In two recent papers exact Hermite-Gaussian solutions to relativistic wave equations have been obtained for both electromagnetic and particle beams that include Gouy phase. The solutions for particle beams correspond to those of the Schrödinger equation in the non-relativistic limit. Here, distinct canonical and kinetic 4-momentum operators will be defined for quantum particles in matter wave beams. The kinetic momentum is equal to the canonical momentum minus the fluctuating terms resulting from the transverse localization of the beam. Three results are obtained. First, the total energy of a particle for each beam mode is calculated. Second, the localization terms couple into the canonical 4-momentum of the beam particles as a Lorentz covariant quantum 4-potential originating at the waist. The quantum 4-potential plays an analogous role in relativistic Hamiltonian quantum mechanics to the Bohm potential in the non-relativistic quantum Hamilton-Jacobi equation. Third, the orbital angular momentum (OAM) operator must be defined in terms of canonical momentum operators. It is further shown that kinetic 4-momentum does not contribute to OAM indicating that OAM can therefore be regarded as a pure manifestation of quantum 4-potential.
Integration of Dirac's Efforts to construct Lorentz-covariant Quantum Mechanics
保罗·A. M.狄拉克在洛伦兹协变世界中构造定域量子系统。1927年，他指出时间-能量不确定性应该包含在洛伦兹协变图像中。在1945年，他试图用一个在空间和时间变量中都局部化的可归一化高斯函数来构造洛伦兹群的表示。1949年，他提出了瞬时形式，以排除类时振荡。他还介绍了光锥坐标系洛伦兹助推。同样在1949年，他指出非齐次洛伦兹群的李代数可以作为洛伦兹协变世界中的不确定关系。综合这三篇论文，就有可能产生一个可以进行洛仑兹变换的谐振子波函数。此外，狄拉克在1963年考虑了两个耦合振子，导出了$O(3,\,2)$ de Sitter群的生成元的李代数，该群有十个生成元。它被证明是可能的收缩这个群的非均匀洛伦兹群的10个生成元，构成了量子力学的基本对称性在爱因斯坦的洛伦兹协变世界。
