Understanding Ward Identity and Its Connection to Gauge Invariance

[myqphy]

This is a question about Ward-Takahashi Identity.I go through the materials presented about Ward Identity in Peskin's book. there are two sections where mentioned this identity. First, in section 5.5, when the author discussed photon polarization sums. Second, in section7.4, where the author present a proof of Ward identity based on Feynman diagram.

Accroding to the methodology the author used. it seems that the discussion based on the classical theorem of current conservation is somehow "cheating" (see the first paragraph of chapter 7), for in section 5.5, they just supposed that the current conservation can be directly extended to quantum case. Although it is somehow cheating, we can clearly see the connection between Ward identity, current conservation and more important, gauge invariance.

Quite contrary, from the derivation which is not"cheating" in section 7.4. I can not clearly see the connection between gauge invariance and current conservation. Even more, by using the methodology of the author, it seems that Proca field(if we assume there is minmum coupling between Proca field and Dirac field), which is not gauge invariant, still obeys the Ward Identity. So I was totally confused of it.

So my question is, how can I find the connection between gauge invariance and Ward Identity from the derivation in section 7.7?

Than you very much

 

[eljose79]

for your question about the Ward-Takahashi Identity. The Ward Identity is a fundamental theorem in quantum field theory that relates the current conservation to the gauge invariance of a theory. The Ward-Takahashi Identity is an extension of this theorem that applies to non-abelian gauge theories.

In section 5.5, the author uses the classical theorem of current conservation to establish the connection between Ward Identity and gauge invariance. This may seem like "cheating" because it assumes that the classical theorem can be directly extended to the quantum case. However, this is a valid approach as it allows us to see the connection between these concepts.

In section 7.4, the author presents a derivation of the Ward Identity based on Feynman diagrams. This approach does not explicitly use the classical theorem of current conservation, but it still shows the connection between gauge invariance and Ward Identity. The proof relies on the fact that gauge invariance is a symmetry of the theory and is reflected in the structure of Feynman diagrams.

The connection between gauge invariance and Ward Identity can be seen more clearly in section 7.7. Here, the author shows that the Ward Identity can be derived from the invariance of the action under local gauge transformations. This demonstrates that gauge invariance is a fundamental property of the theory and is intimately related to the Ward Identity.

Regarding your confusion about the Proca field, it is important to note that the Ward Identity is a general theorem that applies to all gauge theories, regardless of whether the fields are gauge invariant or not. The Ward Identity is a consequence of the underlying symmetry of the theory, and it does not depend on the specific form of the fields.

In summary, the connection between gauge invariance and the Ward Identity can be seen through various approaches, such as the classical theorem of current conservation, Feynman diagrams, and the invariance of the action under local gauge transformations. I hope this helps clarify your understanding of the Ward-Takahashi Identity.