Instantons functional form

The uncertainty principle requires that any extremum path should be spread, and the next step in our calculation is to find the prefactor in (3.54) by incorporating small fluctuations around the instanton solution, in the spirit of the usual steepest descent method. Following Callan and Coleman [1977], let us represent an arbitrary path in the form x t) = xins(T) + Sx(t) and expand the action functional up to quadratic terms in S (t), assuming that these deviations are small ... 

The main idea of improvement is to find a direction in the space of trajectories along which the classical action. Equation (6.109), decreases. In order to do so, we use z(t) to define the r-dependent functions < [z(t)] that form the basis of expansion in the vicinity of qo[z(r)]. Namely, we look for a better instanton trajectory in the form... 

The iteration continues until we obtain the converged results. The initial guess of the instanton path can, of course, be taken in various ways, but it naturally affects the number of iterations required. Note, however, that the minimization of the action functional can be realized in a rather small subspace of paths formed by 4> (z). The dimension of this subspace is N X Nb, where N is the number of coordinates and Nb is the number of basis functions

polyatomic molecules presented in the next chapter. 

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)]... 

We derive the pre-exponential factor 5 in a convenient form in the Cartesian coordinates. The derivation is quite similar to the case of tunneling splitting. We consider the functional integral of the harmonic fluctuations around the instanton trajectory... 

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