质量算符和五维概率分布空间的规范场模型

1, 质量算符

$\hat{m}=-ih\frac{\partial }{\partial \tau }$

$\hat{E}=ih\frac{\partial }{\partial t}$

$\hat{p_{i}}=-ih\frac{\partial }{\partial x^{i}}$

2,　引入质量算符的动机

3, 算符之间的对易关系

x_{a}=(\vec{r},t,\tau )" style="font-family:宋体;text-align:right;           $a=1,2,3,4,5$

$g^{ab}=\begin{pmatrix} &-1 &0 &0 &0 &0 \\ &0 &-1 &0 &0 &0 \\ &0 &0 &-1 &0 &0 \\ &0 &0 &0 &1 &0 \\ &0 &0 &0 &0 &-1 \end{pmatrix}$  ，

$\partial_{a}=(\vec{\partial},\frac{\partial }{\partial t},\frac{\partial }{\partial \tau } )=(\partial _{\mu },\partial _{\tau });$

$\partial _{a}\partial ^{a}=-\partial_{i}\partial _{i}+\frac{\partial^{2} }{\partial t^{2}}-\frac{\partial ^{2}}{\partial \tau^{2} }.$

5维变量的自由Dirac方程，

$i\gamma ^{a}\partial _{a}\Psi=(i\gamma ^{\mu }\partial _{\mu }+i\partial_{\tau} )\Psi=0$

$L_{0}=\bar{\Psi} \gamma ^{a}\partial _{a}\Psi.$

4, 规范原理

$\Psi \rightarrow {\Psi }'=e^{i\alpha (x,y,z,t,\tau )}\Psi$

$A_{a}=(A_{1},A_{2},A_{3},\varphi ,\phi )=(A_{\mu },\phi )$

$\phi =\phi (x,y,z,t,\tau )$

$%uFF26_{ab}=\partial _{a}A_{b}-\partial _{b}A_{a}$ $F_{ab}=\partial _{a}A_{b}-\partial _{b}A_{a}$

$L_{0}=-\frac{1}{4}F_{ab}F^{ab}$

$D_{a}\equiv \partial _{a}-ieA_{a}$

$L=\bar{\Psi}\gamma ^{a}D_{a}\Psi-\frac{1}{4}F_{ab}F^{ab}$

http://www.npr.ac.cn/qikan/manage/wenzhang/2014-1-23.pdf

https://blog.sciencenet.cn/blog-96769-860080.html

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