Final answer:
The transformation of the path integral measure for fermions by the inverse of the Jacobian ensures conservation of probability and invariance under field transformations, reflecting the nature of fermions as described by anti-commuting variables in quantum field theory.
Step-by-step explanation:
In quantum field theory, the path integral measure for fermions transforms as DΨ→(DetU)⁻¹DΨ, where DetU represents the determinant of a transformation matrix, or the Jacobian, which describes how a function, field, or spatial volume changes under a continuous transformation. The choice to have the path integral measure transform by the inverse of the determinant is not merely a definition or convention; it encapsulates a deeper principle of the conservation of probability and invariance under changes of variables, specifically for transformations involving fermionic fields. When performing a change of variables in the path integral formulation, the measure must transform in a way that reflects the 'volume' element of the integration in the space of field configurations, and since fermions are described by anti-commuting variables, the transformation involves the inverse of the determinant to account for these properties. This is conceptually similar to how the expectation value of momentum in quantum mechanics requires incorporating the proper operator, and computing integrals generally requires correct expressions for differential elements (like dl, dA, or dV) tailored to the physical situation.