Ward - Takahashi identities and Noether's theorem in quantum field theory

A gap in the mathematical logic in derivations in quantum field theory arises as consequence of variation before quantization. To close this gap the present paper introduces a mathematically rigorous variational calculus for operator fields. Using quantization before variation it is demonstrated that the so-called naive results are correct; in particular both Noether's theorem and the Ward-Takahashi identities retain full validity in quantum field theory.