Algebraic implications of composability of physical systems
type: Item
instance of: scholarly article
title: Algebraic implications of composability of physical systems
zbMATH Open document ID: 0358.17005
DOI: 10.1007/BF01617995
author: Emil Grgin
author: Aage Petersen
published in: Communications in Mathematical Physics
publication date: 1976-01-01T00:00:00Z
review text: From the examples provided by classical and quantum mechanics, the authors abstract the concept of a two-product algebra, i.e., of an ordered triple \(\{\mathcal H, \tau, \alpha\}\) where \(\mathcal H\) is a vector space over a (commutative) field \(\mathcal F\), \(\tau\) is a multiplication on \(\mathcal H\) and \(\alpha\) is a Lie multiplication on \(\mathcal H\) which is derivative with respect to \(\tau\). From the manner in which physical system compose in classical and quantum mechanics, they abstract then the concept of a composition class, i.e., of a set \(\mathcal J\) of two-product algebras equipped with a multiplication making it into a monoid with the neutral element \(\{\mathcal F, \tau, 0\}\), where \(\tau\) is the multiplication on \(\mathcal F\). They show that all composition classes are obtainable from a family \(\{\mathcal J_a\}\), labelled by an index \(a \in \mathcal J\) defined modulo the set of all squares in \(\mathcal F\), such that the multiplication \(\tau\) is symmetric on every element of \(\mathcal J_a\) for all \(a\). The composition class \(\mathcal J_0\) is said to be classical and, for \(a \neq 0\), \(\mathcal J_a\) is said to be quantal because the case of quantum mechanics is obtained with \(\mathcal F = \mathcal R\) and \(a = (n/2)^2\).
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 81R05
zbMATH DE Number: 3558004
MaRDI profile type: Publication
cites work: Duality of observables and generators in classical and quantum mechnics
cites work: Phase-Space Formulation of the Dynamics of Canonical Variables
full work available at URL: https://doi.org/10.1007/bf01617995
OpenAlex ID: W2068980724
description: scientific article; zbMATH DE number 3558004