南部-後藤作用量是玻色弦理論中最簡單的作用量之一。這個作用量以南部陽一朗和後藤鉄男(日語:後藤鉄男/ごとうてつお Gotō Tetsuo)這兩個日本物理家的名字命名。[1]
南後作用量等於世界面的面積:
若
![{\displaystyle -ds^{2}=-(c\,dt)^{2}+dx^{2}+dy^{2}+dz^{2},\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb158f20a4ead597fcf1724e0516f01a8de364ab)
相對論的作用量是下面的泛函:
![{\displaystyle S=-mc\int ds.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/072784c69e561836c90aa3dc38103023a38c3542)
最小作用量原理說經典方程說泛函導數等於0:
![{\displaystyle \delta S=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be02bab96c7b908ef7d379d67f1d3aca2fdfde29)
量子相對論用泛函積分
![{\displaystyle Z=\int \exp(iS)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1487fa8906f70eff10dd52ddcb57c007c2341a)
設時空是d+1維的:
![{\displaystyle x=(x^{0},x^{1},x^{2},\ldots ,x^{d}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2381fd2e1bebcbb3a27c8395f4b20038e0cffdc6)
(
,
)是世界面的參數。
![{\displaystyle X(\tau ,\sigma )=(X^{0}(\tau ,\sigma ),X^{1}(\tau ,\sigma ),X^{2}(\tau ,\sigma ),\ldots ,X^{d}(\tau ,\sigma )).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0289a1a40130052355f65367b575a502bb76ff)
設
是
維時空的距離函數,則
![{\displaystyle g_{ab}=\eta _{\mu \nu }{\frac {\partial X^{\mu }}{\partial y^{a}}}{\frac {\partial X^{\nu }}{\partial y^{b}}}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b79b692f61284445687b2d01184a288b16d609)
是世界面的距離函數。
而
。世界面的面積
是
![{\displaystyle \mathrm {d} {\mathcal {A}}=\mathrm {d} ^{2}\Sigma {\sqrt {-g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efb250659c7345e582f75e730d8070464087003e)
其中
,
。若
![{\displaystyle {\dot {X}}={\frac {\partial X}{\partial \tau }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efccfa86959cfca38d2d914649e67752b4294990)
![{\displaystyle X'={\frac {\partial X}{\partial \sigma }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d75ae027bb50dd4013302fdca73b63111296f42b)
則距離函數
是
![{\displaystyle g_{ab}=\left({\begin{array}{cc}{\dot {X}}^{2}&{\dot {X}}\cdot X'\\X'\cdot {\dot {X}}&X'^{2}\end{array}}\right)\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/baba7f25a6c4709250b8b8a32a04b8a2aa44447e)
![{\displaystyle g={\dot {X}}^{2}X'^{2}-({\dot {X}}\cdot X')^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75fd2134f205ecc1a349411ed1e1acabed0389b5)
南後作用是[2][3]
![{\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e40ac71af9f855dee24333948a52cc2bb004210)
使用上文的距離函數
![{\displaystyle {\mathcal {S}}=-{\frac {1}{2\pi \alpha '}}\int \mathrm {d} ^{2}\Sigma {\sqrt {{\dot {X}}^{2}-{X'}^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1155022752851e0fcfa5c93b101210b179848773)
或
![{\displaystyle {\mathcal {S}}=-{\frac {1}{4\pi \alpha '}}\int \mathrm {d} ^{2}\Sigma ({\dot {X}}^{2}-{X'}^{2}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f54ed7606571183cf2a5c3b8b6e68bf83aaa70d8)
這是上文相對論作用量的二維推廣。
- ^ Nambu, Yoichiro, Lectures on the Copenhagen Summer Symposium (1970), unpublished.
- ^ Zwiebach, Barton. A First Course in String Theory. Cambridge University Press. 2003. ISBN 978-0521880329.
- ^ 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) (頁面存檔備份,存於互聯網檔案館) (also available online (頁面存檔備份,存於互聯網檔案館))
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