# 諾特定理

## 應用

• 對於物理系統對於空間平移的不變性（換言之，物理定律不隨著空間中的位置而變化）給出了動量的守恆律；
• 對於轉動的不變性給出了角動量的守恆律；
• 對於時間平移的不變性給出了著名的能量守恆定律

## 證明

• 經典力學上，哈密頓表述中，M是一個一維流形R，代表時間而目標空間是廣義位置的空間餘切叢
• 場論中，M是時空流形，而目標空間是場在任何給定可取的值的集合。例如，如果有m個純量場，φ1,...,φm，則目標流形是Rm。若流形是一個實向量場，則目標流形同構R3

${\displaystyle S:{\mathcal {C}}\rightarrow \mathbb {R} ,}$

${\displaystyle {\mathcal {L}}(\varphi ,\partial _{\mu }\varphi ,x)}$

${\displaystyle S[\varphi ]\equiv \int _{M}d^{n}x{\mathcal {L}}[\varphi (x),\partial _{\mu }\varphi (x),x].}$

${\displaystyle {\frac {\delta }{\delta \phi (x)}}S[\phi ]=0}$

${\displaystyle Q\left[\int _{N}d^{n}x{\mathcal {L}}\right]=\int _{\partial N}ds_{\mu }f^{\mu }[\phi (x),\partial \phi ,\partial \partial \phi ,...]}$

${\displaystyle Q[{\mathcal {L}}(x)]=\partial _{\mu }f^{\mu }(x)}$

 ${\displaystyle Q\left[\int _{N}d^{n}x{\mathcal {L}}\right]}$ ${\displaystyle =\int _{N}d^{n}x\left[{\frac {\partial {\mathcal {L}}}{\partial \phi }}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}\right]Q[\phi ]+\int _{\partial N}ds_{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]}$ ${\displaystyle =\int _{\partial N}ds_{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ].}$

${\displaystyle \partial _{\mu }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu }\right]=0.}$

${\displaystyle J^{\mu }\equiv {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu }}$

### 評論

${\displaystyle \int _{\partial N}ds_{\mu }J^{\mu }=0.}$

${\displaystyle Q_{1}[{\mathcal {L}}]=\partial _{\mu }f_{1}^{\mu }}$

${\displaystyle Q_{2}[{\mathcal {L}}]=\partial _{\mu }f_{2}^{\mu }}$

(這個是否離殼或僅僅在殼成立無關緊要）。則，

${\displaystyle [Q_{1},Q_{2}][{\mathcal {L}}]=Q_{1}[Q_{2}[{\mathcal {L}}]]-Q_{2}[Q_{1}[{\mathcal {L}}]]=\partial _{\mu }f_{12}^{\mu }}$

${\displaystyle j_{12}^{\mu }=\left({\frac {\partial }{\partial (\partial _{\mu }\phi )}}{\mathcal {L}}\right)(Q_{1}[Q_{2}[\phi ]]-Q_{2}[Q_{1}[\phi ]])-f_{12}^{\mu }.}$

### 證明的一般化

${\displaystyle Q[{\mathcal {L}}]=\partial _{\mu }f^{\mu }}$

（離殼或僅僅在殼都可以）。則，

${\displaystyle q[\epsilon ][S]=\int d^{d}xq[\epsilon ][{\mathcal {L}}]}$
${\displaystyle =\int d^{d}x\left({\frac {\partial }{\partial \phi }}{\mathcal {L}}\right)\epsilon Q[\phi ]+\left[{\frac {\partial }{\partial (\partial _{\mu }\phi )}}{\mathcal {L}}\right]\partial _{\mu }(\epsilon Q[\phi ])}$
${\displaystyle =\int d^{d}x\epsilon \partial _{\mu }{\Bigg \{}f^{\mu }-\left[{\frac {\partial }{\partial (\partial _{\mu }\phi )}}{\mathcal {L}}\right]Q[\phi ]{\Bigg \}}}$

${\displaystyle \partial _{\mu }\left[f^{\mu }-\left[{\frac {\partial }{\partial (\partial _{\mu }\phi )}}{\mathcal {L}}\right]Q[\phi ]-2\left[{\frac {\partial }{\partial (\partial _{\mu }\partial _{\nu }\phi )}}\right]\partial _{\nu }Q[\phi ]+\partial _{\nu }\left[\left[{\frac {\partial }{\partial (\partial _{\mu }\partial _{\nu }\phi )}}{\mathcal {L}}\right]Q[\phi ]\right]-\,\cdots \right]=0.}$

