[Paper Review] Parasupersymmetry and N-fold Supersymmetry in Quantum Many-Body Systems I. General Formalism and Second Order

Toshiaki Tanaka|arXiv (Cornell University)|Oct 30, 2006

Quantum Mechanics and Non-Hermitian Physics78 references4 citations

TL;DR

This paper proposes a representation-free formulation of parafermionic algebra and parasupersymmetry in quantum many-body systems, showing that every parasupersymmetric system of order $ p $ contains $ \mathcal{N} $-fold supersymmetric pairs with $ \mathcal{N} \leq p $, implying weak quasi-solvability and isospectrality. It introduces quasi-parasupersymmetry as a less restrictive alternative, with explicit second-order examples demonstrating generalized $ 2 $-fold superalgebras and isospectral Hamiltonians.

ABSTRACT

We propose an elegant formulation of parafermionic algebra and parasupersymmetry of arbitrary order in quantum many-body systems without recourse to any specific matrix representation of parafermionic operators and any kind of deformed algebra. Within our formulation, we show generically that every parasupersymmetric quantum system of order p consists of N-fold supersymmetric pairs with N

Motivation & Objective

• To develop a general, representation-independent formulation of parafermionic algebra and parasupersymmetry in quantum many-body systems.
• To clarify the structural relationship between parasupersymmetry and $ \mathcal{N} $-fold supersymmetry, showing that the former implies the latter with $ \mathcal{N} \leq p $.
• To introduce and analyze a new symmetry, quasi-parasupersymmetry, which is less restrictive than parasupersymmetry and distinct from $ \mathcal{N} $-fold supersymmetry.
• To construct explicit second-order parasupersymmetric models, including one-body and two-body systems, and demonstrate their isospectral and quasi-solvable properties.

Proposed method

• Formulate parafermionic algebra using fundamental axioms: nilpotency $ (\psi^{-})^{p+1} = 0 $, $ (\psi^{+})^{p+1} = 0 $, and a generalized anti-commutation relation involving all powers up to $ p $.
• Define component parasupercharges $ Q_k^\pm $ and Hamiltonians $ H_k $, and derive closed-form commutation and non-linear relations between them.
• Express parasupersymmetric conditions entirely in terms of component Hamiltonians and supercharges, avoiding matrix representations or deformed algebras.
• Introduce quasi-parasupersymmetry via relaxed conditions on the supercharge products, allowing for non-trivial higher-order differential operators.
• Construct three explicit second-order models: one equivalent to the Rubakov–Spiridonov one-body system, and two two-body systems with folded supersymmetries.
• Verify isospectrality and weak quasi-solvability by analyzing the algebraic structure and spectral properties of the Hamiltonians.

Experimental results

Research questions

• RQ1 How can parafermionic algebra and parasupersymmetry be formulated without relying on matrix representations or deformed oscillator algebras?
• RQ2 What is the precise relationship between parasupersymmetry of order $ p $ and $ \mathcal{N} $-fold supersymmetry with $ \mathcal{N} \leq p $?
• RQ3 Can a less restrictive symmetry than parasupersymmetry be defined, and how does it differ from $ \mathcal{N} $-fold supersymmetry in one-body systems?
• RQ4 Do explicit second-order parasupersymmetric models exhibit isospectrality and weak quasi-solvability, and can they support generalized $ 2 $-fold superalgebras?
• RQ5 Can the formalism be extended to position-dependent mass systems or multiple parafermionic variables?

Key findings

• Every parasupersymmetric quantum system of order $ p $ contains $ \mathcal{N} $-fold supersymmetric pairs with $ \mathcal{N} \leq p $, ensuring isospectrality and weak quasi-solvability.
• The parasupersymmetric conditions are expressible in closed form using only component Hamiltonians $ H_k $ and supercharges $ Q_k^\pm $, independent of representation.
• The first example of a second-order parasupersymmetric system is essentially equivalent to the one-body Rubakov–Spiridonov model and admits a generalized $ 2 $-fold superalgebra.
• The two-body models exhibit two folded supersymmetries, and their Hamiltonians are isospectral due to the underlying $ \mathcal{N} $-fold structure.
• Quasi-parasupersymmetry is introduced as a distinct, less restrictive symmetry; in second-order cases, it is equivalent to parasupersymmetry, but it differs from $ \mathcal{N} $-fold supersymmetry in one-body systems.
• The formalism is extendable to position-dependent mass systems via von Roos-type Hamiltonians, with explicit construction of component supercharges involving mass functions and derivatives.