Determination of Instanton Trajectory

Let us recapitulate the recipe to find the instanton in the case of double well potential developed in Chapter 6. We consider the classical motion on the inverted potential with the Lagrangian [see Equation (6.107)], 

By solving t/z/t/r = z(z), we can know the time-dependence ofz(r) and the trajectory qo[z(r)] as a function of time. In order to improve the path, we use the variational principle to minimize the classical action in the space of z-parametrized paths. We look for a better instanton trajectory in the form [see Equation (6.113)] 

In the case of decay problem, one cannot fix both ends of the trajectory, but we can reformulate the problem in the following way. Let us try to find one-half of the instanton trajectory that runs for the semi-infinite time interval [-oo, 0] from the 

Now the procedure for the calculation of instanton can be summarized as follows. Without loss of generality we set q = 0. 

At the kth step of iteration the approximate instanton path is supposed to be given by 

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