Path Integral over Variables of Normal Motion

4 Path Integral over Variables of Normal Motion 

The quasiclassical amplitude in the momentum representation does not suffer from this feature, because boundary conditions Pz h) = Pu Pzitz) = P2z determine the unique classical trajectory for the typical scattering potentials. Such amplitude cannot be obtained as a Fourier transform of the quasiclassical propagator in coordinate representation. So it is necessary to modify the stationary phase method for the evaluation of the path integral in momentmn representation. 

For the evaluation of this integral the method used in the Appendix 1 to the book by R. Rajaraman (Rajaraman 1982) has been modified. It is convenient to append the integration over the final momentum  

In order to get rid of —p /2mg in the exponent the functional cliange of variable may be used  

The Jacobian of this vaiiable change is J = [/(t2)//(ti)) (Gel fand and Yaglom 1960). Thus obtained path integral may be evrduated with the help of finite approximation T)k = ( ) the change of variables bi = 6 = (% — r)k-i)l /vk, 

---