Formulation in Terms of Cartesian Coordinates

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 

Here det[- ] is understood as the infinite product of the eigenvalues of the corresponding differential operator with the Dirichlet boundary condition at t oo. The prime in the first numerator indicates that the zero eigenvalue of[—9 /9t - - 1 

The longitudinal one-dimensional factor [first ratio of determinants in Equation (8.17)1 has been evaluated by Shulman [205] for the quartic potential. The analogous method can be used in the general case to lead to [see also Equations (2.108), (2.109), and (2.129)] 

Putting all the terms together, we finally obtain the decay rate as 

It should be noted that 5o is the action along the instanton from r = -oo to t = oo and thus is two times of the action from the potential minimum to the turning point. 

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