Peskin and Schroeder in Section 7.4 in their book state the Ward identity in the following way:
Ward identity. If
M(k) = ε_μ(k) M^μ(k)
is the probability amplitude of a QED process involving an external photon with 4-momentum k and the polarization vector ε_μ(k), then if the polarization vector of the photon would be "longitudinal", that is,
ε_μ(k) = k_μ
for all coordinates μ, then the probability amplitude M(k) is zero. This is written using the Einstein notation as
k_μ M^μ(k) = 0.
In this we assume that the probability amplitude of the process can be written as a sum of the amplitudes for each polarization vector component of the photon.
Suppose that the photon entering the process is a real photon. We know that its polarization has to be normal relative to its spatial momentum and 4-momentum. If the Ward identity would not hold, then by running the process backward we would be able to produce a longitudinally polarized real photon, which is a contradiction. Thus, we proved that the Ward identity must hold for real photons.
However, if the photon is an internal photon line which carries pure spatial momentum between an electron and a proton in a scattering experiment, then the "polarization" of the photon is not defined, or it is longitudinal. What does the Ward identity say in that case?
Maybe we should not talk about polarization but state the Ward identity as a rule about Feynman diagrams. In the book of Peskin and Schroeder, Section 7.4, the identity is proved by induction for Feynman rules.
The identity seems to be a consequence of the fact that the value of the electron propagator only depends on how "far" off-shell is the electron. For example, in the following diagrams, the electron in the internal line is "as much" off-shell in both diagrams.
| photon
| q
e- ----------------------------
| photon
| k
| photon
| q
e- -----------------------------
| photon
| k
The electron enters and leaves the diagrams on-shell.
The Ward identity in the vacuum polarization integral
The enclosed picture was copied from the link.
In the link, the Ward identity used in a vacuum polarization calculation is "proved" from the formula above. The integrals over all 4-momenta k on
1 / (γ • (k - q) - m)
and
1 / (γ • k - m)
are identical if one changes the variable k - q in the first formula to k. One can claim that the difference of the integrals is zero, which would prove the Ward identity.
Both integrals diverge badly. If we integrate over a volume which is symmetric with respect to 4-momenta 0 and q, then the integrals really have the same value.
If one uses an integration volume which is asymmetric, the difference of the integrals does not converge.
Question. Is it fair to say that the difference of the integrals is zero?
In this blog we have been claiming that the vacuum polarization integral really calculates the part which is wiped off by destructive interference. Is there some reason why we should use an integration volume which is symmetric with respect to 0 and q?
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