Editing Quantum chaos (section)

==== Rough sketch on how to arrive at the Gutzwiller trace formula ====
# Start with the semiclassical approximation of the time-dependent Green's function (the Van Vleck propagator).
# Realize that for caustics the description diverges and use the insight by Maslov (approximately Fourier transforming to momentum space (stationary phase approximation with h a small parameter) to avoid such points and afterwards transforming back to position space can cure such a divergence, however gives a phase factor). 
# Transform the Greens function to energy space to get the energy dependent Greens function (again approximate Fourier transform using the stationary phase approximation). New divergences might pop up that need to be cured using the same method as step 3
# Use <math>d(E)=-\frac{1}{\pi}\Im(\operatorname{Tr}(G(x,x^\prime,E))</math> (tracing over positions) and calculate it again in stationary phase approximation to get an approximation for the density of states <math>d(E)</math>.
Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it. In step 4 it restricts you to orbits where initial and final momentum are the same i.e. periodic orbits. Often it is nice to choose a coordinate system parallel to the direction of movement, as it is done in many books.