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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.

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The main problem with the Nambu-Goto path integral is how to obtain a consistent path integral measure (PIM). When people say that it is ill-defined (and not just difficult to work with), they presumably mean that various naive choices of PIMs are inconsistent. The easiest way to obtain a consistent PIM is to go to the Hamiltonian formulation, cf. e.g. this Phys.SE post. One may argue that the Hamiltonian formulation of the Nambu-Goto string is equivalent to the Polyakov string, and hence consistent, cf. e.g. my Phys.SE answer here.

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