Classical massless QED has axial current conservation. When quantizing the theory, we expect that suddenly ∂μˆjμ5≠0 (as an operator equality).
I have two questions regarding this:
Why is it then, that when trying to prove indeed that ∂μˆjμ5≠0 in the quantum version, we go on to try to prove that ∂μ⟨Ω|ˆj5mu(x)ˆjν(y)ˆjλ(z)|Ω⟩≠0
? I understand that the above relation would imply the violation of a naive Ward-identity, but what's the connection between axial current conservation and Ward-identities? Is it the case that if the Ward identity is violated then necessarily the current cannot be conserved?
When calculating ⟨Ω|ˆj5mu(x)ˆjν(y)ˆjλ(z)|Ω⟩, it appears that the ˆj's are treated as external photon lines, as in this image:
why is that? I would naively calculate such a transition amplitude by plugging in functional derivatives w.r.t. to the Fermion and anti-Fermion sources, for example, like so: ⟨Ω|ˆj5mu(z)ˆjν(x)ˆjλ(y)|Ω⟩=γμα1α2γ5α2α3γνα4α5γλα6α6((−1iδδηα1(z))(1iδδˉηα3(z))(−1iδδηα4(x))(1iδδˉηα5(x))(−1iδδηα6(y))(1iδδˉηα7(y))ZQEDZQED)|sources=0
This post imported from StackExchange Physics at 2014-08-07 15:35 (UCT), posted by SE-user PPR