It's a well known elementary fact that the Nambu-Goto action SNG=T∫dτdσ√(∂τXμ)2(∂σXμ)2−(∂σXμ∂τXμ)2 and Polyakov action SP=−T2∫dτdσ√hhabημν∂aXμ∂bXν are equivalent at the classical level. More precisely, by solving δSP/δhab=0 for hab and plugging it back to SP, we get SNG.
However, my question is whether they are equivalent at the quantum level or not. That is, if let ZP[J]:=∫D[hab]D[Xμ]exp(−iSP[hab,Xμ]+i∫dτdσJμXμ)∫D[hab]D[Xμ]exp(−iSP[hab,Xμ]) and ZNG[J]:=∫D[Xμ]exp(−iSNG[Xμ]+i∫dτdσJμXμ)∫D[Xμ]exp(−iSNG[Xμ]), do we also have ZP[J]=ZNG[J]?
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