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I was wondering if their is a general strategy for computing path integrals with a mix of boundary and bulk integral actions. Do people use divergence theorem to convert the action into bulk integrals, or is there some other trick?

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The usual Feynman rules may still be derived. If there are both boundary fields and bulk fields, one needs to include the bulk-to-boundary propagators, too. The same problem has to be solved in the AdS/CFT correspodence. Search for "Witten diagrams" to see how the Feynman diagrams are ultimately done in this context.

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