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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.