非对称粒子-反粒子狄拉克方程：一次量子化
Asymmetric particle-antiparticle Dirac equation: first quantization
AI总结：
本文提出非对称狄拉克方程，使同波数的粒子与反粒子具有不同能量和动量，证明其洛伦兹协变性、与标准狄拉克方程的联系，并求解氢原子及非相对论极限。
摘要：
我们推导出一个类狄拉克方程，即非对称狄拉克方程，其中具有相同波数的粒子和反粒子具有不同的能量和动量。我们证明该方程在固有洛伦兹变换（boost 和空间旋转）下是洛伦兹协变的，并确定了其波函数相应的变换规律。我们得到了非对称狄拉克方程与标准狄拉克方程之间的形式联系，并证明通过适当调节本波动方程的自由参数，可以使其重现通常狄拉克方程的预言。我们证明，在非对称狄拉克方程的理论框架中，粒子的静止质量是一组四个参数的函数，这些参数在固有洛伦兹变换下是相对论不变量。这四个参数类似于标准狄拉克方程中出现的质量。我们证明，为了保证非对称狄拉克方程在宇称和时间反演操作（非固有洛伦兹变换）以及电荷共轭操作下的协变性，这四个参数改变符号的方式与四矢量的四个分量完全相同。然而，质量作为这些参数平方的函数，仍然是不变量。我们还广泛研究了非对称狄拉克方程的自由粒子平面波解，并推导出其能量、螺旋度和自旋投影算符以及若干 Gordon 恒等式。在对非对称狄拉克方程应用最小耦合方案后，我们在当前框架下求解了氢原子，这也使我们能够适当地得到其非相对论极限。
英文摘要：
We derive a Dirac-like equation, the asymmetric Dirac equation, where particles and antiparticles sharing the same wave number have different energies and momenta. We show that this equation is Lorentz covariant under proper Lorentz transformations (boosts and spatial rotations) and also determine the corresponding transformation law for its wave function. We obtain a formal connection between the asymmetric Dirac equation and the standard Dirac equation and we show that by properly adjusting the free parameters of the present wave equation we can make it reproduce the predictions of the usual Dirac equation. We show that the rest mass of a particle in the theoretical framework of the asymmetric Dirac equation is a function of a set of four parameters, which are relativistic invariants under proper Lorentz transformations. These four parameters are the analog to the mass that appears in the standard Dirac equation. We prove that in order to guarantee the covariance of the asymmetric Dirac equation under parity and time reversal operations (improper Lorentz transformations) as well as under the charge conjugation operation, these four parameters change sign in exactly the same way as the four components of a four-vector. The mass, though, being a function of the square of those parameters remains an invariant. We also extensively study the free particle plane wave solutions to the asymmetric Dirac equation and derive its energy, helicity, and spin projection operators as well as several Gordon's identities. The hydrogen atom is solved in the present context after applying the minimal coupling prescription to the asymmetric Dirac equation, which also allows us to appropriately obtain its non-relativistic limit.
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