查看“︁南部-后藤作用”︁的源代码

'''南部-后藤作用量'''是[[玻色弦理論|玻色弦理论]]中最简单的[[作用量]]之一。这个作用量以[[南部阳一郎|南部阳一朗]]和[[後藤鉄男]]（{{Jp|j=後藤鉄男|hg=ごとうてつお|rm=Gotō Tetsuo|lead=yes}}）这两个[[日本]]物理家的名字命名。<ref>Nambu, Yoichiro, ''Lectures on the Copenhagen Summer Symposium'' (1970), unpublished.</ref>

南后作用量等于[[世界面]]的[[面积]]：

<math>\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}.</math>

== 狭义相对论的作用量 ==
若

:<math>-ds^2 = -(c \, dt)^2 + dx^2 + dy^2 + dz^2, \ </math>

[[相对论]]的作用量是下面的[[泛函]]：

:<math>S = -mc \int ds.</math>

[[最小作用量原理]]说经典方程说[[泛函导数]]等于0：<blockquote><math>\delta S = 0.</math></blockquote>量子相对论用[[泛函积分]]<blockquote><math> Z = \int \exp(iS) </math></blockquote>

== 世界面 ==
设时空是d+1维的：

:<math>x = (x^0, x^1, x^2, \ldots, x^d).</math>

(<math>\tau</math>, <math>\sigma</math>)是世界面的参数。

:<math>X (\tau, \sigma) = (X^0(\tau,\sigma), X^1(\tau,\sigma), X^2(\tau,\sigma), \ldots, X^d(\tau,\sigma)).</math>

设 <math> \eta_{\mu \nu} </math> 是 <math>(d+1)</math>维时空的[[距离函数]]，则

:<math> g_{ab} = \eta_{\mu \nu} \frac{\partial X^\mu}{\partial y^a} \frac{\partial X^\nu}{\partial y^b} \ </math>

是[[世界面]]的距离函数。<math> a,b = 0,1 </math> 而 <math> y^0 = \tau, y^1 = \sigma </math>。[[世界面]]的[[面积]] <math> \mathcal{A} </math> 是

:<math> \mathrm{d} \mathcal{A} = \mathrm{d}^2 \Sigma \sqrt{-g} </math>

其中<math>\mathrm{d}^2\Sigma = \mathrm{d}\sigma \, \mathrm{d}\tau</math> ， <math> g = \mathrm{det} \left( g_{ab} \right) \ </math>。若

:<math>\dot{X} = \frac{\partial X}{\partial \tau}</math>

:<math>X' = \frac{\partial X}{\partial \sigma},</math>

则[[距离函数]] <math> g_{ab} </math>是

:<math> g_{ab} = \left( \begin{array}{cc} \dot{X}^2 & \dot{X} \cdot X' \\ X' \cdot \dot{X} & X'^2 \end{array} \right) \ </math>
:<math> g = \dot{X}^2 X'^2 - (\dot{X} \cdot X')^2 </math>

== 南后作用 ==
南后作用是<ref>{{Cite book|last=Zwiebach|first=Barton|authorlink=Barton Zwiebach|title=A First Course in String Theory|publisher=[[Cambridge University Press]]|date=2003|isbn=978-0521880329}}</ref><ref>See Chapter 19 of
[[Hagen Kleinert|Kleinert's]] standard textbook on ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 5th edition, [http://www.worldscibooks.com/physics/7305.html World Scientific (Singapore, 2009)] {{Wayback|url=http://www.worldscibooks.com/physics/7305.html |date=20090424041920 }} (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=11 online] {{Wayback|url=http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=11 |date=20090101070536 }})</ref>

:<math>\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}.</math>

使用上文的距离函数

:<math>\mathcal{S} = -\frac{1}{2\pi\alpha'} \int \mathrm{d}^2 \Sigma \sqrt{{\dot{X}} ^2 - {X'}^2},</math>

或

:<math>\mathcal{S} = -\frac{1}{4\pi\alpha'} \int \mathrm{d}^2 \Sigma ({\dot{X}}^2 - {X' }^2).</math>

这是上文相对论作用量的二维推广。

== 相关 ==

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

== 参考文献 ==
{{Reflist}}

{{弦理论}}
[[Category:弦理论]]
[[Category:有未审阅翻译的页面]]