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However, it seems in path integrals, you integrate over the field and its conjugate, so they can take on values that are not the complex conjugate of each other. But you can write the measure in terms of two real fields, so it would seem that in the integrand of a path integral, the field and its conjugate will always take on values that are the complex conjugate of each other!

Furthermore, Weinberg mentions in his QFT book that in Euclidean space, the field and its conjugate must be treated as independent.

So does the answer to the question of whether the field and its complex conjugate are independent of each other depend on Minkowski or Euclidean space?