Formulation of the Spinor Field in the Presence of a Minimal Length Based on the Quesne-Tkachuk Algebra

TL;DR

This paper formulates the Dirac equation within a Lorentz-covariant deformed algebra introducing a minimal length, revealing two mass states and establishing an upper bound on minimal length near the electron's Compton wavelength.

Contribution

It develops a Lagrangian formulation for spinor fields under Quesne-Tkachuk algebra, deriving a modified Dirac equation with higher derivatives and analyzing minimal length bounds.

Findings

Modified Dirac equation describes two massive particles.

Physically acceptable masses require eta<1/8m^2c^2.

Upper bound for minimal length near 3^{-13} m.

Abstract

In 2006 Quesne and Tkachuk (J. Phys. A: Math. Gen. {\bf 39}, 10909, 2006) introduced a (D+1)-dimensional $(\beta,\beta')$-two-parameter Lorentz-covariant deformed algebra which leads to a nonzero minimal length. In this work, the Lagrangian formulation of the spinor field in a (3+1)-dimensional space-time described by Quesne-Tkachuk Lorentz-covariant deformed algebra is studied in the case where $\beta'=2\beta$ up to first order over deformation parameter $\beta$. It is shown that the modified Dirac equation which contains higher order derivative of the wave function describes two massive particles with different masses. We show that physically acceptable mass states can only exist for $\beta<\frac{1}{8m^{2}c^{2}}$. Applying the condition $\beta<\frac{1}{8m^{2}c^{2}}$ to an electron, the upper bound for the isotropic minimal length becomes about $3 \times 10^{-13}m$. This value is near to the reduced Compton wavelength of the electron $(\lambda_c = \frac{\hbar}{m_{e}c} = 3.86\times 10^{-13} m)$ and is not incompatible with the results obtained for the minimal length in previous investigations.