Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1177δ−Functions determine the Lagrange multipliersωv = ∆x 1x ′ 1and ωu = ∆x 0 − x′0 ∆x 1x ′ ,1giving the final result∫ψ(x) = D n−1 p > D n−1˜x > exp(i[−p > ∆x > −()−If x ′0 = 0, (6.288) implies u > 0.∆x 0 − x′0 ∆x 1x ′ 1E p (˜x, p > ) + ∆x 1x ′ p > x ′ >])ψ(˜x).16.6.5 Brane DynamicsWe turn to the general case of action (6.284). We denote the σ 0 , σ 1 , . . . , σ pderivatives of x by x ,0 , x ,1 , . . . , x ,p . Again we assume that the vector x µ ,0 istime–like, and vectors x µ ,i are space–like (here and hereafter in this chapteri, k, l = 1, . . . , p while a, b = 0, . . . , p). Now in the action (6.284)g(σ) =1(p + 1)! ɛ a 0...a pɛ b0...bp x α0,b 0x α0,a0 · · · x αp,b px αp,ap ,with ɛ being the unit antisymmetric Levi–Civita symbol, and the canonicalmomentum is [Golovnev and Prokhorov (2005)]p µ =−γ(−1)pp! √ (−1) p g ɛ 0a 1...a pɛ b0...bp x µ,b0 x ,b1 · x ,a1 . . . x ,bp · x ,ap .Evidently p µ x µ ,i = 0 due to antisymmetry of ɛ, and using the equalityɛ a0...ap g(σ) = ɛ b0...bp x ,b0 · x ,a0 . . . x ,bp · x ,apone getsp 2 = (−1) p γ 2 ζ(x), with ζ(x) = detx µ ,i x µ,k.So, with the loss of information about the sign of p 0 , the constraints arep µ x µ ,i = 0, (i = 1, 2, . . . , p), (6.297)p 2 − (−1) p γ 2 ζ(x) = 0. (6.298)From (6.298) we have p 0 = ±E p withE p =√−→p 2 + (−1)p γ 2 ζ(x).