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Symmetry

Special Issue: “Symmetries in Quantum Mechanics”

G. Jordan Maclay

Published Sep 3, 2021• 0 citations
Publisher

Abstract

This Special Issue “Symmetries in Quantum Mechanics” describes research using two of the most fundamental probes we have in nature [...]
DOI
10.3390/sym13091620

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Particles, Localization and "Collapse"

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Hegerfeldt's theorem shows that the localization of a particle (or field) is not possible under very general assumptions of quantum mechanics and special relativity. This theorem, and further developments and associated theorems force penetrating questions on the proper understanding of the theory and its use. In particular, the use of the collapse postulate is called into question. The existence of particles and even localized fields are called into question. And, this in turn challenges local measurements. It generally forces one to consider the profound problems and even wrong results that arise from the collapse postulate. We give a qualitatively new example of such a wrong result. Researchers and students alike are bothered by these fundamental issues. Students' ability to learn the material, and even their interest in physics, can be compromised by such disturbing conundra that seem to make physics, or at least QFT and its QM limit, seem incomprehensible and even inconsistent. This article resolves these issues in a way accessible to undergraduates, by applying the natural (minimal) interpretation of quantum mechanics (QM) (the ensemble interpretation) and explicating and clarifying the basic principles and premises used in QM and the Hegerfeldt theorem and its application to measurements.

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Several recent studies have suggested that incompatible variables, which play an essential role in quantum mechanics (QM), are, somewhat surprisingly, not necessarily unique to QM. To investigate this possibility and obtain a better understanding of two central postulates of QM, namely the commutator postulate and the Born postulate, we introduce a classical probabilistic theoretical framework which is more general than QM and contains QM as a special case. We call this framework the General Incompatible Variables (GIV) theory, and we show that not only QM systems but also any probabilistic systems (classical or quantal) that possess incompatible variables will exhibit the quantal properties of uncertainty and interference, and we illustrate this with a roulette-like classical system (which we call the Arrow) that shows precisely these properties. We show that QM emerges naturally from the GIV framework when the fundamental variables are taken to be symmetries, so that the incompatibility of the QM variables is actually just the incompatibility of the corresponding symmetries (or their generators). Specifically, when the variables are taken to be the elements of the Poincare group (the symmetry of spacetime) we find that the commutator postulate and the Born postulate follow automatically, and are therefore no longer postulates. QM thus emerges as a special case of GIV theories for which the variables are symmetries.

arXiv:2206.00008•May 31, 2022•0 citations

Symmetries in Quantum Mechanics

Walter Greiner, Berndt Müller

Springer eBooks•Jan 1, 1994•24 citations
Hegerfeldt's theorem shows that the localization of a particle (or field) is not possible under very general assumptions of quantum mechanics and special relativity. This theorem, and further developments and associated theorems force penetrating questions on the proper understanding of the theory and its use. In particular, the use of the collapse postulate is called into question. The existence of particles and even localized fields are called into question. And, this in turn challenges local measurements. It generally forces one to consider the profound problems and even wrong results that arise from the collapse postulate. We give a qualitatively new example of such a wrong result. Researchers and students alike are bothered by these fundamental issues. Students' ability to learn the material, and even their interest in physics, can be compromised by such disturbing conundra that seem to make physics, or at least QFT and its QM limit, seem incomprehensible and even inconsistent. This article resolves these issues in a way accessible to undergraduates, by applying the natural (minimal) interpretation of quantum mechanics (QM) (the ensemble interpretation) and explicating and clarifying the basic principles and premises used in QM and the Hegerfeldt theorem and its application to measurements.
Several recent studies have suggested that incompatible variables, which play an essential role in quantum mechanics (QM), are, somewhat surprisingly, not necessarily unique to QM. To investigate this possibility and obtain a better understanding of two central postulates of QM, namely the commutator postulate and the Born postulate, we introduce a classical probabilistic theoretical framework which is more general than QM and contains QM as a special case. We call this framework the General Incompatible Variables (GIV) theory, and we show that not only QM systems but also any probabilistic systems (classical or quantal) that possess incompatible variables will exhibit the quantal properties of uncertainty and interference, and we illustrate this with a roulette-like classical system (which we call the Arrow) that shows precisely these properties. We show that QM emerges naturally from the GIV framework when the fundamental variables are taken to be symmetries, so that the incompatibility of the QM variables is actually just the incompatibility of the corresponding symmetries (or their generators). Specifically, when the variables are taken to be the elements of the Poincare group (the symmetry of spacetime) we find that the commutator postulate and the Born postulate follow automatically, and are therefore no longer postulates. QM thus emerges as a special case of GIV theories for which the variables are symmetries.