The introduction of spinor fields and the problem of the positivity of the energy. Need for anti-commutators. Calculus in a Grassmannian manifold, the Berezin integral. Clifford algebras. Quantum mechanics and quantum field theory with fermions. The use of path integrals.
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.