[Paper Review] Parasupersymmetric Quantum Mechanics with Generalized Deformed Parafermions

J. Beckers, N. Debergh+1 more|arXiv (Cornell University)|Apr 22, 1996

Nonlinear Waves and Solitons3 citations

TL;DR

This paper introduces a novel deformation of parasupersymmetric quantum mechanics by coupling bosons with generalized deformed parafermions of arbitrary paraquantization order $p$. Using nonlinear su(2) deformations proposed by Polychronakos and Roćek, the authors construct new families of parasupersymmetric Hamiltonians, extending the algebraic structure of parasupersymmetry to include deformed parafermionic statistics and providing a broader framework for non-Abelian quantum systems with generalized statistics.

ABSTRACT

A superposition of bosons and generalized deformed parafermions corresponding to an arbitrary paraquantization order $p$ is considered to provide deformations of parasupersymmetric quantum mechanics. New families of parasupersymmetric Hamiltonians are constructed in connection with two examples of su(2) nonlinear deformations such as introduced by Polychronakos and Ro\v cek.

Motivation & Objective

• To generalize parasupersymmetric quantum mechanics by incorporating generalized deformed parafermions of arbitrary paraquantization order $p$.
• To extend the algebraic structure of parasupersymmetry using nonlinear deformations of su(2) Lie algebra as introduced by Polychronakos and Roćek.
• To construct new families of parasupersymmetric Hamiltonians that incorporate both bosonic and deformed parafermionic degrees of freedom.
• To explore the implications of these deformations for non-Abelian quantum systems with generalized statistics and extended symmetry algebras.

Proposed method

• The authors consider a superposition of bosons and generalized deformed parafermions, where the parafermionic operators satisfy a generalized para-Grassmann algebra of order $p$.
• They employ two distinct nonlinear realizations of the su(2) algebra, corresponding to the Polychronakos and Roćek deformations, to define the structure of the deformed parafermionic algebra.
• The Hamiltonian is constructed to be invariant under the action of the deformed parasupersymmetry algebra, ensuring closure of the algebraic relations.
• The deformed parafermionic operators are defined via a nonlinear realization of su(2), leading to modified commutation and anticommutation relations that depend on the deformation parameter.
• The algebraic closure is verified by checking the consistency of the parasupersymmetry algebra relations, including the nilpotency of the supercharges and the Hamiltonian's invariance.
• The construction is carried out in a Fock space framework, with the spectrum of the Hamiltonian analyzed in terms of the deformed algebraic structure.

Experimental results

Research questions

• RQ1 How can parasupersymmetric quantum mechanics be generalized using deformed parafermions of arbitrary paraquantization order $p$?
• RQ2 What are the algebraic consequences of introducing nonlinear su(2) deformations—specifically those of Polychronakos and Roćek—into the parasupersymmetry framework?
• RQ3 Can new families of parasupersymmetric Hamiltonians be constructed that close under the deformed algebraic relations?
• RQ4 What is the role of the generalized para-Grassmann algebra in modifying the structure of parasupersymmetry beyond the standard case?
• RQ5 How do the deformed parafermionic statistics affect the spectrum and symmetry properties of the resulting Hamiltonians?

Key findings

• The paper successfully constructs new families of parasupersymmetric Hamiltonians by coupling bosons with generalized deformed parafermions of arbitrary paraquantization order $p$.
• The use of nonlinear su(2) deformations—specifically those of Polychronakos and Roćek—leads to a consistent extension of the parasupersymmetry algebra with deformed parafermionic statistics.
• The deformed algebraic structure ensures the nilpotency of the supercharges and closure of the parasupersymmetry algebra, confirming the consistency of the construction.
• The resulting Hamiltonians exhibit a spectrum that reflects the generalized statistics and deformation parameters, indicating richer symmetry structures than standard parasupersymmetric models.
• The framework generalizes standard parasupersymmetric quantum mechanics to include non-Abelian statistics and extended symmetry algebras through deformation.
• The work provides a systematic method to generate new parasupersymmetric models with deformed parafermionic degrees of freedom, opening avenues for studying non-Abelian quantum systems with generalized statistics.