The Berezin integral is differentiation with respect to a Grassmann variable: and . A finite-dimensional fermionic Gaussian integral gives a determinant or Pfaffian.
For independent Grassmann variables and an invertible ordinary matrix , the linear change transforms the Berezin integral measure by the inverse Jacobian determinant. Indeed, the highest-degree monomial transforms by , and coefficient extraction must undo that factor. For paired variables and the two factors cancel. The order of odd variables and differentials fixes the overall sign and must be kept consistent.
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The Berezin integral is a concept from mathematical physics, particularly in the field of supersymmetry and quantum mechanics. It is a type of integral used for integrating functions of Grassmann variables, which are anticommuting variables that the Berezin calculus is built around. Grassmann variables play a fundamental role in the formulation of supersymmetry, where they are used to describe fermionic fields or states.