# 电荷守恒定律

$Q(t_2)=Q(t_1) + Q_{IN} - Q_{OUT}$

——班傑明·富蘭克林[6]

## 電磁學表述

$I=-\oint_\mathbb{S} \mathbf{J} \cdot \mathrm{d}^2\mathbf{r}$

$I=-\int_\mathbb{V} \nabla\cdot\mathbf{J}\ \mathrm{d}^3r$

$Q=\int_\mathbb{V} \rho\ \mathrm{d}^3r$

$\frac{\mathrm{d}Q}{\mathrm{d} t}=I=\int_\mathbb{V} \frac{\partial \rho}{\partial t}\ \mathrm{d}^3r$

$\int_\mathbb{V}\frac{\partial \rho}{\partial t}+\mathbf{\nabla} \cdot \mathbf{J}\ \mathrm{d}^3r=0$

$\frac{\partial \rho}{\partial t}+\nabla \cdot \mathbf{J} =0$

$\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\epsilon_0\frac{\partial\mathbf{E}}{\partial t}$

$\nabla\cdot(\nabla\times\mathbf{B})=0=\mu_0\nabla\cdot\mathbf{J}+\mu_0\epsilon_0\frac{\partial(\nabla\cdot\mathbf{E})}{\partial t}$

$\nabla\cdot\mathbf{E}=\rho/\epsilon_0$

$\frac{\partial \rho}{\partial t}+\mathbf{\nabla} \cdot \mathbf{J}=0$

## 規範不變性

### 電磁學

$\phi' = \phi - \frac {\partial \Lambda}{\partial t}$
$\mathbf{A}' = \mathbf{A} + \nabla \Lambda$

$\mathbf{E}' =-\nabla \phi' - \frac {\partial \mathbf{A}'}{\partial t}=-\nabla \phi - \frac {\partial \mathbf{A}}{\partial t}=\mathbf{E}$
$\mathbf{B}' =\nabla\times\mathbf{A}'=\nabla\times\mathbf{A}=\mathbf{B}$

### 諾特定理

\begin{align}\mathcal{L} & =-\ \frac{1}{16\pi}F_{\alpha\beta}F^{\alpha\beta}-\ \frac{1}{c}J_{\alpha}A^{\alpha} \\ & =-\ \frac{1}{16\pi}(\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha})(\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha})-\ \frac{1}{c}J_{\alpha}A^{\alpha} \\ \end{align}

$A'^{\alpha}=A^{\alpha}+\partial^{\alpha}\Lambda$

\begin{align}\mathcal{L}' & =-\ \frac{1}{16\pi}[\partial_{\alpha}(A_{\beta}+\partial_{\beta}\Lambda)-\partial_{\beta}(A_{\alpha}+\partial_{\alpha}\Lambda)]\ [\partial^{\alpha}(A^{\beta}+\partial^{\beta}\Lambda)-\partial^{\beta}(A^{\alpha}+\partial^{\alpha}\Lambda)] -\ \frac{1}{c}J_{\alpha}(A^{\alpha}+\partial^{\alpha}\Lambda) \\ & =-\ \frac{1}{16\pi}(\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha})(\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha}) -\ \frac{1}{c}J_{\alpha}(A^{\alpha}+\partial^{\alpha}\Lambda) \\ & =\mathcal{L} -\ \frac{1}{c}J_{\alpha}\partial^{\alpha}\Lambda \\ \end{align}

$\mathcal{I}'-\mathcal{I}=-\ \frac{1}{c}\int_{\mathbb{V}} J_{\alpha}\partial^{\alpha}\Lambda \mathrm{d}^4x =-\ \frac{1}{c}\int_{\mathbb{V}} \partial^{\alpha}(J_{\alpha}\Lambda)\mathrm{d}^4x +\ \frac{1}{c}\int_{\mathbb{V}} \Lambda\partial^{\alpha}J_{\alpha} \mathrm{d}^4x$

$\mathcal{I}'-\mathcal{I}= \frac{1}{c}\int_{\mathbb{V}} \Lambda\partial^{\alpha}J_{\alpha} \mathrm{d}^4x$

$\partial^{\alpha}J_{\alpha}=0$

### 規範場論

$\mathcal{L}=i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-mc^2\overline{\psi}\psi$

$\psi'=\psi e^{i\theta}$

\begin{align}\mathcal{L}' & =i\hbar c\overline{\psi'}\gamma^{\mu}\partial_{\mu}\psi'-mc^2\overline{\psi'}\psi' \\ & =i\hbar c\overline{\psi} e^{-i\theta}\gamma^{\mu}\partial_{\mu}(\psi e^{i\theta})-mc^2\overline{\psi} e^{-i\theta}\psi e^{i\theta} \\ & =i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-mc^2\overline{\psi}\psi \\ & =\mathcal{L} \\ \end{align}

$\mathcal{L}'=\mathcal{L}-\hbar c(\partial_{\mu}\theta)\overline{\psi}\gamma^{\mu}\psi$

$\mathcal{L}_1=i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-mc^2\overline{\psi}\psi -q\overline{\psi}\gamma^{\mu}\psi A_{\mu}$

$\mathcal{L}'_1=\mathcal{L}_1-\hbar c(\partial_{\mu}\theta)\overline{\psi}\gamma^{\mu}\psi+q\overline{\psi}\gamma^{\mu}\psi\partial_{\mu}\Lambda$

$\mathcal{L}_P =-\ \frac{1}{16\pi}(\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha})(\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha})+ \frac{m^2c^2}{8\pi\hbar^2}A^{\nu}A_{\nu}$

$\mathcal{L}_2=i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-mc^2\overline{\psi}\psi -\ \frac{1}{16\pi}(\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha})(\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha})-q\overline{\psi}\gamma^{\mu}\psi A_{\mu}$

$\mathcal{L}_2=i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-mc^2\overline{\psi}\psi -\ \frac{1}{16\pi}(F_{\alpha\beta}F^{\alpha\beta})-\frac{1}{c}J^{\mu}A_{\mu}$

$\partial^{\mu}F_{\mu\nu}-\frac{4\pi}{c}J^{\mu}=0$

## 實驗證據

 $e\to \nu_e\gamma$ 平均壽命大於4.6×1026年（90% 置信水平）。[17]

 $e\to$任意粒子 平均壽命大於6.4×1024年（68% 置信水平）[20] $n\to p\nu\bar{\nu}$ 對於所有中子衰變事件，電荷不守恆衰變的發生率低於8×10−27（68% 置信水平）[21]

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20. ^ This is the most stringent of several limits given in Table 1 of this paper.
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