Deformed angular momentum algebra within the real Hilbert space

TL;DR

This paper introduces deformed angular momentum algebras derived from generalized position operators within a real Hilbert space, showing they differ from standard Hermitian algebras but still produce consistent quantum expectation values.

Contribution

It develops complex and quaternionic angular momentum algebras with modified commutation relations, expanding the mathematical framework of quantum angular momentum.

Findings

Deformed algebras produce expectation values similar to standard quantum mechanics.

The algebraic structures differ notably in their commutation relations.

Deformed algebras remain valid for describing angular momentum in a real Hilbert space.

Abstract

Starting from generalized position operators, we derive complex and quaternionic angular momentum operators along with their commutation algebra as well. These algebras differ from the standard Hermitian ones, especially in terms of commutation relations involving partial and total angular momentum operators. Despite these differences, the effective quantum expectation values obtained from slightly deformed algebras align with those from the conventional Hermitian algebra. This suggests that even though the wave functions and resulting dynamics differ from standard quantum Hermitian behavior, these deformed algebras can still be effectively understood as valid angular momentum algebras.