# Request for a clarification about the Ward identity

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• HomogenousCow
In summary, the conversation discusses the Ward identity in Peskin & Schroeder's book, specifically how it relates to the path integral and the perturbative terms in the green's functions. The Fourier transform of the three point function is also mentioned, as well as the role of the photon and fermion propagators. The Ward identity states that if the photon propagator and constant factors are removed and the remaining part is contracted with the photon momentum, it will result in a relation similar to the one obtained from the path integral. The conversation ends with a question on how to relate the two together, particularly when the photon field is at the same position as the current.
HomogenousCow
TL;DR Summary
Request for clarification on the Ward identity
Hi I've been reading Peskin & Schroeder lately and I have some confusions over the ward identity.

So I think I understand how the identity works at a practical level but not exactly where it comes from. To illustrate my questions (which are difficult to state generally), I will make use the example of the three point function ##G^\mu (x_1,x_2,x_3)=\langle \Omega |T{ A^\mu (x_1)\psi (x_2) \bar \psi (x_3)}|\Omega\rangle##. The Fourier transform of this function looks like

$$\tilde{G}^\mu(p_1,p_2,p_3) = (2\pi)^4 \delta^4 (p_1 - p_2 + p_3) D^{\mu\nu}(p_2-p_3)\tilde{S}(p_2)i\Gamma_{\nu} (p_2,p_3)\tilde{S}(p_3),$$ where ##\Gamma## are the 1PI graphs, ##D## and ##S## are the photon and fermion propagators respectively. The Ward identity says if we throw out the photon progator and the constant factors before it, and then contract the remaining part with the photon momentum ##q = p_2 - p_3##, we'll get

$$q_\mu \tilde{S}(p_2)i\Gamma^\mu(p_2,p_3)\tilde{S}(p_3) \sim e(\tilde{S}(p_2) - \tilde{S}(p_3)).$$

Okay this is all good but I don't understand how this relates to the version of the Ward-Takahashi identity obtained from the path integral. Specifically, how can the above be dervied from the relation $$i\partial_\mu \langle 0 |T{ j^\mu (x)\psi (x_2) \bar \psi (x_3)}|0\rangle = \mathsf{Contact\ terms}$$ when the perturbative terms in the greens functions take the form $$\langle 0 |T{A^\nu(x_1)\psi (x_2) \bar \psi (x_3)}[\int d^4x A^\mu (x) j_\mu (x)]^n|0\rangle.$$ I can't see how you can relate the two together since in the latter case there is a photon field at the same position as the current.

## 1. What is the Ward identity?

The Ward identity is a fundamental concept in quantum field theory that relates the symmetries of a physical system to its conservation laws. It states that for any continuous symmetry of a system, there exists a corresponding conserved quantity.

## 2. Why is it important to request a clarification about the Ward identity?

Requesting a clarification about the Ward identity is important because it ensures a clear understanding of the concept and its implications. This can help in further research and application of the Ward identity in different contexts.

## 3. How is the Ward identity derived?

The Ward identity is derived from the Noether's theorem, which states that for every continuous symmetry of a physical system, there exists a corresponding conserved quantity. The Ward identity is a manifestation of this theorem in quantum field theory.

## 4. What are some examples of the Ward identity in action?

One example of the Ward identity in action is in the theory of electromagnetism, where the symmetry of gauge invariance leads to the conservation of electric charge. Another example is in the theory of quantum chromodynamics, where the symmetry of color gauge invariance leads to the conservation of color charge.

## 5. How does the Ward identity relate to other fundamental concepts in physics?

The Ward identity is closely related to other fundamental concepts in physics, such as symmetries, conservation laws, and Noether's theorem. It also has implications in various areas of physics, including particle physics, quantum field theory, and condensed matter physics.

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