Localization of Non-Relativistic Particles

Madrid, Rafael de la

doi:10.1007/s10773-006-9320-z

Cited by 3 publications

References 14 publications

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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J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete.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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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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There are no particles, there are only fields

Hobson¹

2013

American Journal of Physics

120

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Quantum foundations are still unsettled, with mixed effects on science and society. By now it should be possible to obtain consensus on at least one issue: Are the fundamental constituents fields or particles? As this paper shows, experiment and theory imply unbounded fields, not bounded particles, are fundamental. This is especially clear for relativistic systems, implying it's also true of non-relativistic systems. Particles are epiphenomena arising from fields. Thus the Schroedinger field is a space-filling physical field whose value at any spatial point is the probability amplitude for an interaction to occur at that point. The field for an electron is the electron; each electron extends over both slits in the 2-slit experiment and spreads over the entire pattern; and quantum physics is about interactions of microscopic systems with the macroscopic world rather than just about measurements. It's important to clarify this issue because textbooks still teach a particles- and measurement-oriented interpretation that contributes to bewilderment among students and pseudoscience among the public. This article reviews classical and quantum fields, the 2-slit experiment, rigorous theorems showing particles are inconsistent with relativistic quantum theory, and several phenomena showing particles are incompatible with quantum field theories.Comment: To be published in Am. J. Phy

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Revising the applicability of the Stone-von Neumann theorem to scattering a quantum particle on a one-dimensional potential barrier

Chuprikov¹

2016

Preprint

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It is shown that the Stone-von Neumann theorem is inapplicable to scattering a quantum nonrelativistic particle on a one-dimensional "short-range" potential barrier, since the unboundedness of the position operator plays here a crucial role. The Shcrödinger representation associated with this process is reducible: long before and long after the scattering event the space of its asymptotes represents the direct sum of the subspaces of left and right asymptotes. There is a dichotomous-contextinduced superselection rule (SSR), in which the role of a superselection operator is played by the Pauli matrix σ 3 and the role of superselection (coherent) sectors is played by the above subspaces. By the SSR any superposition of states from different coherent sectors is a mixed state, and splitting the incident wave packet into the transmitted and reflected parts is nothing but a conversion of a pure state into a mixed one. The average values of any observable can be defined only for the transmission and reflection subprocesses. The former evolves within a single coherent sector of the Hilbert space, in the momentum representation; while the latter evolves within a single coherent sector of this space, in the coordinate representation.

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Localization of Non-Relativistic Particles

Cited by 3 publications

References 14 publications

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

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

There are no particles, there are only fields

Revising the applicability of the Stone-von Neumann theorem to scattering a quantum particle on a one-dimensional potential barrier

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