@Skewered Cigar @Peter Woit
You can construct a phase space from the Lagrangian. You start with the Lagrangian and vary it (as usual), but don’t throw away the boundary term (not as usual). This boundary term is a (pre)symplectic potential, and a second variation gives a conserved (pre)symplectic form on the space of solutions of the theory. You can try this with usual quantum mechanics viewed as a field in 1-dimensions
The whole “pre” stuff is about whether this symplectic form is degenerate or not; in any gauge theory it will have degenerate directions. However, you can take functions on the phase space which are constant along the degenerate directions, and define a Poisson bracket on such functions. Quantization then be done is the usual way, along with the usual issues of operator ordering for nonlinear functions, e.g. any Hamiltonian of interest.
Of course, all of this is much easier said than done, but in principle the Lagrangian gives you, at least, all the classical structure needed for quantization.
Kartik Prabhu