# Functional integral (semiclassic formula)

• tpm

#### tpm

If to calculate the propagator K(x,x') (vaccuum)for a theory so:

$$(i\hbar \frac{\partial}{\partial t}\Psi - H\Psi )K(x,x')=\delta (x-x')$$ (1)

we use the functional integral approach:

$$K(x,x')=<0|e^{iS[x]/\hbar }|0>$$

my question is, let's suppose we use the semiclassical WKB approach to calculate $$K_{WKB} (x,x')$$ my question is ¿does the classical propagator satisfies the Schöedinguer equation (1) or as an approximation i'd like to know if the semiclassical propagator satisfies a Hamilton-Jacobi type equation... thanks.

hello
the solution of Hamilton-Jacobi equation is the generators of the canonical transformations (such as the action (the phase) it self) you can see articles: Barut , another : Boujdaa

The WKB approximation is a first order expansion in powers of h-bar, so that's how it's related. To zeroth order, you obtain the Hamilton-Jacobi equation, and to first order I believe is where you get the WKB approximation, if I recall correctly.