Ivancevic_Applied-Diff-Geom

1172 Applied Differential Geometry: A Modern IntroductionIn the continuous limit we definekAN → σ 1,∆ −→ x kA/N → −→ k (σ 1 ),m|∆ −→ x k | → γ.Here σ 1 ∈ [0, A]; usually one takes A = π, but for our purposes it may benatural to consider A as the string length and|∆ −→ x k | = A N , |−→ k | = 1.In any case we haveS = −γN∑∫k=0with−→ k =∂ −→ x (x 0 , σ 1 )∂σ 1,√dx 0 ∆l 1 − −→ ∫v 2 k⊥ → −γ−→ v =∂ −→ x (x 0 , σ 1 )∂x 0 ,The string length is equal to∫L = | −→ k |dσ 1 .dx 0 | −→ k |dσ 1√1 − −→ v 2 ⊥,−→ v ⊥ = −→ v − (−→ v −→ k )k 2 −→ k .After that we parameterize the world–sheetx 0 = x 0 (σ 0 , σ 1 ),by introducing a new parameter σ 0x 0 t and σ 0 τ). We get−→ x =−→ x (σ0 , σ 1 )(again the standard notations areẋ ≡ ∂x(σ 0, σ 1 )= (1, −→ v )ẋ 0 , x ′ ≡ ∂x(σ 0, σ 1 )= (x ′ 0 , −→ k + −→ v x ′ 0 ),∂σ 0 ∂σ∫1and S = −γ d 2 σẋ 0 | −→ √k | 1 − −→ v 2 ⊥.The last expression is equal to the Nambu–Goto action:∫ √S = −γ d 2 σ (ẋx ′ ) 2 − ẋ 2 x ′2 .Note that one could start with it and get∫ √S = −γ dx 0 dl 1 − −→ 2v ⊥ ,which is a continuous limit of (6.290).