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Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation
Symmetry ( IF 2.2 ) Pub Date : 2020-07-06 , DOI: 10.3390/sym12071121
Maen Salman , Trond Saue

Four-component relativistic atomic and molecular calculations are typically performed within the no-pair approximation where negative-energy solutions are discarded. These states are, however, needed in QED calculations, wherein, furthermore, charge conjugation symmetry, which connects electronic and positronic solutions, becomes an issue. In this work, we shall discuss the realization of charge conjugation symmetry of the Dirac equation in a central field within the finite basis approximation. Three schemes for basis set construction are considered: restricted, inverse, and dual kinetic balance. We find that charge conjugation symmetry can be realized within the restricted and inverse kinetic balance prescriptions, but only with a special form of basis functions that does not obey the right boundary conditions of the radial wavefunctions. The dual kinetic balance prescription is, on the other hand, compatible with charge conjugation symmetry without restricting the form of the radial basis functions. However, since charge conjugation relates solutions of opposite value of the quantum number κ , this requires the use of basis sets chosen according to total angular momentum j rather than orbital angular momentum l. As a special case, we consider the free-particle Dirac equation, where opposite energy solutions are related by charge conjugation symmetry. We show that there is additional symmetry in that solutions of the same value of κ come in pairs of opposite energy.

中文翻译:

狄拉克方程有限基近似中的电荷共轭对称性

四分量相对论原子和分子计算通常在无对近似内进行,其中负能量解被丢弃。然而,在 QED 计算中需要这些状态,此外,连接电子和正电子解决方案的电荷共轭对称性成为一个问题。在这项工作中,我们将讨论在有限基近似内的中心场中狄拉克方程的电荷共轭对称性的实现。考虑了三种基组构造方案:受限、逆和双动平衡。我们发现电荷共轭对称可以在受限和逆动力学平衡处方内实现,但只能通过一种特殊形式的基函数不服从径向波函数的正确边界条件。另一方面,对偶动力学平衡公式与电荷共轭对称性兼容,而不限制径向基函数的形式。然而,由于电荷共轭涉及量子数 κ 的相反值的解,这需要使用根据总角动量 j 而不是轨道角动量 l 选择的基组。作为一种特殊情况,我们考虑自由粒子狄拉克方程,其中相反的能量解通过电荷共轭对称性相关。我们表明,在具有相同 κ 值的解成对出现相反能量时,存在额外的对称性。由于电荷共轭涉及量子数 κ 的相反值的解,这需要使用根据总角动量 j 而不是轨道角动量 l 选择的基组。作为一种特殊情况,我们考虑自由粒子狄拉克方程,其中相反的能量解通过电荷共轭对称性相关。我们表明,在具有相同 κ 值的解成对出现相反能量时,存在额外的对称性。由于电荷共轭涉及量子数 κ 的相反值的解,这需要使用根据总角动量 j 而不是轨道角动量 l 选择的基组。作为一种特殊情况,我们考虑自由粒子狄拉克方程,其中相反的能量解通过电荷共轭对称性相关。我们表明,在具有相同 κ 值的解成对出现相反能量时,存在额外的对称性。
更新日期:2020-07-06
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