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《THE DIRAC EQUATION》CHAPTER 4( SUMMARY )
[align=justify]In order to make the Dirac operator well defined and to ensure the existence of unique solutions for the initial value problem, the potential function V(x) must have a certain regularity. In Sect. 4.3 we formulate conditions establishing the self-adjointness of the Dirac operator H = H_0 + V. Concerning the local behavior, potentials with 1/r-singularities are admitted only for coupling constants yless than c/2. Stronger singularities can only be dealt with for special matrix potentials. In the case of an electrostatic Coulomb potential one can find physically distinguished self-adjoint extensions for y < c (nuclear charges < 137). The behavior of the potentials at infinity is not restricted in contrast to the Schrodinger case. This can be understood as a consequence of the finite propagation speed of Dirac particles. In Sect. 4.3.4 we show that the Dirac operator in an external field has the same essential spectrum as the free Dirac operator provided the potential vanishes at infinity.
For static potentials the self-adjointness of the Dirac operator is sufficient to solve the initial value problem by Stone's theorem. For time-dependent potentials we need some additional assumptions. A short discussion of these problems is given in Sect. 4.4. The situation is particularly simple if the time dependence is the result of a gauge or symmetry transformation. The limitations of the theory are most clearly indicated by the Klein paradox. This phenomenon occurs whenever the interaction is so strong as to cause transitions from electron states to positron states. We show that in the presence of high potential steps the Dirac equation has solutions which violate the principle of charge conservation. This paradoxical situation has no completely satisfactory physical explanation within a one particle interpretation.
The limitations of the theory are most clearly indicated by the Klein paradox. This phenomenon occurs whenever the interaction is so strong as to cause transitions from electron states to positron states. We show that in the presence of high potential steps the Dirac equation has solutions which violate the principle of charge conservation. This paradoxical situation has no completely satisfactory physical explanation within a one particle interpretation.
The explicit solution of the Dirac equation with an external field is of course largely facilitated by the presence of symmetries. Most important for applications (e.g., the hydrogen atom) are spherically symmetric potentials, which we treat in Sect. 4.6. In this case the Hilbert space can be decomposed into an orthogonal direct sum of partial wave subspaces, which are the simultaneous eigenspaces of the angular momentum operators J^2,J_3 and the operator K which describes the spin-orbit coupling. On each subspace the Dirac equation is equivalent to a two dimensional system of ordinary differential equations of first order (the radial DiraC equation). Some general results on the spectral properties of the radial Dirac operator are reviewed in Sect. 4.6.6.
In Sect. 4.7 we present some results on the behavior of eigenvalues. We prove the relativistic virial theorem, which gives simple criteria for the absence of embedded eigenvalues in certain regions of the continuous spectrum.
《狄拉克方程》第四章提要
[align=justify] 本章我们介绍作为乘法算符V的含时空参数x的4*4矩阵值函数的一种势场。根据他们在庞加莱变换下的特性，场可以分标量势，赝标量势，矢量势或赝矢量势，张量力或赝张量力（见4.1和4.2节）。矢量势描述电磁力，标量势可能用作夸克禁闭模型，而张量力对于描述反常电磁矩却是必不可少的.
为了给狄拉克算子以准确定义并确保初值问题的解的唯一性，势函数Ｖ(x)必须有一定的规律性。在4.3节，我们阐述建立狄拉克算符H=H_0+V的自伴矩阵的条件。关于局部性，呈1/r规律的势已被证明的仅适合于耦合小于c/2的常数。更强的奇异性仅能被用于处理特殊的矩阵势。在静电库仑势y/r的情形下人们可以发现自然的y< c(即核电荷数<137)的著名自伴随矩阵。与Schr鰀inger情形相比，无穷势是不受限制的。这可被理解为狄拉克粒子的有限性传播的一个结果。在4.3.4节，我们证明了狄拉克算子在外场中有着对应着无穷远处时零势能极限的自由狄拉克算子的相同的本质谱。
对于静态势，狄拉克算子自伴随矩阵根据运用斯通定理足以求解初值问题。对于含时势我们需要一些附加的假设。这些问题简短的讨论见于4.4节。如果含时关系是测量或对称变换的结果，这种情形将尤其简单。Klein矛盾最清楚地表现出理论的局限性。这现象发生在交互作用如此强以致引起从电子态到正电子转变的任何时候。我们证明在高势能步幅的情形下Dirac方程存在违返电荷守恒律的解，这种荒谬的结果没有任何令人满意的的物理解释在描述单个粒子的时候。
Klein佯谬最最清楚地表明了Dirac理论的局限性。这一现象发生在相互作用如此强因此致使从(负)电子态到正电子态转变的任何时候。我们阐明在高势阶跃下狄拉克方程存在违反电荷守恒定律的解。这一荒谬的情形在单粒描述中没有真正令人满意的解释（这些矛盾的各种解释，通常都不是符合客观逻辑的——Sunroom注）。
外加场中Dirac方程的显性解当然主要是对称性下得出的。大部分重要应用对象（例如氢原子）是球对称势，这一部见4.6节。在此情形下希尔伯特空间可被分解为部分波子空间的直角方向的和，这子空间是本征角动量算符J^2,J_3和描述自旋轨道耦合算符K同时存在的本征空间。在每一个子空间中狄拉克方程等效为一阶线性微分方程组的两个空间系统（径向Dirac方程）。有关径向Dirac算子的谱特征的一些普遍方法在4.6.4节中作了回顾。
在4.7节里，我们介绍一些关于本征值特性的结果。我们证明相对论维里定理，这定理对于连续谱空间里本征值的缺乏给出了简单的标准。 | 
发表于 2007-5-22 07:59:07
