Star product for deformed oscillator algebra Aq(2, ν)

An analogue of the Moyal star product is presented for the deformed oscillator algebra. It contains several homotopy-like additional integration parameters in the multiplication kernel generalizing the differential Moyal star-product formula exp[iεαβ ∂α∂β]. Using Pochhammer formula [1], integration over these parameters is carried over a Riemann surface associated with the expression of the type zx(1 − z)y where x and y are arbitrary real numbers.

Minimal bosonization of double-graded quantum mechanics

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.

Quantization of κ-deformed Dirac equation

In this paper, we study the quantization of Dirac field theory in the [Formula: see text]-deformed space–time. We adopt a quantization method that uses only equations of motion for quantizing the field. Starting from [Formula: see text]-deformed Dirac equation, valid up to first order in the deformation parameter [Formula: see text], we derive deformed unequal time anticommutation relation between deformed field and its adjoint, leading to undeformed oscillator algebra. Exploiting the freedom of imposing a deformed unequal time anticommutation relations between [Formula: see text]-deformed spinor and its adjoint, we also derive a deformed oscillator algebra. We show that deformed number operator is the conserved charge corresponding to global phase transformation symmetry. We construct the [Formula: see text]-deformed conserved currents, valid up to first order in [Formula: see text], corresponding to parity and time-reversal symmetries of [Formula: see text]-deformed Dirac equation also. We show that these conserved currents and charges have a mass-dependent correction, valid up to first order in [Formula: see text]. This novel feature is expected to have experimental significance in particle physics. We also show that it is not possible to construct a conserved current associated with charge conjugation, showing that the Dirac particle and its antiparticle satisfy different equations in [Formula: see text] space–time.

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

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.