ua nt - p h / 06 04 11 8 v 2 9 A ug 2 00 6 Lorentz - covariant deformed algebra with minimal length and application to the ( 1 + 1 ) - dimensional Dirac oscillator

The D-dimensional (β, β ′)-two-parameter deformed algebra introduced by Kempf is generalized to a Lorentz-covariant algebra describing a (D + 1)-dimensional quan-tized spacetime. In the D = 3 and β = 0 case, the latter reproduces Snyder algebra. The deformed Poincaré transformations leaving the algebra invariant are identified. It is shown that there exists a nonzero minimal uncertainty in position (minimal length). The Dirac oscillator in a (1 + 1)-dimensional spacetime described by such an algebra is studied in the case where β ′ = 0. Extending supersymmetric quantum mechanical and shape-invariance methods to energy-dependent Hamiltonians provides exact bound-state energies and wavefunctions. Physically acceptable states exist for β < 1/(m 2 c 2). A new interesting outcome is that, in contrast with the conventional Dirac oscillator, the energy spectrum is bounded.