General Canonically Invariant Formulation

The result of the preceding subsection is not very convenient for practical use, because it requires the solution of the differential equation Equation (8.21) that contains the Hessian of the potential in terms of the transverse local coordinates together with the corresponding curvatures. One can obtain a more useful canonically invariant form that does not rely on any local coordinates. To do this, we first rewrite the first integral in Equation (8.27) in terms of the time x along the instanton trajectory. Using the transformation 

This is readily checked by making the coordinate transformation Equation (8.12) and using the properties of The quantity Wb has the simple physical meaning as the main exponent factor of the semiclassical wave function exp[-Wb(q)], which coincides with the ground state of the harmonic oscillator in the potential well. In the same way as before, we introduce the second derivative matrix 

The initial condition follows from the existence of the limit = A(t -oo) and reads 

To proceed further it is convenient to rotate the frame of reference in such a way that tiiv(Xo) = SiN-This simply means that the coordinate qn is chosen along the tangent to the instanton trajectory at the turning point. Then, det H is equal to the minor of the N, N) element of A and we obtain [cf. Equation (6.80)] 

Using Equations (8.35) and (8.37) and the fact that tin are the components of the unit tangent vector along the instanton trajectory, we finally obtain the decay rate as 

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