基于Quesne-Tkachuk代数的极小长度旋量场的表示
Formulation of the Spinor Field in the Presence of a Minimal Length Based on the Quesne-Tkachuk Algebra
摘要：
2006年Quesne和Tkachuk（J. Phys. A：Math. Gen. { \bf 39}，10909，2006）引入了一个（D+1）维$(β,β')$ -两参数Lorentz-协变变形代数，它导致了一个非零的最小长度。本文研究了在形变参数$β$上一阶$β'=2β$的情况下，由Quesne-Tkachuk Lorentz-协变形变代数描述的（3+1）维时空中旋量场的拉格朗日形式.结果表明，含有波函数高阶导数的修正狄拉克方程描述了两个质量不同的重粒子。我们表明，物理上可接受的质量状态只能存在$β<\frac{1}{8m^{2}c^{2}}$。将条件$β<\frac{1}{8m^{2}c^{2}}$应用于电子，各向同性最小长度的上限变为约$3 \times 10^{-13}m$。这个值接近于电子$(λ_c = \frac{\hbar}{m_{e}c} = 3.86\times 10^{-13} m)$的约化康普顿波长，并且与以前研究中对最小长度所得到的结果并不矛盾。
英文摘要：
In 2006 Quesne and Tkachuk (J. Phys. A: Math. Gen. {\bf 39}, 10909, 2006) introduced a (D+1)-dimensional $(β,β')$-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 $β'=2β$ up to first order over deformation parameter $β$. 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 $β<\frac{1}{8m^{2}c^{2}}$. Applying the condition $β<\frac{1}{8m^{2}c^{2}}$ 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 $(λ_c = \frac{\hbar}{m_{e}c} = 3.86\times 10^{-13} m)$ and is not incompatible with the results obtained for the minimal length in previous investigations.
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