# Ward Identity in Srednicki QFT book

• wphysics
In summary, the conversation discussed the Ward-Takahashi identity and its proof using the path integral method. The first part of the proof was summarized, where it was mentioned that the term \delta\phi_a(x) can be dropped and integrated over x. The reason for this was explained to be because it is arbitrary and does not affect the integral. The expert also clarified that the linearity of the Klein-Gordon wave operator allows it to be pulled out of the path integral, and that perturbation theory may be necessary for calculating the path integral of the Klein-Gordon operator inside the time-ordered product.
wphysics
Hello guys, I am working on Ch22 "Continuous symmetries and conserved currents" of Srednicki QFT book.

I am trying to understand how to prove the Ward-Takahashi identity using path integral method, done in page 136 of Srednicki.

I understood everything up to Equation 22.22, which is
$$0 = \int \mathcal{D}\phi e^{iS} \int d^4 x \bigg[i \frac{\delta S}{\delta \phi_a (x) }\phi_{a_1}(x_1) \cdots \phi_{a_n}(x_n) + \sum_{j=1}^n \phi_{a_1}(x_1)\cdots \delta_{aa_j} \delta^4 (x-x_j) \cdots \phi_{a_n}(x_n) \bigg]\delta\phi_a (x)$$

Here, $\delta\phi_a(x)$ is arbitrary, so we can drop this and integration over x. So, what we get is
$$0 = \int \mathcal{D}\phi e^{iS}\bigg[i \frac{\delta S}{\delta \phi_a (x) }\phi_{a_1}(x_1) \cdots \phi_{a_n}(x_n) + \sum_{j=1}^n \phi_{a_1}(x_1)\cdots \delta_{aa_j} \delta^4 (x-x_j) \cdots \phi_{a_n}(x_n) \bigg]$$

Let's consider the free real scalar field and n=1. Then, $\frac{\delta S}{\delta \phi_a (x) }$ is just $(\partial_x^2-m^2)\phi(x)$. So, the above equation becomes
$$0 = \int \mathcal{D}\phi e^{iS}\bigg[i(\partial_x^2-m^2)\phi(x) \phi(x_1) + \delta^4(x-x_1) \bigg]$$
So, this is equivalent to
$$(-\partial_x^2 + m^2) i <0|T\phi(x)\phi(x_1) |0> = \delta^4(x-x_1)$$
Here comes my question.
The author argued that the Klein-Gordon wave operator should sit outside the time-ordered product is clear from the path integral form of Eq(22.22).
I guess that this is due to that we can pull the Klein-Gordon wave operator out of path integral. Is this guess right? Even though this is right, I would like to have clearer explanation for this.

My second question is what kinds of path integral I need to calculate
$$i <0|T(\partial_x^2 - m^2) \phi(x)\phi(x_1) |0>$$
where the Klein-Gordon wave operator is inside of time ordered operator.
Because if my guess is right, the Klein-Gordon wave operator can always be pull out of path integral, so it seems that there is no way to calculate this from the path integral method.
Here, I assume that $\phi(x)$ is not a solution of classical equation, so the Klein-Gordon wave operator acting on this field can be non zero.

Hello! Thank you for sharing your progress and questions on your current work. I am also familiar with the Ward-Takahashi identity and its proof using the path integral method, and I would be happy to provide some clarification and additional information on your questions.

Firstly, your understanding of the first part of the proof is correct. The reason why we can drop the term \delta\phi_a(x) and integrate over x is because it is arbitrary and does not affect the integral. This is a common technique in path integral calculations, where we can manipulate the integrand as long as we do not change the boundary conditions or the overall value of the integral.

To answer your first question, yes, you are correct in your guess that the reason why the Klein-Gordon wave operator can be pulled out of the path integral is because of its linearity. This property allows us to manipulate it and pull it out of the integral without changing the overall value. This is also why we can calculate the path integral of the Klein-Gordon operator inside the time-ordered product, as it can be pulled out and treated as a constant factor.

For your second question, you are correct in assuming that there is no way to calculate the path integral of the Klein-Gordon operator inside the time-ordered product. This is because, as you mentioned, the Klein-Gordon operator acting on a non-classical field can result in a non-zero value. Therefore, we cannot simply pull it out of the path integral and treat it as a constant factor. In these cases, we need to use other methods, such as perturbation theory, to calculate the path integral.

I hope this answers your questions and helps clarify your understanding of the Ward-Takahashi identity proof using the path integral method. If you have any further questions or need any additional clarification, please don't hesitate to ask. Good luck with your work!

## 1. What is the Ward Identity in Srednicki's QFT book?

The Ward Identity is a fundamental principle in quantum field theory that relates the behavior of a quantum field under infinitesimal gauge transformations to the associated conserved current. It plays a crucial role in understanding the behavior of quantum fields and their interactions.

## 2. How is the Ward Identity derived in Srednicki's QFT book?

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. In the context of QFT, the Ward Identity is derived by considering the infinitesimal gauge transformations of the fields and using the equations of motion to relate them to the conserved currents.

## 3. What is the significance of the Ward Identity in QFT?

The Ward Identity is a crucial tool in QFT as it allows us to relate the behavior of quantum fields to their associated conserved currents. This helps us to better understand the symmetries of the theory and how they are related to conservation laws. It also plays a key role in the study of anomalies and the renormalization of QFT.

## 4. Can the Ward Identity be generalized to non-Abelian gauge theories?

Yes, the Ward Identity can be generalized to non-Abelian gauge theories. In fact, it is an essential tool in understanding the behavior of non-Abelian gauge theories, such as the Standard Model of particle physics. The generalization involves considering the transformation of the fields under gauge transformations that are not purely abelian.

## 5. How does the Ward Identity relate to the BRST symmetry in Srednicki's QFT book?

The BRST symmetry is a mathematical tool used in QFT to study the behavior of gauge theories. It is closely related to the Ward Identity, as both involve the transformation of fields under infinitesimal gauge transformations. In fact, the BRST symmetry is a consequence of the Ward Identity, and understanding one can lead to a better understanding of the other.

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