I subconsciously feel not fully comfortable about Wick rotating or analytic continuation from Euclidean to Minkowski space. I simply wonder whether there is any subtlety here, and when we need to be conscious whether (i) this continuation can be dangerous or the continuation cannot be done; or (ii) the two theories(Euclidean and Minkowski) may not be the same?

For example, if you read Polchinski's String theory, most of worldsheet and CFT business is done in Euclidean signature. But in Chap 3 of Polchinsk's at p.83, says ``The same procedure works for the Polyakov action if we write the metric in terms of a tetrad, and make the same rotation. This provides a formal justification for the equivalence of the Minkowski and Euclidean path integrals. It has been shown by explicit calculation that they define the same amplitudes, respectively in the light-cone and conformal gauges.''

But right at p.83 footnote, says ''In more than two dimensions, things are not so simple because the Hilbert action behaves in a more complicated way under the rotation. No simple rotation damps the path integral. In particular, the meaning of the Euclidean path integral for four-dimensional gravity is very uncertain.''

Is Polchinski providing an example here? Can someone explain in what (generic) scenarios there will be subtlety about the analytic continuation?

I subconsciously feel not fully comfortable about Wick rotating or analytic continuation from Euclidean to Minkowski space. I simply wonder whether there is any subtlety here, and when we need to be conscious whether (i) this continuation can be dangerous or the continuation cannot be done; or (ii) the two theories(Euclidean and Minkowski) may not be the same?

For example, if you read Polchinski's String theory, most of worldsheet and CFT business is done in Euclidean signature. But in Chap 3 of Polchinsk's at p.83, says

``The same procedure works for the Polyakov action if we write the metric in terms of a tetrad, and make the same rotation. This provides a formal justification for the equivalence of the Minkowski and Euclidean path integrals. It has been shown by explicit calculation that they define the same amplitudes, respectively in the light-cone and conformal gauges.''

But right at p.83 footnote, says

''In more than two dimensions, things are not so simple because the Hilbert action behaves in a more complicated way under the rotation. No simple rotation damps the path integral. In particular, the meaning of the Euclidean path integral for four-dimensional gravity is very uncertain.''

Is Polchinski providing an example here? Can someone explain in what (generic) scenarios there will be subtlety about the analytic continuation?