scholarly journals FRACTIONAL SUPERSYMMETRIC QUANTUM MECHANICS, TOPOLOGICAL INVARIANTS AND GENERALIZED DEFORMED OSCILLATOR ALGEBRAS

2003 ◽  
Vol 18 (07) ◽  
pp. 515-525 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Quantum Mechanics ◽  
Fock Space ◽  
Supersymmetric Quantum Mechanics ◽  
Topological Invariants ◽  
Oscillator Algebra ◽  
Irreducible Components ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra ◽  
Witten Index

Fractional supersymmetric quantum mechanics of order λ is realized in terms of the generators of a generalized deformed oscillator algebra and a ℤλ-grading structure is imposed on the Fock space of the latter. This realization is shown to be fully reducible with the irreducible components providing λ sets of minimally bosonized operators corresponding to both unbroken and broken cases. It also furnishes some examples of ℤλ-graded uniform topological symmetry of type (1, 1, …, 1) with topological invariants generalizing the Witten index.

scholarly journals REDUCIBILITY AND BOSONIZATION OF PARASUPERSYMMETRIC AND ORTHOSUPERSYMMETRIC QUANTUM MECHANICS

2002 ◽  
Vol 17 (14) ◽  
pp. 839-849 ◽  
Author(s):  
C. QUESNE ◽  
N. VANSTEENKISTE
Keyword(s):  
Quantum Mechanics ◽  
Fock Space ◽  
Oscillator Algebra ◽  
Irreducible Components ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

Order-p parasupersymmetric and orthosupersymmetric quantum mechanics are shown to be fully reducible when they are realized in terms of the generators of a generalized deformed oscillator algebra and a ℤp+1-grading structure is imposed on the Fock space. The irreducible components provide p + 1 sets of bosonized operators corresponding to both unbroken and broken cases. Such a bosonization is minimal.

scholarly journals PSEUDOSUPERSYMMETRIC QUANTUM MECHANICS: GENERAL CASE, ORTHOSUPERSYMMETRIES, REDUCIBILITY, AND BOSONIZATION

2003 ◽  
Vol 18 (02) ◽  
pp. 271-292 ◽  
Author(s):  
C. QUESNE ◽  
N. VANSTEENKISTE
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Constant Magnetic Field ◽  
Explicit Solution ◽  
Quantum Mechanical ◽  
Oscillator Algebra ◽  
Quantum Mechanical System ◽  
Irreducible Components ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

Pseudosupersymmetric quantum mechanics (PsSSQM), based upon the use of pseudofermions, was introduced in the context of a new Kemmer equation describing charged vector mesons interacting with an external constant magnetic field. Here we construct the complete explicit solution for its realization in terms of two superpotentials, both equal or unequal. We prove that any orthosupersymmetric quantum mechanical system has a pseudosupersymmetry and give conditions under which a pseudosupersymmetric one may be described by orthosupersymmetries of order two. We propose two new matrix realizations of PsSSQM in terms of the generators of a generalized deformed oscillator algebra (GDOA) and relate them to the cases of equal or unequal superpotentials, respectively. We demonstrate that these matrix realizations are fully reducible and that their irreducible components provide two distinct sets of bosonized operators realizing PsSSQM and corresponding to nonlinear spectra. We relate such results to some previous ones obtained for a GDOA connected with a C3-extended oscillator algebra (where C3 = ℤ3) in the case of linear spectra.

scholarly journals Minimal bosonization of double-graded quantum mechanics

2021 ◽  
Vol 36 (33) ◽  
Author(s):  
C. Quesne
Keyword(s):  
Quantum Mechanics ◽  
Central Element ◽  
Degree Of Freedom ◽  
Supersymmetric Quantum Mechanics ◽  
Oscillator Algebra ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

The superalgebra of [Formula: see text]-graded supersymmetric quantum mechanics is shown to be realizable in terms of a single bosonic degree of freedom. Such an approach is directly inspired by a description of the corresponding [Formula: see text]-graded superalgebra in the framework of a Calogero–Vasiliev algebra or, more generally, of a generalized deformed oscillator algebra. In the case of the [Formula: see text]-graded superalgebra, the central element [Formula: see text] has the property of distinguishing between degenerate eigenstates of the Hamiltonian.

scholarly journals SPECTRUM DEGENERACY OF GENERAL (p=2)-PARASUPERSYMMETRIC QUANTUM MECHANICS AND PARASUPERSYMMETRIC TOPOLOGICAL INVARIANTS

