Can the semiclassical propagator be defined using a Hamilton-Jacobi equation?

[Karlisbad]

If we have SE (or other differential equation) defining the propagator by:

$$(\frac{\partial}{\partial t}-H(q,p))G(x,s)=\delta (x-s)$$

then my question is..can you define a "semiclassical" operator?..i mean in the sense that it would solve approximately the equation (1) above but in WKB representation, ... does it satisfy some kind of Hamilton-Jacobi equation?..

 

[dextercioby]

Can you define "t,x,s,q,p" in your eq.? By the looks of it, it doesn't make too much sense to me.

Daniel.

 

[Karlisbad]

Sorry "dextercioby" ^_^ perhaps..is it clearer if i put $$G(x,x')$$ where here x means $$x=(x,y,z,t)$$

My question is that if you can't solve SE (in most cases) how you can expect to obtain G?..Path integral formulation allows you to calculate G(x,x') but only in the "Semiclassical" approach, by expanding the functional near its classical solution...my question is..DOes The semiclassical-propàgator satisfy a Hamilton-Jacobi equation?