We analyze a modified Dirac equation based on a noncommutative structure in phase space originating from a generalized uncertainty principle with a minimum length. The noncommutative structure induces generalized momenta and contributions to the energy levels of the standard Dirac equation. Applying techniques of perturbation theory, we find the lowest-order corrections to the energy levels and eigenfunctions of the Dirac equation in three dimensions for a spherically symmetric linear potential and for a square-well times triangular potential along one spatial dimension. We find that the corrections due to the noncommutative contributions may be of the same order as the relativistic ones, leading to an upper bound on the parameter fixing the minimum length induced by the generalized uncertainty principle.

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具有广义动量的狄拉克方程能级的微扰计算

我们分析了基于相空间中的非对易结构的修正狄拉克方程，该结构源自具有最小长度的广义不确定性原理。非对易结构引入了广义动量和对标准狄拉克方程能级的贡献。应用微扰理论技术，我们找到了狄拉克方程的能级和本征函数在三个维度上的最低阶校正，用于球对称线性势和沿一个空间维度的方阱乘以三角势。我们发现由于非交换贡献引起的修正可能与相对论贡献的修正量级相同，从而导致由广义不确定性原理引起的固定最小长度的参数的上限。