南部-后藤作用

若

${\displaystyle -ds^{2}=-(c\,dt)^{2}+dx^{2}+dy^{2}+dz^{2},\ }$

相对论的作用量是下面的泛函：

${\displaystyle S=-mc\int ds.}$

最小作用量原理说经典方程说泛函导数等于0：

${\displaystyle \delta S=0.}$

量子相对论用泛函积分

${\displaystyle Z=\int \exp(iS)}$

设时空是d+1维的：

${\displaystyle x=(x^{0},x^{1},x^{2},\ldots ,x^{d}).}$

(${\displaystyle \tau }$, ${\displaystyle \sigma }$)是世界面的参数。

${\displaystyle X(\tau ,\sigma )=(X^{0}(\tau ,\sigma ),X^{1}(\tau ,\sigma ),X^{2}(\tau ,\sigma ),\ldots ,X^{d}(\tau ,\sigma )).}$

设 ${\displaystyle \eta _{\mu \nu }}$ 是 ${\displaystyle (d+1)}$维时空的距离函数，则

${\displaystyle g_{ab}=\eta _{\mu \nu }{\frac {\partial X^{\mu }}{\partial y^{a}}}{\frac {\partial X^{\nu }}{\partial y^{b}}}\ }$

是世界面的距离函数。${\displaystyle a,b=0,1}$ 而 ${\displaystyle y^{0}=\tau ,y^{1}=\sigma }$。世界面的面积 ${\displaystyle {\mathcal {A}}}$ 是

${\displaystyle \mathrm {d} {\mathcal {A}}=\mathrm {d} ^{2}\Sigma {\sqrt {-g}}}$

其中${\displaystyle \mathrm {d} ^{2}\Sigma =\mathrm {d} \sigma \,\mathrm {d} \tau }$ ， ${\displaystyle g=\mathrm {det} \left(g_{ab}\right)\ }$。若

${\displaystyle {\dot {X}}={\frac {\partial X}{\partial \tau }}}$

${\displaystyle X'={\frac {\partial X}{\partial \sigma }},}$

则距离函数 ${\displaystyle g_{ab}}$是

${\displaystyle g_{ab}=\left({\begin{array}{cc}{\dot {X}}^{2}&{\dot {X}}\cdot X'\\X'\cdot {\dot {X}}&X'^{2}\end{array}}\right)\ }$

${\displaystyle g={\dot {X}}^{2}X'^{2}-({\dot {X}}\cdot X')^{2}}$

南后作用是

${\displaystyle {\mathcal {S}}=-{\frac {1}{2\pi \alpha '}}\int \mathrm {d} ^{2}A=-{\frac {1}{2\pi \alpha '}}\int \mathrm {d} ^{2}\Sigma {\sqrt {-g}}=-{\frac {1}{2\pi \alpha '}}\int \mathrm {d} ^{2}\Sigma {\sqrt {({\dot {X}}\cdot X')^{2}-({\dot {X}})^{2}(X')^{2}}}.}$

使用上文的距离函数

${\displaystyle {\mathcal {S}}=-{\frac {1}{2\pi \alpha '}}\int \mathrm {d} ^{2}\Sigma {\sqrt {{\dot {X}}^{2}-{X'}^{2}}},}$

或

${\displaystyle {\mathcal {S}}=-{\frac {1}{4\pi \alpha '}}\int \mathrm {d} ^{2}\Sigma ({\dot {X}}^{2}-{X'}^{2}).}$

这是上文相对论作用量的二维推广。

• 拉格朗日场论
• 原时
• 世界面
• 宇宙弦（cosmic string）
• 泊里雅科夫作用量
• 泛函积分