Field redefinition invariance in quantum field theory

The issue of field redefinition invariance of path integrals in quantum field theory is re-examined. A paradox is presented involving the reduction to an effective quantum-mechanical theory of a (d + 1)-dimensional free scalar field in a Minkowskian space-time 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.