Incompatibility of Frequency Splitting and Spatial Localization: A Quantitative Analysis of Hegerfeldt’s Theorem

Finster, Felix; Paganini, Claudio F.

doi:10.1007/s00023-022-01215-8

Ann. Henri Poincaré

2022

DOI: 10.1007/s00023-022-01215-8

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Incompatibility of Frequency Splitting and Spatial Localization: A Quantitative Analysis of Hegerfeldt’s Theorem

Felix Finster

Claudio F. Paganini

Abstract: We prove quantitative versions of the following statement: If a solution of the $$1+1$$ 1 + 1 -dimensional wave equation has spatially compact support and consists mainly of positive frequencies, then it must have a significant high-frequency component. Similar results are proven for the $$3+1$$ 3 + 1 -dimensi… Show more

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2024

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Cited by 2 publications

References 26 publications

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Quantum particle localization observables on Cauchy surfaces of Minkowski spacetime and their causal properties

De Rosa,

Moretti

2024

Lett Math Phys

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We introduce and study a general notion of spatial localization on spacelike smooth Cauchy surfaces of quantum systems in Minkowski spacetime. The notion is constructed in terms of a coherent family of normalized POVMs, one for each said Cauchy surface. We prove that a family of POVMs of this type automatically satisfies a causality condition which generalizes Castrigiano’s one and implies it when restricting to flat spacelike Cauchy surfaces. As a consequence, no conflict with Hegerfeldt’s theorem arises. We furthermore prove that such families of POVMs do exist for massive Klein–Gordon particles, since some of them are extensions of already known spatial localization observables. These are constructed out of positive definite kernels or are defined in terms of the stress–energy tensor operator. Some further features of these structures are investigated, in particular the relation with the triple of Newton–Wigner selfadjoint operators and a modified form of Heisenberg inequality in the rest 3-spaces of Minkowski reference frames.

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Quantum particle localization observables on Cauchy surfaces of Minkowski spacetime and their causal properties

De Rosa,

Moretti

2024

Lett Math Phys

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Towards a probabilistic foundation of relativistic quantum theory: the one-body Born rule in curved spacetime

Reddiger,

Poirier

2024

Quantum Stud.: Math. Found.

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In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.

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Incompatibility of Frequency Splitting and Spatial Localization: A Quantitative Analysis of Hegerfeldt’s Theorem

Cited by 2 publications

References 26 publications

Quantum particle localization observables on Cauchy surfaces of Minkowski spacetime and their causal properties

Quantum particle localization observables on Cauchy surfaces of Minkowski spacetime and their causal properties

Towards a probabilistic foundation of relativistic quantum theory: the one-body Born rule in curved spacetime

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