I have found a lot of places saying that the Nambu-Goto action is ill-defined, that the squareroot exponential is a complicated thing to make sense of in a path-integral and so on. Then people go on and introduce the Polyakov action to make sense of things, and this is fine with me.
My question is, can anyone share the details why the Nambu-Goto action is ill-defined? I cannot seem to find any rigorous analysis showing that path-integrals with squareroots are problematic. I am looking for a proof of the ill-defined nature of the Nambu-Goto path-integral, or at least a fairly detailed description.