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  • 1
    Electronic Resource
    Electronic Resource
    On supersymmetries in nonrelativistic quantum mechanics (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 152-160 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: One-dimensional nonrelativistic systems are studied when time-independent potential interactions are involved. Their supersymmetries are determined and their closed subsets generating kinematical invariance Lie superalgebras are pointed out. The study of even supersymmetries is particularly enlightened through the already known symmetries of the corresponding Schrödinger equation. Three tables collect the even, odd, and total supersymmetries as well as the invariance (super)algebras.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529954
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  • 2
    Electronic Resource
    Electronic Resource
    On supersymmetric harmonic oscillators and the Green–Cusson Ansätze (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 3094-3100 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: New properties on parabosonic and parafermionic Cusson's realizations of the Green's Anzätze are first obtained. Then paraproperties are shown to have a specific interest for our real world, where bosons and fermions have to be superposed if Nature is supersymmetric, these results being obtained on supersymmetric harmonic oscillators. In particular, from the Green–Cusson Ansätze, we construct a class of supersymmetric Hamiltonians that are more general than the ones issued from standard or spin–orbit coupling considerations. We explicitly consider the simplest (previously unknown) super-Hamiltonian constructed from paraproperties in quantum mechanics and establish its invariance Lie superalgebra.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529056
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  • 3
    Electronic Resource
    Electronic Resource
    Parastatistics, supersymmetry and parasupercoherent states (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 1513-1523 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Coherent states are constructed in p=2-parasupersymmetric quantum mechanics in connection with the two relative para-Bose and para-Fermi sets of trilinear structure relations. They are called parasupercoherent states. Parasupersymmetric operators [such as the Hamiltonian and (two) annihilation operators] are introduced and discussed. In particular, the parasuperspectrum is determined and compared with the recent results obtained by Rubakov and Spiridonov [Mod. Phys. Lett. A 3, 1337 (1988)]. The superalgebra contents subtended by osp(2/2) and osp(3/2) are analyzed and exploited in order to get constants of motion in both contexts through parallel properties on the supersymmetric harmonic oscillator.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528694
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  • 4
    Electronic Resource
    Electronic Resource
    Lie structures in parasupersymmetric quantum mechanics: II. The spin-orbit coupling supersymmetrization procedure (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 1815-1821 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The spin-orbit coupling supersymmetrization procedure is considered in parasupersymmetric quantum mechanics describing oscillatorlike systems. The superposition of bosonic and parafermionic degrees of freedom is mainly studied in the cases of two and three spatial dimensions but the n-dimensional case is also determined. New Lie superalgebras and Lie parasuperalgebras are pointed out: They differ from those already obtained in connection with the standard procedure of supersymmetrization studied in Part I of this series [J. Beckers and N. Debergh, J. Math. Phys. (square, solid)(square, solid), (square, solid)(square, solid)(square, solid) (1991)].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529246
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  • 5
    Electronic Resource
    Electronic Resource
    Superposition formulas for nonlinear superequations (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 2528-2534 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Nonlinear superequations, for which the general solution can be expressed algebraically in terms of a finite number of particular solutions, are obtained. They are based on the orthosymplectic supergroup OSP(m,2n) and its action on a homogeneous superspace. Superposition formulas are discussed for the cases m=1, n arbitrary, and m=2, n=1. For OSP(2,2) the number of particular solutions needed to reconstruct the general solution depends on the dimension of the underlying Grassmann algebra, whereas for OSP(1,2n) it does not.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528997
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  • 6
    Electronic Resource
    Electronic Resource
    On generalized coherent states with maximal symmetry for the harmonic oscillator (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 1732-1738 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Generalized coherent states for the one-dimensional harmonic oscillator with maximal symmetry, i.e., admitting the semidirect sum so(2,1) (D'Alembertian) h(2) as the largest invariance Lie algebra pointed out by Niederer are constructed. The normalization of such states as well as their completeness property are determined and discussed. They are also analyzed in the subcontexts of the so(2,1) algebra and of the Heisenberg h(2) algebra. General considerations on Heisenberg relations, on minimal dispersions, and on quantum mechanical entropies are presented in connection with the uncertainty principle.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528261
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  • 7
    Electronic Resource
    Electronic Resource
    Nonlinear equations with superposition formulas and the exceptional group G2. II. Classification of the equations (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 520-529 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Nonlinear equations with superposition formulas are obtained, corresponding to the action of the complex and real forms of the exceptional Lie group G2 on the homogeneous spaces G2/H. The isotropy group of the origin H is taken as one of the maximal parabolic subgroups of G2, or as one of the maximal reductive subgroups, leaving some vector space V∈C7 (or V∈R7) invariant. The parabolic subgroups, as well as the simple subgroups SL(3,C), SU(3), SL(3,R) or SU(2,1) lead to equations with quadratic or quartic nonlinearities. The semisimple subgroups SL(2,C)⊗SL(2,C), SU(2)⊗SU(2), and SU(1,1)⊗SU(1,1) lead to equations with quadratic nonlinearities and additional nonlinear constraints on the independent variables.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527636
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  • 8
    Electronic Resource
    Electronic Resource
    Lie structures in parasupersymmetric quantum mechanics: I. The standard supersymmetrization procedure (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 1808-1814 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Parasupersymmetric quantum mechanics leads to the construction of new Lie structures that are called Lie parasuperalgebras. They are characterized by double commutation relations between even and odd generators and include as particular substructures the usual Lie superalgebras. Oscillatorlike systems in one spatial dimension are especially exploited in order to put such structures in evidence but n-dimensional extensions are easily determined. These developments in the standard supersymmetrization procedure will be considered first and the spin orbit coupling one will be considered in a subsequent article [J. Beckers and N. Debergh, J. Math. Phys. (square, solid)(square, solid), (square, solid)(square, solid)(square, solid) (1991)].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529245
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  • 9
    Electronic Resource
    Electronic Resource
    On pararelativistic quantum oscillators (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 3387-3392 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Different choices of matrices characterizing p=2 parafermions are analyzed in connection with the description of relativistic spin-one particles through the Kemmer formulation. The free and interacting cases are considered and the relations between parasupersymmetry and Kemmer theory are enhanced as it is also the case between supersymmetry and Dirac theory. In that way the oscillatorlike context leads to the characterization of pararelativistic oscillators.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529886
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  • 10
    Electronic Resource
    Electronic Resource
    On chains of orthosymplectic Lie superalgebras and the n-dimensional quantum harmonic oscillator (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 1655-1661 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Triplets of embedded orthosymplectic Lie superalgebras are singled out and analyzed in terms of their respective even and odd root systems. The corresponding chains of embeddings are considered for arbitrary integers m and n (for both m≠n and m=n). It is shown that the second member inside each triplet is always in a one-to-one correspondence with the semidirect sum of the third member and the associated Heisenberg superalgebra that is characterized by the same entries m and n. For arbitrary m=n, invariance superalgebras of the n-dimensional harmonic oscillator are recovered and their one-to-one correspondence associated with a recent "character reversal'' phenomenon linking these invariance superstructures is rigorously explained. Specific and general cases as well as conclusions are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528250
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