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414 chapter 15Dht it bt aBx iAX at bt jCx it aBx b = x Nx jAX ax j’Figure 15.9 Left: A path through the space-time lattice that starts and ends at x a = x b . Theaction is an integral over this path, while the path integral is a sum of integrals over all paths.The dotted path BD is a transposed replica of path AC. Right: The dashed path joins the initialand final times in two equal time steps; the solid curve uses N steps each of size ε. The positionof the curve at time t j defines the position x j .Lattice path integration is based on the composition theorem for propagators:∫G(b, a)= dx j G(x b ,t b ; x j ,t j )G(x j ,t j ; x a ,t a ) (t a