For positive , the displayed identity follows by differentiating at zero.
Integer powers of a partition function represent independent copies with the same disorder. Integer moments alone do not justify analytic continuation or interchanging disorder averages and thermodynamic limits.[ref]
Citation Classic commentary on Theory of spin glasses by Sam Edwards. Edwards, 1985 commentary, replica averaging and acknowledged mathematical limitations
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The "Replica Trick" is a method used in theoretical physics, particularly in statistical mechanics and quantum field theory, to analyze systems with a large number of degrees of freedom. The technique is commonly associated with the study of disordered systems, like spin glasses, and it helps in calculating averages over disorder configurations.