1996 ◽  
Vol 11 (06) ◽  
pp. 1057-1076 ◽  
Author(s):  
ALI MOSTAFAZADEH
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Topological Invariant ◽  
Supersymmetric Quantum Mechanics ◽  
Topological Invariants ◽  
Witten Index

A thorough analysis of the general features of (p=2)-parasupersymmetric quantum mechanics is presented. It is shown that for both the Rubakov–Spiridonov (RS) formulation and the Beckers–Debergh (BD) formulation of (p=2)-parasupersymmetric quantum mechanics, the degeneracy structure of the energy spectrum can be derived using the defining parasuperalgebras. Thus the results of this article are independent of the details of the Hamiltonian. In fact, they are valid for arbitrary systems based on arbitrary-dimensional coordinate manifolds. In particular, the RS and BD systems possess identical degeneracy structures. For a subclass of RS (alternatively, BD) systems, a new topological invariant is introduced. This is a counterpart of the Witten index of supersymmetric quantum mechanics.

scholarly journals FUNCTIONAL INTEGRAL REPRESENTATIONS AND GOLDEN–THOMPSON INEQUALITIES IN BOSON–FERMION SYSTEMS

2013 ◽  
Vol 25 (08) ◽  
pp. 1350015
Author(s):  
ASAO ARAI
Keyword(s):  
Quantum Mechanics ◽  
Integral Representation ◽  
Fock Space ◽  
Type Inequality ◽  
Functional Integral ◽  
Integral Representations ◽  
Supersymmetric Quantum Mechanics ◽  
Fermion Systems ◽  
Witten Index ◽  
Functional Integral Representation

For a general class of boson–fermion Hamiltonians H acting in the tensor product Hilbert space L2(ℝn) ⊗ ∧(ℂr) of L2(ℝn) and the fermion Fock space ∧(ℂr) over ℂr(n, r ∈ ℕ), we establish, in terms of an n-dimensional conditional oscillator measure, a functional integral representation for the trace Tr (F ⊗ zN f e-tH)(F ∈ L∞(ℝn), z ∈ ℂ∖{0}, t > 0), where N f is the fermion number operator on ∧(ℂr). We prove a Golden–Thompson type inequality for | Tr (F ⊗ zN f e-tH)|. Also we discuss applications to a model in supersymmetric quantum mechanics and present an improved version of the Golden–Thompson inequality in supersymmetric quantum mechanics given by Klimek and Lesniewski ([Lett. Math. Phys.21 (1991) 237–244]). An upper bound for the number of the supersymmetric states is given as well as a sufficient condition for the spontaneous supersymmetry breaking. Moreover, we derive a functional integral representation for the analytical index of a Dirac type operator on ℝn (Witten index) associated with the supersymmetric quantum mechanical model.

Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach

2014 ◽  
Vol 29 (06) ◽  
pp. 1450028 ◽  
Author(s):  
S. Aghaei ◽  
A. Chenaghlou
Keyword(s):  
Dirac Equation ◽  
Quadratic Algebra ◽  
Two Dimensional ◽  
Oscillator Algebra ◽  
Quadratic Algebras ◽  
Energy Eigenvalues ◽  
Vector Potentials ◽  
Algebra Approach ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.

scholarly journals THREE-PARAMETER (TWO-SIDED) DEFORMATION OF HEISENBERG ALGEBRA

2012 ◽  
Vol 27 (21) ◽  
pp. 1250114 ◽  
Author(s):  
A. M. GAVRILIK ◽  
I. I. KACHURIK
Keyword(s):  
Deformation Parameter ◽  
Particle Number ◽  
Heisenberg Algebra ◽  
Unusual Property ◽  
Oscillator Algebra ◽  
Number Operator ◽  
The Third ◽  
Defining Relation ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

A three-parametric two-sided deformation of Heisenberg algebra (HA), with p, q-deformed commutator in the L.H.S. of basic defining relation and certain deformation of its R.H.S., is introduced and studied. The third deformation parameter μ appears in an extra term in the R.H.S. as pre-factor of Hamiltonian. For this deformation of HA we find novel properties. Namely, we prove it is possible to realize this (p, q, μ)-deformed HA by means of some deformed oscillator algebra. Also, we find the unusual property that the deforming factor μ in the considered deformed HA inevitably depends explicitly on particle number operator N. Such a novel N-dependence is special for the two-sided deformation of HA treated jointly with its deformed oscillator realizations.

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