Snyder Noncommutativity and Pseudo-Hermitian Hamiltonians from a Jordanian Twist

TL;DR

This paper explores how Jordanian twists deform quantum mechanics, leading to Snyder noncommutativity and pseudo-Hermitian Hamiltonians, with a focus on the algebraic structure and multi-particle bosonic systems.

Contribution

It introduces a novel deformation scheme using Jordanian twists that results in Snyder noncommutativity and pseudo-Hermitian Hamiltonians within a Hopf algebra framework.

Findings

Deformation of quantum mechanics via Jordanian twist induces Snyder noncommutativity.

Deformed Hamiltonians are shown to be pseudo-Hermitian, aligning with Mostafazadeh's framework.

Multi-particle sector consists of bosonic particles with a consistent algebraic structure.

Abstract

Nonrelativistic quantum mechanics and conformal quantum mechanics are deformed through a Jordanian twist. The deformed space coordinates satisfy the Snyder noncommutativity. The resulting deformed Hamiltonians are pseudo-Hermitian Hamiltonians of the type discussed by Mostafazadeh. The quantization scheme makes use of the so-called "unfolded formalism" discussed in previous works. A Hopf algebra structure, compatible with the physical interpretation of the coproduct, is introduced for the Universal Enveloping Algebra of a suitably chosen dynamical Lie algebra (the Hamiltonian is contained among its generators). The multi-particle sector, uniquely determined by the deformed 2-particle Hamiltonian, is composed of bosonic particles.