Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1145A natural intrinsic candidate for S is the area of x(Σ), which gives theso–called Nambu–Goto action 20∫ ∫Area (x (Σ)) = dµ γ = n 2 ξ √ det γ mn , (6.234)Σwhich depends only upon g and x, but not on g [Goto (1971)]. However,the transition amplitudes derived from the Nambu–Goto action are notwell–defined quantum–mechanically.Otherwise, we can take as starting point the so–called Polyakov action 21∫∫S[x, g] = κ (dx, ∗dx) g = κ dµ g g mn ∂ m x µ ∂ n x ν g µν (x), (6.235)ΣΣwhere κ is the string tension (a positive constant with dimension of inverselength square). The stationary points of S with respect to g are at g 0 = e φ γfor some function φ on Σ, and thus S[x, g 0 ] ∼ Area (x (Σ)).The Polyakov action leads to well–defined transition amplitudes, get byintegration over the space Met(Σ) of all positive metrics on Σ for a giventopology, as well as over the space of all maps Map(Σ, M). We can definethe path integralAmp =∑topologiesΣ∫Met(Σ)Σ∫1D[x] e −S[x,g,g] ,N(g) Map(Σ,M)where N is a normalization factor, while the measures D[g] and D[x] areconstructed from Diff + (Σ) and Diff(M) invariant L 2 norms on Σ andM. For fixed metric g, the action S is well–known: its stationary pointsare the harmonic maps x: Σ → M (see, e.g., [Eells and Lemaire (1978)]).However, g here varies and in fact is to be integrated over. For a generalmetric g, the action S defines a nonlinear sigma model, which is renormalizablebecause the dimension of Σ is 2. It would not in general berenormalizable in dimension higher than 2, which is usually regarded asan argument against the existence of fundamental membrane theories (see[Deligne et. al. (1999)]).20 Nambu–Goto action is the starting point of the analysis of string behavior, usingthe principles of ordinary Lagrangian mechanics. Just as the Lagrangian for a free pointparticle is proportional to its proper timei.e., the ‘length’ of its world–line, a relativisticstring’s Lagrangian is proportional to the area of the sheet which the string traces as ittravels through space–time.21 The Polyakov action is the 2D action from conformal field theory, used in stringtheory to describe the world–sheet of a moving string.