Semiheaps and Ternary Algebras in Quantum Mechanics Revisited

TL;DR

This paper revisits semiheaps and ternary algebras in quantum mechanics, exploring their construction on Hilbert spaces and how quantum symmetries induce algebraic homomorphisms.

Contribution

It introduces a new discussion on how symmetries of quantum systems induce homomorphisms in semiheaps and ternary algebras.

Findings

Semiheaps can be constructed on Hilbert spaces.

Symmetries induce homomorphisms of semiheaps.

Ternary algebras relate to quantum symmetries.

Abstract

We re-examine the appearance of semiheaps and (para-associative) ternary algebras in quantum mechanics. In particular, we review the construction of a semiheap on a Hilbert space and the set of bounded operators on a Hilbert space. The new aspect of this work is a discussion of how symmetries of a quantum system induce homomorphisms of the relevant semiheaps and ternary algebras.