Questions: Ward-Takahashi Identities

Question 1

The simplest Ward identity in QED states that q_mu M^mu = 0, where M^mu is any amplitude with an external photon of momentum q. What is the physical content of this equation?

• A. It says that photons cannot be created or destroyed
• B. It says that the longitudinal polarization of the photon does not contribute to any physical amplitude — this is the quantum-level guarantee that the photon has only two physical polarization states, enforced by gauge invariance
• C. It says that all amplitudes with photons are zero
• D. It says that the photon momentum is always zero

Answer: (B)

In a covariant gauge, the photon propagator includes contributions from all four polarizations (including longitudinal and timelike). The Ward identity q_mu M^mu = 0 guarantees that when you contract any amplitude with the photon's four-momentum (which projects onto the longitudinal/scalar polarizations), you get zero. This means unphysical polarizations do not contribute to physical observables, which is essential for unitarity. It also implies that replacing a polarization vector epsilon^mu by q^mu in any amplitude gives zero — a useful computational check.

Question 2

The Ward-Takahashi identity for the QED vertex is q_mu Gamma^mu(p+q, p) = S^{-1}(p+q) - S^{-1}(p), where Gamma^mu is the full (all-orders) vertex function and S is the full electron propagator. What does this relation imply about renormalization?

• A. It implies that the vertex function is finite
• B. It implies Z_1 = Z_2: the vertex renormalization factor equals the electron field renormalization factor, so that the charge renormalization comes entirely from the photon field (Z_3)
• C. It implies that the electron mass does not renormalize
• D. It implies that all QED divergences cancel

Answer: (B)

The Ward-Takahashi identity connects the vertex function (renormalized by Z_1) to the electron propagator (renormalized by Z_2). Since both sides must renormalize consistently, the identity requires Z_1 = Z_2. This means the net charge renormalization is e_R = e_0 sqrt(Z_3), independent of Z_1 and Z_2. The physical consequence is that all charged particles have their charges renormalized by the same factor (through the vacuum polarization), guaranteeing the universality of electric charge. This is why the proton charge equals the positron charge to extraordinary precision.

Question 3

Ward identities can be derived from gauge invariance of the path integral. This derivation is independent of perturbation theory and holds to all orders.

• T. True
• F. False

Answer: True

In the path integral formulation, Ward identities follow from the invariance of the functional integral under gauge transformations. The measure D[psi]D[psi-bar]D[A] and the action S are both gauge-invariant, so an infinitesimal gauge transformation gives zero when applied to Z[J]. This produces an exact functional identity among Green's functions — the Ward-Takahashi identity — that holds non-perturbatively. The derivation does not assume weak coupling or the validity of perturbation theory. This is why Ward identities are among the most reliable structural results in QFT.

Question 4

Explain why the Ward identity protects the photon mass from receiving radiative corrections, and what would happen if this protection failed.

Model answer: The vacuum polarization tensor Pi^{mu nu}(q) must satisfy the Ward identity q_mu Pi^{mu nu} = 0, which forces Pi^{mu nu}(q) to be proportional to (q^mu q^nu - g^{mu nu} q^2) — it is transverse. This means Pi^{mu nu}(0) = 0: there is no constant (momentum-independent) contribution to the photon self-energy, which would act as a mass term. The photon propagator is modified from 1/q^2 to 1/(q^2(1 - Pi(q^2))), where Pi(q^2) vanishes at q^2 = 0, ensuring the pole stays at q^2 = 0 (massless photon). If the Ward identity were violated (by a gauge anomaly, for example), nothing would prevent Pi(0) from being nonzero, and the photon would acquire a mass, dramatically changing electromagnetism.

This is the most physically consequential application of the Ward identity. The masslessness of the photon — and hence the infinite range of the electromagnetic force — is not an accident or a fine-tuning but a consequence of gauge invariance, protected to all orders in perturbation theory by the Ward identity.