# 电荷守恒定律

（重定向自電荷守恆

${\displaystyle Q(t_{2})=Q(t_{1})+Q_{IN}-Q_{OUT}}$

——班傑明·富蘭克林[6]

## 電磁學表述

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

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

${\displaystyle Q=\int _{\mathbb {V} }\rho \ \mathrm {d} ^{3}r}$

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

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

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

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$

${\displaystyle 0\equiv \nabla \cdot (\nabla \times \mathbf {B} )=\mu _{0}\nabla \cdot \mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial (\nabla \cdot \mathbf {E} )}{\partial t}}}$

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

## 規範不變性

### 電磁學

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

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

### 諾特定理

{\displaystyle {\begin{aligned}{\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{aligned}}}

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

{\displaystyle {\begin{aligned}{\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{aligned}}}

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

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

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

### 規範場論

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

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

{\displaystyle {\begin{aligned}{\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{aligned}}}

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

${\displaystyle {\mathcal {L}}_{1}=i\hbar c{\overline {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -mc^{2}{\overline {\psi }}\psi -q{\overline {\psi }}\gamma ^{\mu }\psi A_{\mu }}$

${\displaystyle {\mathcal {L}}'_{1}={\mathcal {L}}_{1}-\hbar c(\partial _{\mu }\theta ){\overline {\psi }}\gamma ^{\mu }\psi +q{\overline {\psi }}\gamma ^{\mu }\psi \partial _{\mu }\Lambda }$

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

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

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

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

## 實驗證據

 ${\displaystyle e\to \nu _{e}+\gamma }$ 平均壽命大於4.6×1026年（90% 置信水平）。[17]

 ${\displaystyle e\to }$ 任意粒子 平均壽命大於6.4×1024年（68% 置信水平）[20] ${\displaystyle n\to p+\nu +{\bar {\nu }}}$ 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]

## 參考文獻

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