标题： 狄拉克粒子，自旋和光子 显示英文标题

标题： Dirac particles, spin and photons

摘要： 我们用相空间$X=T^* R^{1,3}\times C^2_L\times C^2_R$中移动的点来描述自旋为相对论粒子，其中$T^* R^{1,3}=R^{1,3}\times R^{1,3}$是坐标和动量的空间，$C^2_L$和$C^2_R$是洛伦兹群类型$(\frac12 , 0)$和$(0, \frac12)$的表示空间。 从相对论力学中具有洛伦兹不变的哈密顿函数$H$在相空间$X$上过渡到量子力学中的哈密顿算子$\hat H$，我们引入了两个复共轭线丛$L_C^+$和$L_C^-$在$X$上。 量子粒子被引入为沿空间$C^2_L\times C^2_R$解析的丛$L_C^+$的截面$\Psi_+$，反粒子是沿内部自旋空间$C^2_L\times C^2_R$反解析的丛$L_C^-$的截面$\Psi_-^{}$。 The wave functions $\Psi_\pm$ are characterized by conserved charges $q_{\sf{v}}=\pm 1$ associated with the structure group U(1)$_{\sf{v}}$ of the bundles $L_C^\pm$. Wave functions $\Psi_\pm$ are governed by relativistic analogue of the Schrödinger equation. 我们展示如何从这些方程在自旋空间坐标$C^2_L\times C^2_R$中函数$\Psi_\pm^{}$的零阶、一阶和二阶展开中得到自旋$s=0$(Klein-Gordon)、自旋$s=\frac12$(Dirac) 和自旋$s=1$(Proca 场) 的场。 这些场的 Klein-Gordon、Dirac 和 Proca 方程来源于扩展相空间$T^* R^{1,3}\times C^2_L\times C^2_R$上的薛定谔方程。 利用这些结果，我们还引入描述第一量子化光子的方程。 我们证明，考虑场 $\Psi_\pm$ 的电荷 $q_{\sf{v}}=\pm 1$ 会改变内积和电流的定义，这消除了相对论量子力学中的负能和负概率。 显示英文摘要

摘要： We describe relativistic particles with spin as points moving in phase space $X=T^* R^{1,3}\times C^2_L\times C^2_R$, where $T^* R^{1,3}=R^{1,3}\times R^{1,3}$ is the space of coordinates and momenta, and $C^2_L$ and $C^2_R$ are the spaces of representation of the Lorentz group of type $(\frac12 , 0)$ and $(0, \frac12)$. Passing from relativistic mechanics with a Lorentz-invariant Hamiltonian function $H$ on the phase space $X$ to quantum mechanics with a Hamiltonian operator $\hat H$, we introduce two complex conjugate line bundles $L_C^+$ and $L_C^-$ over $X$. Quantum particles are introduced as sections $\Psi_+$ of the bundle $L_C^+$ holomorphic along the space $C^2_L\times C^2_R$, and antiparticles are sections $\Psi_-^{}$ of the bundle $L_C^-$ antiholomorphic along the internal spin space $C^2_L\times C^2_R$. The wave functions $\Psi_\pm$ are characterized by conserved charges $q_{\sf{v}}=\pm 1$ associated with the structure group U(1)$_{\sf{v}}$ of the bundles $L_C^\pm$. Wave functions $\Psi_\pm$ are governed by relativistic analogue of the Schr\"odinger equation. We show how fields with spin $s=0$ (Klein-Gordon), spin $s=\frac12$ (Dirac) and spin $s=1$ (Proca fields) arise from these equations in the zeroth, first, and second order expansions of the functions $\Psi_\pm^{}$ in the coordinates of the spin space $C^2_L\times C^2_R$. The Klein-Gordon, Dirac and Proca equations for these fields follow from the Schr\"odinger equation on the extended phase space $T^* R^{1,3}\times C^2_L\times C^2_R$. Using these results, we also introduce equations describing first quantized photons. We show that taking into account the charges $q_{\sf{v}}=\pm 1$ of the fields $\Psi_\pm$ changes the definitions of the inner products and currents, which eliminates negative energies and negative probabilities from relativistic quantum mechanics.