# Ward identity from Ward-Takahashi identity?

• center o bass
In summary: This argument can be generalized to any physical process involving two particles. The Ward-Takahashi identity states that the total momentum of the particles is equal to the sum of their momenta. So if you have a physical process involving two particles, and take the sum of their momenta, you will get the Ward-Takahashi identity.
center o bass
The Ward-Takahashi identity for the simplest QED vertex function states that

$$q_\mu \Gamma^\mu (p + q, p) = S^{-1}(p+q) - S^(p)^{-1}.$$

Often the 'Ward-identity' is stated as, if one have a physical process involving an external photon with the amplitude

$$M = \epsilon_\mu M^\mu$$

then

$$q_\mu M^\mu = 0$$
if q is the momentum of the external photon. One can argue on that the latter identity is true because of current conservation, but can one show that it follows from the Ward-Takahashi identity above? If so how?

I agree with you on that one. Electron to a photon and an electron is not a physical prosess at all. However suppose that one of the the electrons are not on shell by being coupled to a subdiagram; one could for example have photon +electron coming in - vertex function - virtual electron - vertex function - photon +electron. Now one of the electrons are virtual with a momentum equal to the sum of the incoming photon and electron.

(sorry, I deleted my previous post because I realized you were looking for something more general. I think that what I'm writing below is a better answer to your question.)

Here's my understanding, which is based on a reading of Peskin and Schroeder section 7.4.

You start with your statement of the Ward-Takahashi identity, which is true for the electron vertex, and proceed to show that it is also true for any physical process, not just the simple 3-point vertex. That can be done either order-by-order by examining the topology of Feynman diagrams, or more generally by using the functional integral.

Next, you appeal to the argument in the LSZ reduction formula, which says that S-matrix elements are proportional to the residue of the pole of $M$ on the mass shell of the external particles. If the $M$ on the left is on-shell, then neither $M(p+q)$ or $M(p)$ on the right are on-shell, so neither have a pole in the right place to contribute to the S-matrix. Thus, the right-hand side is zero when you extract out the poles to compute the S-matrix.

## 1. What is the Ward identity from Ward-Takahashi identity?

The Ward identity from Ward-Takahashi identity is a mathematical relationship between Green's functions and their corresponding vertex functions in quantum field theory. It is a consequence of the Ward-Takahashi identity, which is a symmetry in the theory.

## 2. Why is the Ward identity from Ward-Takahashi identity important?

The Ward identity from Ward-Takahashi identity is important because it provides a powerful tool for analyzing and understanding symmetries in quantum field theory. It allows for the calculation of physical quantities and their relationships to symmetry transformations.

## 3. How is the Ward identity from Ward-Takahashi identity derived?

The Ward identity from Ward-Takahashi identity is derived from the Ward-Takahashi identity, which is a consequence of the gauge invariance of the quantum field theory. This identity is obtained by considering the behavior of the Green's functions under infinitesimal transformations of the fields.

## 4. What is the difference between the Ward identity and the Ward-Takahashi identity?

The Ward identity from Ward-Takahashi identity is a specific mathematical relationship derived from the more general Ward-Takahashi identity. The Ward-Takahashi identity is a symmetry in quantum field theory, while the Ward identity is a consequence of this symmetry.

## 5. How is the Ward identity from Ward-Takahashi identity used in practice?

The Ward identity from Ward-Takahashi identity is used in practice to simplify and constrain the calculations involved in quantum field theory. It allows for the simplification of physical quantities and can also be used to check the consistency of a theory and its symmetries.

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