﻿ 局域规范变换中协变导数D<sub>μ</sub>的作用新探

# 局域规范变换中协变导数Dμ的作用新探A New Study on the Role of Covariant Derivative Dμ in Local Gauge Transformation

Abstract: It is not accurate to say that electromagnetic action is represented by introducing gauge field through covariant derivative. In fact, the local gauge transformation of charged free particles in-troduces electromagnetic fields. The introduction of gauge field by covariant derivative eli-minates the effect of electromagnetic field introduced by local gauge transformation. This is a physical and mathematical operation to restore the free motion state of charged particles and Lo-rentz covariant to ensure the form invariance of Dirac equation.

1. 整体规范变换

$\psi \to {\psi }^{\prime }={\text{e}}^{-i\alpha }\psi$

$\stackrel{¯}{\psi }\to {\stackrel{¯}{\psi }}^{\prime }=\stackrel{¯}{\psi }{\text{e}}^{i}{}^{\alpha }$

${\partial }_{\mu }\psi \to {\partial }_{\mu }{\psi }^{\prime }={\text{e}}^{-i\alpha }{\partial }_{\mu }\psi$

$\psi$$\stackrel{¯}{\psi }$${\partial }_{\mu }\psi$ 按相同规律变换。即可得到洛伦兹协变形式的狄拉克方程 [1]

$\left(i{r}^{\mu }{\partial }_{\mu }-m\right)\psi =0$

2. 局域规范变换

$\psi \to {\psi }^{\prime }={\text{e}}^{-i\alpha \left(x\right)}\psi$

$\stackrel{¯}{\psi }\to {\stackrel{¯}{\psi }}^{\prime }={\text{e}}^{i\alpha \left(x\right)}\stackrel{¯}{\psi }$

${\partial }_{\mu }\psi \to {\partial }_{\mu }{\psi }^{\prime }={\partial }_{\mu }\left[{\text{e}}^{-i\alpha \left(x\right)}\psi \right]={\text{e}}^{-i\alpha \left(x\right)}{\partial }_{\mu }\psi -i{\partial }_{\mu }\alpha \left(x\right){\text{e}}^{-i\alpha \left(x\right)}\psi \ne {\text{e}}^{-i\alpha \left(x\right)}{\partial }_{\mu }\psi$

${\partial }_{\mu }\to {D}_{\mu }={\partial }_{\mu }-ie{A}_{\mu }$

$\left(i{r}^{\mu }{D}_{\mu }-m\right)\psi =0$

3. 规范变换协变导数的物理意义

$\left(i{r}^{\mu }{D}_{\mu }-m\right)\psi =0$

4. 结论

1) 通过协变导数引进规范场，让粒子作自由运动，能保证狄拉克方程形式不变。

2) 带电粒子作自由运动，就可以保证局域规范变换中封闭曲面流和荷的守恆。

3) 对自旋波函数ψ(x)作局域规范变换，自旋与轨道之间就引进了偶合作用。

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[3] Zhao, G.Q. (2019) Meaning of the Wave Function and the Origin of Probability in Quantum Mechanics. International Journal of Quantum Foundations, 1, 32-45. https://www.ijqf.org/archives/5710

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