Algebraic implications of composability of physical systems (Q1239238)

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 \(a = (n/2)^2\).