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The uniqueness of the lagrangian, wholesale, is what I had in mind! Weinberg and friends made arguments for example about the uniqueness of GR’s lagrangian and that for other fields (conditional of course on the exclusion of things like higher derivatives)
Yea, i think it's a super interesting question. The restriction to being gauge symmetric and Renormalizable + Poincare invariant really restricts what you can write down.
