I've learnt quantum field theory for a semester but I still can't understand the role of classical equation of motion in QFT.
I have looked up for several books. They all discuss classical field theory. And they turn to the quantum part without leaving comments. It seems the following things have nothing to do with it. Then why discuss it?
I can vaguely tell that maybe the solution space of classical of motion has something to do with the Hilbert space (maybe through mode expansion?) but I'm not sure.
I don't work in high energy physics. Maybe it's a trivial question but please help me.
edited: I came up with this idea when learning the Non-Abelian Chern-simons field theory via the quantum hall effect notes by David Tong. He calculates (in Page 189) the ground space degeneracy of chern simons action by determining the solutions of classical equation of motion. It seems he finds the classical degree of freedom and quantizes them.
So I think if it is the case. Then what the quantum field theory does is find the degree of freedom of classical motion (the solution of classical equation) and quantize them by creating of annihilating operators. But I don't know whether it is compatible with the canonical quantization or not.
Best Answer
There is an intimate connection between classical and quantum physics, see e.g. Bohr's correspondence principle, Ehrenfest theorem, the WKB approximation & the Schwinger-Dyson (SD) equations for starters.
One particular case of the SD equations $\langle \frac{\delta S}{\delta\phi}\rangle=0$ shows that the classical EOMs are satisfied in a quantum averaged sense in the quantum world.
Moreover, one may show that the classical paths give the dominant contributions to the quantum path integral.