Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 11471) M carries a 2−form B ∈ Ω (2) (M). The resulting contribution to theaction is also that of a ‘nonlinear sigma model’∫ ∫S B [x, B] = x ∗ (B) = dx µ ∧ dx ν B µν (x)Σ2) M may carry a dilaton field Φ ∈ Ω (0) (M) so that∫S Φ [x, Φ] = dµ g R g Φ(x).where R g is the Gaussian curvature of Σ for the metric g.3) There may be a tachyon field T ∈ Ω (0) (M) contributing∫S T [x, T ] = dµ g T (x).6.5.8 Transition Amplitude for a Single Point ParticleΣThe transition amplitude for a single point–particle could in fact be get ina way analogous to how we prescribed string amplitudes. Let space–timebe again a Riemannian manifold M, with metric g. The prescription forthe transition amplitude of a particle travelling from a point y ∈ M to apoint y ′ to M is expressible in terms of a sum over all (continuous) pathsconnecting y to y ′ :Amp(y, y ′ ) =ΣΣ∑e −S[path] .pathsjoining y and y ′Paths may be parametrized by maps from C = [0, 1] into M with x(0) = y,x(1) = y ′ . A simple world–line action for a massless particle is get byintroducing a metric g on [0, 1]S[x, g] = 1 ∫dτ g(τ) −1 ẋ µ ẋ ν g µν (x),2Cwhich is invariant under Diff + (C) and Diff(M).Recall that the analogous prescription for the point–particle transitionamplitude is the path integral∫Amp(y, y ′ ) = D[g] 1 ∫D[x] e −S[x,g] .NMet(C)Map(C,M)