## 例子

### 例1：能量守恆

 ${\displaystyle S[x]\,}$ ${\displaystyle =\int {\mathcal {L}}[x(t),{\dot {x}}(t)]dt}$ ${\displaystyle =\int \left\{{\frac {m}{2}}g_{ij}{\dot {x}}^{i}(t){\dot {x}}^{j}(t)-V[x(t)]\right\}dt}$

（也即，在一個彎曲黎曼空間（但不是彎曲時空）中運動的一個牛頓質點，該空間度量為g，質點勢能為V）。

Q為時間平移的生成元。換句話說，${\displaystyle Q[x(t)]={\dot {x}}(t)}$。 [量子場理論學家經常在方程右邊加上一個因子i]。 注意

${\displaystyle Q[{\mathcal {L}}]=mg_{ij}{\dot {x}}^{i}{\ddot {x}}^{j}-{\frac {\partial }{\partial x^{i}}}V(x){\dot {x}}^{i}.}$

${\displaystyle {\frac {d}{dt}}\left[{\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)\right]}$

${\displaystyle f={\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x).}$

 ${\displaystyle j\,}$ ${\displaystyle =\left({\frac {\partial }{\partial {\dot {x}}^{i}}}{\mathcal {L}}\right)Q[x]-f}$ ${\displaystyle =mg_{ij}{\dot {x}}^{j}{\dot {x}}^{i}-\left[{\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)\right]}$ ${\displaystyle ={\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}+V(x).}$

${\displaystyle \sum _{i}\left({\frac {\partial }{\partial {\dot {x}}^{i}}}{\mathcal {L}}\right){\dot {x^{i}}}-{\mathcal {L}}}$

（稱為哈密頓量）是守恆的。

### 例2：線性動量守恆

 ${\displaystyle S[{\vec {x}}]\,}$ ${\displaystyle =\int dt{\mathcal {L}}[{\vec {x}}(t),{\dot {\vec {x}}}(t)]}$ ${\displaystyle =\int dt\left[\sum _{\alpha =1}^{N}{\frac {m_{\alpha }}{2}}({\dot {\vec {x}}}_{\alpha })^{2}-\sum _{\alpha <\beta }V_{\alpha \beta }({\vec {x}}_{\beta }-{\vec {x}}_{\alpha })\right]}$

${\displaystyle Q_{i}[x_{\alpha }^{j}(t)]=\delta _{i}^{j}.}$

${\displaystyle Q_{i}[{\mathcal {L}}]=0}$

${\displaystyle {\vec {f}}=0.}$

${\displaystyle {\vec {J_{i}}}=\sum _{\alpha }\left({\frac {\partial }{\partial {\dot {\vec {x}}}_{\alpha }}}{\mathcal {L}}\right)\cdot {\vec {Q}}[{\vec {x}}_{\alpha }]-{\vec {f}}}$
${\displaystyle =\sum _{\alpha }m_{\alpha }{\dot {\vec {x}}}_{\alpha }^{i}}$

### 例3

 ${\displaystyle S[\phi ]\,}$ ${\displaystyle =\int d^{4}x{\mathcal {L}}[\phi (x),\partial _{\mu }\phi (x)]}$ ${\displaystyle =\int d^{4}x\left({\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -\lambda \phi ^{4}\right)}$

Q為時空縮放的生成元。換句話說，

${\displaystyle Q[\phi (x)]=x^{\mu }\partial _{\mu }\phi (x)+\phi (x).}$

${\displaystyle Q[{\mathcal {L}}]=\partial ^{\mu }\phi \left(\partial _{\mu }\phi +x^{\nu }\partial _{\mu }\partial _{\nu }\phi +\partial _{\mu }\phi \right)-4\lambda \phi ^{3}\left(x^{\mu }\partial _{\mu }\phi +\phi \right).}$

${\displaystyle \partial _{\mu }\left[{\frac {1}{2}}x^{\mu }\partial ^{\nu }\phi \partial _{\nu }\phi -\lambda x^{\mu }\phi ^{4}\right]=\partial _{\mu }\left(x^{\mu }{\mathcal {L}}\right)}$

（其中我們進行了空指標的變換）所以我們置

${\displaystyle f^{\mu }=x^{\mu }{\mathcal {L}}.\,}$

${\displaystyle j^{\mu }=\left[{\frac {\partial }{\partial (\partial _{\mu }\phi )}}{\mathcal {L}}\right]Q[\phi ]-f^{\mu }}$
${\displaystyle =\partial ^{\mu }\phi \left(x^{\nu }\partial _{\nu }\phi +\phi \right)-x^{\mu }\left({\frac {1}{2}}\partial ^{\nu }\phi \partial _{\nu }\phi -\lambda \phi ^{4}\right).}$

（另外：如果要找出該方程的沃德-高橋版本，你會遇到異常問題。）