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The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac's remarkable s o (2 , 3) representation. We show that the orthosymplectic algebra o s p (1 | 4) is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb–Kepler systems, with the o s p (1 | 4) symmetry broken down to the conformal symmetry s o (2 , 3) . The even s o (2 , 3) submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra o s p (1 | 4) creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd s o (2 , 3) -submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary s o (2 , 3) representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal. [ABSTRACT FROM AUTHOR]

A Z 2 × Z 2 -graded Lie superalgebra g is a Z 2 × Z 2 -graded algebra with a bracket [ [ ⋅ , ⋅ ] ] that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, g is not a Lie superalgebra. We construct the most general orthosymplectic Z 2 × Z 2 -graded Lie superalgebra o s p (2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2) in terms of defining matrices. A special case of this algebra appeared already in work of Tolstoy in 2014. Our construction is based on the notion of graded supertranspose for a Z 2 × Z 2 -graded matrix. Since the orthosymplectic Lie superalgebra o s p (2 m + 1 | 2 n) is closely related to the definition of parabosons, parafermions and mixed parastatistics, we investigate here the new parastatistics relations following from o s p (2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2) . Some special cases are of particular interest, even when one is dealing with parabosons only. [ABSTRACT FROM AUTHOR]

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac’s remarkable so(2,3) representation. We show that the orthosymplectic algebra osp(1|4) is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb–Kepler systems, with the osp(1|4) symmetry broken down to the conformal symmetry so(2,3). The even so(2,3) submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra osp(1|4) creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd so(2,3)-submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary so(2,3) representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal.

I formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P, Q satisfy the R-deformed Heisenberg relations, and find representations of P, Q in which the angular momentum l and the dimension d, which can by any real positive number, appear as parameters. These representations lead to corresponding representations of paraboson operators which can be used, for example, to solve the time-dependent harmonic oscillator for any d>0 using the method of Lewis and Riesenfeld. I develop algebraic properties of Weyl-ordered polynomials in P, Q by viewing them as tensor operators with respect to the Lie algebra sl2, and also discuss the q-analogue deformation of these properties. [Copyright &y& Elsevier]

• New formulas are obtained for the grand partition function (GPF) of a system of n parabosons of any order of statistics p. • New formulas are obtained for the GPF of a system of m parafermions of any order of statistics p. • The expressions for the GPFs are suitable for the analysis of thermodynamic properties of the parabosons and parafermions. • The average number of particles on a given orbital and in paraboson and parafermion systems are analyzed and illustrated. • A new relation is discovered between the GPFs of n -paraboson and of (n − p)-parafermion systems of order p , in a special case. New formulas are given for the grand partition function of paraboson systems of order p with n orbitals and parafermion systems of order p with m orbitals. These formulas allow the computation of statistical and thermodynamic functions for such systems. We analyze and discuss the average number of particles on an orbital, and the average number of particles in the system. For some special cases (identical orbital energies, or equidistant orbital energies) we can simplify the grand partition functions and describe thermodynamic properties in more detail. Some specific properties are also illustrated in plots of thermodynamic functions. [ABSTRACT FROM AUTHOR]