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 quantized space-time. 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 space-time 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.