南部-后藤作用

维基百科，自由的百科全书

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

狭义相对论的作用量

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

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

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

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

世界面

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

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

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

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

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

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

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

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

南后作用

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

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

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

参考文献

1. ^ Nambu, Yoichiro, Lectures on the Copenhagen Summer Symposium (1970), unpublished.
2. ^ Zwiebach, Barton. A First Course in String Theory. Cambridge University Press. 2003. ISBN 978-0521880329.
3. ^ See Chapter 19 of Kleinert's standard textbook on Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th edition, World Scientific (Singapore, 2009)页面存档备份，存于互联网档案馆） (also available online页面存档备份，存于互联网档案馆）)