Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1173We can also propose another discrete analogue of the Nambu–Gotostring. Let’s consider the following action:N∑∫S = −m dσ 0√ẋ µ k⊥ẋµk⊥,k=0where ẋ µ k⊥ is the part of ẋµ k perpendicular to xµ k+1 − xµ k. The continuouslimit isN → ∞,with the invariant intervalWe haveẋ µ ⊥ = ẋµ − ẋν x ′ νx ′ρ x ′ x ′µ ,ρ∫ √S = −γ dτ|ds|kAN → σ 1,m|∆s k | → γ,(∆s k ) 2 = (x µ k+1 − xµ k )(x µ k+1− x µk ).( dsdσ 1) 2= x ′2 ,ẋ 2 − (ẋx′ ) 2x ′ 2and∫= −γ dσ 0 dσ 1√(ẋx ′ ) 2 − ẋ 2 x ′2 .In contrast to the previous paragraph the presented discrete theory hasthe relativistic invariant form from the very beginning but even the senseof ẋ ⊥ depends upon the parametrization of the world–sheet. In the gaugeσ 0 = x 0 these two approaches coincide.6.6.3 A BraneNow we shall prove our statement for p > 1. We consider (N +1) p particlesarranged into some pD lattice with the position vectors −→ x i1i 2...i pand theaction [Golovnev and Prokhorov (2005)]∫S = −mdx 0N ∑i 1=0· · ·N∑i p=0√1 − −→ v 2 i 1,...i p⊥ ,where−→ v i1,...i p⊥ is the component of−→ v i1...i pperpendicular to−→ x i1...i k +1...i p− −→ x i1...i k ...i pfor all k. In continuous limit we demandAi kN → σ k and m ∆V→ γ with ∆V being volume of a cell of the lattice.The action takes the form∫ √S = −γ dx 0 dV 1 − −→ v 2 ⊥,