Axiomatics of particle interactions
type: Item
instance of: scholarly article
title: Axiomatics of particle interactions
zbMATH Open document ID: 0810.53059
DOI: 10.1007/BF00979492
author: George W. Mackey
published in: International Journal of Theoretical Physics
publication date: 1995-04-09T00:00:00Z
review text: The author's aim is to present a connection between the Yang-Mills approach on gauge fields and the mathematical formalism developed by the author in the last forty years in the field of quantum mechanics. In section 2, the author reviews two of his discoveries which helped him ``to understand why the quantum mechanics of particles takes the form that it does''. The first of these results concerns the von Neumann formalism in which observables are associated with selfadjoint operators in a Hilbert space and pure states with unit vectors in the same Hilbert space. The author obtained the association between real-valued observables and selfadjoint operators by first obtaining an association between such observables and projection-valued measures on the real line and then using the spectral theorem of Hilbert to change this into an association between real observables and selfadjoint operators. It was proved that non-real valued observables do not correspond to selfadjoint operators [\textit{G. W. Mackey}, The mathematical foundations of quantum mechanics, W. A. Benjamin (1963; Zbl 0114.440)]. The other result was in some sense a justification of the famous Heisenberg commutation relations in the particular case of a single free particle using the imprimitivity theorem. In the next section (3), the author continues the exposition of his theory by generalizing the results quoted above to the case of a finite number of interacting particles. For such a system, the operators defining the velocity of the \(j\)-th particle are the commutators \(i[H,X_{1j}]\), \(i[H,X_{2j}]\) and \(i[H,X_{3j}]\), where \(H\) is a selfadjoint operator such that \(t \to e^{-iHt}\) is the one-parameter unitary group defining the time evolution of the system and \(X_{1j}\), \(X_{2j}\), \(X_{3j}\) are the usual multiplicative operators. Then the dynamical operator \(H\) reads \[ H = {1\over 2\mu} \sum_{j} \Biggl\{ {1\over i}\frac{\partial}{\partial X_{1j}} + A_ 1)^ 2 + ({1\over i}\frac {\partial}…
Mathematics Subject Classification ID: 53Z05
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81S05
zbMATH DE Number: 575477
zbMATH Keywords: Yang-Mills theory
zbMATH Keywords: von Neumann formalism
zbMATH Keywords: observables
zbMATH Keywords: Heisenberg commutation relations
zbMATH Keywords: imprimitivity theorem
zbMATH Keywords: classical limit
zbMATH Keywords: equations of motion
reviewed by: Serban Misicu
MaRDI profile type: Publication
cites work: Feynman’s proof of the Maxwell equations
cites work: Quantum Mechanics and Hilbert Space
cites work: Q5723020
cites work: Infinite-dimensional group representations
cites work: Q5562579
cites work: Q4185057
cites work: Conservation of Isotopic Spin and Isotopic Gauge Invariance
full work available at URL: https://doi.org/10.1007/bf00979492
OpenAlex ID: W2074965684
description: scientific article; zbMATH DE number 575477