Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1181CLASSICAL STRINGYGEOMETRYQUANTUM STRINGYGEOMETRYα ′ /R 2CLASSICAL RIEMANNIANGEOMETRY↑→g sQUANTUM RIEMANNIANGEOMETRYFig. 6.23 The deformation from classical Riemannian geometry to quantum stringygeometry (see text for explanation).6.7.2 Green–Schwarz Bosonic Strings and BranesHere, we briefly describe the world–sheet dynamics of the Green–Schwarzbosonic string theory, and (more generally), bosonic p−brane theory, thepredecessor of the current superstring theory (see [Schwarz (1993); Greenet. al. (1987)] for details).World–Line Description of a Point ParticleRecall that a point particle sweeps out a trajectory called world–line inspace–time. This can be described by functions x µ (τ), that describe howthe world–line, parameterized by τ, is embedded in the space–time, whosecoordinates are denoted x µ (µ = 0, 1, 2, 3). For simplicity, let us assumethat the space–time is flat Minkowski space with a Lorentz metric tensor⎛ ⎞η µν =⎜⎝−1 0 0 00 1 0 0⎟0 0 1 0 ⎠ .0 0 0 1Then, the Lorentz–invariant line element (metric form) is given byds 2 = −η µν dx µ dx ν .In normal units ( = c = 1), the action for a particle of mass m is given by∫S = −m ds.