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 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.