南部-后藤作用

南部-后藤作用量是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朗和後藤鉄男（日语：後藤鉄男／ごとうてつお Gotō Tetsuo）这两个日本物理家的名字命名。 南后作用量等于世界面的面积： S = − 1 2 π α ′ ∫ d 2 A = − 1 2 π α ′ ∫ d 2 Σ − g = − 1 2 π α ′ ∫ d 2 Σ ( X ˙ ⋅ X ′ ) 2 − ( X ˙ ) 2 ( X ′ ) 2 . {\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}}}.} 若 − d s 2 = − ( c d t ) 2 + d x 2 + d y 2 + d z 2 ,   {\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： S = − m c ∫ d s . {\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： δ S = 0. {\displaystyle \delta S=0.} 量子相对论用泛函积分 Z = ∫ exp ⁡ ( i S ) {\displaystyle Z=\int \exp(iS)} 设时空是d+1维的： x = ( x 0 , x 1 , x 2 , … , x d ) . {\displaystyle x=(x^{0},x^{1},x^{2},\ldots ,x^{d}).} ( τ {\displaystyle \tau } , σ {\displaystyle \sigma } )是世界面的参数。 X ( τ , σ ) = ( X 0 ( τ , σ ) , X 1 ( τ , σ ) , X 2 ( τ , σ ) , … , X d ( τ , σ ) ) . {\displaystyle X(\tau ,\sigma )=(X^{0}(\tau ,\sigma ),X^{1}(\tau ,\sigma ),X^{2}(\tau ,\sigma ),\ldots ,X^{d}(\tau ,\sigma )).} 设 η μ ν {\displaystyle \eta _{\mu \nu }} 是 ( d + 1 ) {\displaystyle (d+1)} 维时空的距离函数，则 g a b = η μ ν ∂ X μ ∂ y a ∂ X ν ∂ y b   {\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。 a , b = 0 , 1 {\displaystyle a,b=0,1} 而 y 0 = τ , y 1 = σ {\displaystyle y^{0}=\tau ,y^{1}=\sigma } 。世界面的面积 A {\displaystyle {\mathcal {A}}} 是 d A = d 2 Σ − g {\displaystyle \mathrm {d} {\mathcal {A}}=\mathrm {d} ^{2}\Sigma {\sqrt {-g}}} 其中 d 2 Σ = d σ d τ {\displaystyle \mathrm {d} ^{2}\Sigma =\mathrm {d} \sigma \,\mathrm {d} \tau } ， g = d e t ( g a b )   {\displaystyle g=\mathrm {det} \left(g_{ab}\right)\ } 。若 X ˙ = ∂ X ∂ τ {\displaystyle {\dot {X}}={\frac {\partial X}{\partial \tau }}} X ′ = ∂ X ∂ σ , {\displaystyle X'={\frac {\partial X}{\partial \sigma }},} 则距离函数 g a b {\displaystyle g_{ab}} 是 g a b = ( X ˙ 2 X ˙ ⋅ X ′ X ′ ⋅ X ˙ X ′ 2 )   {\displaystyle g_{ab}=\left({\begin{array}{cc}{\dot {X}}^{2}&{\dot {X}}\cdot X'\\X'\cdot {\dot {X}}&X'^{2}\end{array}}\right)\ } g = X ˙ 2 X ′ 2 − ( X ˙ ⋅ X ′ ) 2 {\displaystyle g={\dot {X}}^{2}X'^{2}-({\dot {X}}\cdot X')^{2}} 南后作用是 S = − 1 2 π α ′ ∫ d 2 A = − 1 2 π α ′ ∫ d 2 Σ − g = − 1 2 π α ′ ∫ d 2 Σ ( X ˙ ⋅ X ′ ) 2 − ( X ˙ ) 2 ( X ′ ) 2 . {\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 S = − 1 2 π α ′ ∫ d 2 Σ X ˙ 2 − X ′ 2 , {\displaystyle {\mathcal {S}}=-{\frac {1}{2\pi \alpha '}}\int \mathrm {d} ^{2}\Sigma {\sqrt {{\dot {X}}^{2}-{X'}^{2}}},} 或 S = − 1 4 π α ′ ∫ d 2 Σ ( X ˙ 2 − X ′ 2 ) . {\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 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) （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）)