Final answer:
The Ward identity can be derived from the Noether current of a Lagrangian density assuming the path integral measure is invariant, connecting symmetries to conservation laws and constraints on correlation functions in quantum field theory.
Step-by-step explanation:
The Ward identity can indeed be derived from the Noether current associated with a symmetry of a Lagrangian density, provided that the path integral measure is invariant under that symmetry. This is an important concept in quantum field theory which relates symmetries of the Lagrangian to conservation laws through Noether's theorem, and in turn, to constraints on correlation functions via the Ward identity.
Lagrangian symmetry transformations lead to conserved Noether currents when the equations of motion are satisfied. In quantum field theory, considering the path integral formulation, the invariance of the action under a symmetry transformation leads to the invariance of the generating functional. This invariance places constraints on the correlation functions, which is precisely what the Ward identity describes. When one assumes that the path integral measure is also invariant under the symmetry transformation, we can assert these constraints rigorously. Therefore, one can derive the Ward identities reflecting the symmetry of the system.