This type of problems is often referred to as constrained mechanical system.
It was studied by Dirac, who developed the theory of
constrained quantization. This theory was formalized and further developed by Marseden and Weinstein to what is called "Symplectic reduction".
A particularly illiminating chapter for finite dimensional systems may be found in Marsden and Ratiu's book: "Introduction to mechanics and symmetry".
When the phase space of a dynamical system is a cotangent bundle, one can use the usual methods of canonical quantization, and the corresponding path integral.
However, this formalism does not work in general for nonlinear phase spaces. One important example is when the phase space is defined by a nonlinear surface in a larger linear phase space.
Basically, Given a symmetry of a phase space, one can reduce the problem to a smaller phase space in two stages
- Work on "constant energy surfaces" of the Hamiltonian generating this stymmetry.
- Consider only "invariant observables" on these surfaces.
This procedure reduces by 2 the dimensions of the phase space, and the reduced dimension remains even. One can prove that if the original phase space is symplectic, so will be the reduced phase space.
May be the most simple example is the elimination of the center of mass motion in a two particle system and working in the reduced dynamics.
There is a theorem by Guillemin and Sternberg for certain types of finite dimensional phase spaces which states that quantization commutes with reduction.
That is, one can either quantize the original theory and impose the constraints on the quantum Hilbert space to obtain the "physical" states.
Or, on can reduce the classical theory, then quantize. In this case the reduced Hilbert space is automatically obtained.
The second case is not trivuial because the reduced phase space becomes a non-linear symplectic manifold and in many cases it is not even a manifold (because the group action is not free).
Most of the physics applications treat however field theories which correspond to infinite dimensional phase spaces and there is no counterpart of the Guillemin-Sternberg theorem.
(There are works trying to generalize the theorem to some infinite dimensional spaces by N. P. Landsman).
But in general, the commutativity of the reduction and quantization is used in the physics literature, even though a formal proof is still lacking.
The most known example is the quantization of the moduli space of flat connections in relation to the Chern-Simons theory.
The most known example of constrained dynamics in infinite dimensional spaces is the Yang-Mills theory where the momentum conjugate to $A_0$ vanishes.
It should be mentioned that there is an alternative (and equivalent) approach to treat the constraints and perform the Marsden-Weinstein reduction through BRST, and this is the usual way
in which the Yang-Mills theory is treated. In this approach, the phase space is extended to a supermanifold instead of being reduced. The advantage of this approach is that the resulting supermanifold
is flat and methods of canonical quantization can be used.
In the mentioned case of the scalar field, the phase space may be considered as an infinite numer of copies of $T^{*}\mathbb{R}$. The relation $\Pi = \partial_x \phi$ is the constraint surface.
In a naive dimension count of the reduced phase space dimensions one finds that in every space point the 1+1 dimensions the phase space (The field and its conjugate momentum) are fully reduced by the constraint and its symmetry generator. Thus we are left with a "zero-dimensional" theory. I haven't worked out this example, but I am quite sure that if this case is done carefully we would have been left with a
finite number of residual parameters. This is a sign that this theory is topological - which can be seen through the quantization of the global coefficient".
Update:
In response to Greg's comments here are further references and details.
The following review article (Aspects of BRST Quantization) by J.W. van Holten explains the BRST quantization of electrdynamics and Yang-Mills theory (Faddeev-Popov theory) as constrained mechanical systems.The article contains other example from (finite dimensioal phase space) quantum mechanics as well.
The following article by Phillial Oh. (Classical and Quantum Mechanics of Non-Abelian Chern-Simons Particles) describes the quantization of a (finite dimensional) mechanical system performing the symplectic reduction directly without using BRST. Here, the reduced spaces are coadjoint orbits (such as flag manifolds or projective spaces). The beautiful geometry of these spaces is very well known and this is the reason why the reduction can be performed here directly. For most of the reduced phase spaces, such an explicit knowledge of the geometry is lacking. In field theory, problems such as quantization of the two dimensional Yang-Mills theory possess such an explicit description, but for higher dimensional I don't know of an explicit treatment (besides BRST).
The following article by Kostant and Sternberg, describes the equivalence between the BRST theory and the direct symplectic reduction.
Now, concerning the path integral. I think that most of the recent physics achievements were obtained by means of the path integral, even
if it has some loose points. I can refer you to the following book by Cartier and Cecile DeWitt-Morette, where they treated path integrals on non-flat symplectic manifolds and in addition, they formulated the oscillatory path integral in terms of Poisson processes.
There is a very readable reference by Orlando Alvarez describing the quantization of the global coefficients of topological terms in, Commun. Math. Phys. 100, 279-309 (1985)(Topological Quantization and Cohomology). I think that the Lagrangian given in the question can be treated by the same methods. basically, the quantization of these terms is due to the same physical reason that the product of electric and magnetic charges of magnetic monopoles should be quantized. This is known as the Dirac quantization condition. In the path integral formulation, it follows from the requirement that a gauge transformation should produce a phase shift of multiples of $2\pi$. In geometric quantization, this condition follows from the requirement that the prequantization line bundle should correspond to an integral symplectic form.
Best Answer
In Minkowski spacetime, the action has to be real. Btw, that's necessary for the classical limit to give principle of least action. Yes, such sums are ill-defined, so some might say that the theory is mathematically defined by analytically continuing (Wick rotating) to Euclidean time, where you have nice exponentially decaying weights. You'll get saddle points given by the extrema of the action and you can expand around those solutions and deal with the theory perturbatively.
Think of the path integral in QM. Going from paths of least action (classical mechanics) to a weighted sum over all paths gives us quantum mechanics ("first quantization"). Going from quantum mechanics to QFT involves a sum over all field configurations ("second quantization"). Similarly, the moment you write out the path integral summing over all possible string configurations, you're studying a quantum mechanical system of strings.
Even in QFT when you do the Fadeev-Popov method and introduce ghosts, you have to quantize them so that diagrams with ghosts consistently cancel amplitudes of (some) diagrams with longitudinal gluons. I'm at a loss for a more insightful answer.