the reduced density matrices of coherent states, we show that the fermionic single-particle state which, under constructive bosonization, corresponds to a bosonic coherent state is strictly localized at a single spatial point; however, the energy of this state is infinite. Moreover, since the notion of spatial localization based on reduced density matrices is consistent with relativistic causality, even a particle with infinite energy that is localized at a single spatial point will neither spread outside the light cone nor propagate superluminally. Indeed, the explicit expression for the localized fermionic state derived in our work shows that a massless fermion can preserve its sharply localized profile while propagating at the speed of light, in full agreement with causality.
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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Abstract On plain physical grounds, localization of relativistic quantum particles is extended to the achronal regions of Minkowski spacetime. Achronal localization fulfills automatically the requirements of causality. It constitutes the frame which complies most completely with the principle of causality for quantum mechanical systems. Achronal localization is equivalent to the localization in the regions of the causal logic. Covariant representations of the causal logic are constructed for the systems with mass spectrum of positive Lebesgue measure and every definite spin. Apparently, no representation of the causal logic for a real mass system has been achieved in the past.