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In AdS/CFT one usually hears of normalizable and non-normalizable modes regarding the independent solutions for the different fields. I've found that the "natural" definition of the norm in the case of the scalar field is given by

$-i \int \sqrt{-g} (\phi^* \partial^t \phi-\phi \partial^t \phi ^*) $

Where the integral is along an equal-time slice. I have two questions: Why is that the appropriated definition of norm for a scalar field in a curved spacetime? How does this generalize to higher spin fields?

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