Hamilton-Jacobi expansion of the scattering amplitude

The calculation of the scattering amplitude is reduced to the problem of solving a set of classical Hamilton-Jacobi equations. This allows one to incorporate classical intuition into approximations at a fundamental level. The result is actually an iterative expansion for the scattering amplitude which is expected to be convergent in the high-energy limit. The first term in the expansion is shown to be the Glauber approximation, which is an approximation used extensively in nuclear as well as atomic and particle physics.