The famous equation $E=mc^2$ is a version of particle mass being essentially the magnitude of the (energy-)momentum four-vector in the setting of `relativistic' dynamics, which can be seen as dictated by the Poincaré symmetry adopted as the relativity symmetry. However, as Einstein himself suggested, the naive notion of momentum as mass times velocity may not be right. The Hamiltonian formulation perspective gives exactly such a setting which in the case of motion of a charged particle under an electromagnetic field actually has the right, canonical, momentum four-vector with an evolving magnitude. The important simple result seems to have missed proper appreciation. In relation to that, we present clear arguments against taking the Poincaré symmetry as the fundamental symmetry behind `relativistic' quantum dynamics, and discuss the proper symmetry theoretical formulation and the necessary picture of the covariant Hamiltonian dynamics with an evolution parameter that is, in general, not a particle proper time. In fact, it is obvious that the action of any position operator of a quantum state violates the on-shell mass condition. The phenomenologically quite successful quantum field theories are `second quantized' versions of `relativistic' quantum mechanics. We present a way for some reconciliation of that with our symmetry picture and discuss implications.
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。
