摘要:
狄拉克要求相对论量子力学方程是线性的、且满足相对论质能关系从而得到了狄拉克方程，为此引入了4 × 4矩阵形式的系数，相应的波函数取4-分量的形式。在狄拉克方程的语境中，电子自旋归于相对论效应，从对解的诠释还引出了正电子的概念。然而，为方程中的系数选取4 × 4矩阵形式及其具体表示并没有原理性的支撑，且把波函数理解为4-分量也值得商榷。文章中，作者通过引入4组独立的产生—湮灭算符{(bk,bk+),k=0,1,2,3}，得到狄拉克方程的新表示。这样做不仅规避了为什么要为系数选择4 × 4矩阵的问题，也消解了狄拉克原方程中系数不等价的问题。相对论质能方程要求产生—湮灭算符(bk,bk+)满足反对易关系，这直接决定了由此引入的“自旋”是个二值的内部自由度，这解释了为什么狄拉克理论只有S=1/2的情形。新形式下角动量的外部部分和“自旋”具有相同的形式，角动量和哈密顿量都表现出外部自由度(满足对易关系)与内部自由度(满足反对易关系)之间的耦合。新形式的狄拉克方程消除了对4 × 4矩阵形式的需求，至少会让后续的量子场论计算变得简单。此外，作者注意到狄拉克、薛定谔在1926—1930年这段时期对量子论中的复数共轭、电荷共轭、内部自由度等观念以及它们之间的关联都有深刻、正确的认识。文章还针对相对论量子力学方程诠释所涉及的复共轭、电荷共轭、内部自由度及其与运动变量之间的耦合、方程要素应遵循的代数结构等问题进行简短的讨论，以期有助于正确理解相对论量子力学。
Abstract:
Dirac derived the equation of relativistic quantum mechanics by requiring it be linear and satisfy the relativistic mass-energy relation; to this end, he introduced the coefficients in 4×4 matrices, and, correspondingly, the wavefunction assumes a 4-component form. In the context of Dirac equation, electron spin is attributed to the relativistic effect, and the concept of positron was eventually proposed from the interpretation to the solution. However, the choice of a 4 × 4 matrix form—along with its specific representation—for the coefficients in the equation lacks fundamental justification; furthermore, interpreting the wavefunction as a 4-component entity is also unreasonable. In this paper, by introducing four sets of independent creation and annihilation operators {(bk, bk+), k = 0, 1, 2, 3} we derive a novel representation of the Dirac equation. This approach not only circumvents the question of why 4×4 matrices were chosen for the coefficients, but also resolves the issue of the non-equivalence of the matrix coefficients present in Dirac's original equation. The relativistic mass-energy equation requires anticommutative creation and annihilation operators (bk, bk+, this directly determines that the “spin”thereby introduced is a two-valued internal degree of freedom, which explains why Dirac theory encompasses only the case of S = 1/2.Under this new formalism, the external part of angular momentum and“spin”assume identical forms; furthermore, both the angular momentum and the Hamiltonian exhibit a coupling between the external degree of freedom, which satisfies commutation relation, and internal degree of freedom, which satisfies anticommutation relation. This new formulation of the Dirac equation eliminates the need for 4×4 matrix representations—a feature that, at the very least, promises to simplify subsequent calculations in quantum field theory. Furthermore, we note that during the period from 1926 to 1930, Dirac and Schrödinger possessed a profound and accurate understanding of concepts within quantum theory—such as complex conjugation, charge conjugation, and internal degrees of freedom—as well as the interconnections among them. The current paper also briefly discusses issues pertinent to the interpretation of relativistic quantum mechanical equation—specifically, complex conjugation, charge conjugation, internal degree of freedom and their coupling with dynamical variables, and the algebraic structures that the elements of the Dirac equation must obey—with the aim of facilitating a correct understanding of relativistic quantum mechanics.
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