Field Redefinition Invariance in Quantum Field Theory

The issue of field redefinition invariance of path integrals in quantum field theory is reexamined. A ``paradox'' is presented involving the reduction to an effective quantum-mechanical theory of a $(d+1)$-dimensional free scalar field in a Minkowskian spacetime with compactified spatial coordinates. The implementation of field redefinitions both before and after the reduction suggests that operator-ordering issues in quantum field theory should not be ignored.