WKB Approach in Cartesian Coordinates

The results obtained in the previous subsection of instanton theory can be reproduced in a much easier way with use of the WKB approximation. This is quite helpful to understand the physical meaning of the various quantities and the procedures used. In the same way as we did in the one-dimensional case in Section 2.5.2, we do not need to construct the complex valued Lagrange manifold [7,15,30,37], which constitutes the main obstacle of the conventional WKB theory. Without the energy term, the Hamilton-Jacobi equation can be easily solved and generalization for an 

We expand 1 /(1 -I- Ea ) and sequentially equate the terms according to the orders 

Since along the instanton po(S)d/dS = d/dr, this equation coincides with Equation 

The integration of the transport equation is trivial and for the present purpose we only need the solution on the instanton trajectory 

We note that the only difference from the one-dimensional case comes from the last term. Inserting the expression of the wave function 

---