Monopole star products are non-alternative

TL;DR

This paper demonstrates that the algebraic structures arising in quantum systems with magnetic monopoles cannot be alternative, highlighting fundamental non-associativity in such physical models.

Contribution

It proves that monopole star product algebras are inherently non-alternative, advancing understanding of non-associative quantum algebraic structures.

Findings

Monopole star products are non-alternative.

Associator cannot be completely anti-symmetric in these systems.

Provides a mathematical proof using deformation quantization.

Abstract

Non-associative algebras appear in some quantum-mechanical systems, for instance if a charged particle in a distribution of magnetic monopoles is considered. Using methods of deformation quantization it is shown here, that algebras for such systems cannot be alternative, i.e. their associator cannot be completely anti-symmetric.