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Physical meaning of Ward Identity and computing vertex functions

Following the derivation of Ward Identity by Weinberg book, you get it in the form $$ (l-k)_\mu S'(k)\Gamma^\mu(k,l)S'(l) = i S'(l) - iS'(k) $$ Can anyone explain the physical meaning of this ...
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68 views

How to choose the proper loop correction?

I review my QFT lecture notes and I am having hard times to figure out the significance of Ward identity in vacuum polarization. In class, we calculated one loop correction stated as $$ ...
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How Ward Identity indicate vacuum polarization correction?

In Peskin & Schroeder Chapter 7.5 Renormalization of The Electric Charge, they mention that vacuum polarization correction is $$ iM= (-ie)^2(-1)\int_{}{}\frac{d^4k} ...
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QED and anomaly

I've just started to learn anomalies in quantum field theories. I have a question. How to show that QED is free from vector current anomaly and what would happen if it were not? In other words, how ...
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30 views

Problem in the derivation of the Ward identities

This is a follow up to my previous Phys.SE question. The last equation in that question is $$-a\frac{1}{k^2+i\eta}k^{\lambda}i\Pi_{\lambda\rho}(k)iD^{\rho\nu}(k)=0$$ which we can further simplify as ...
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Why should Ward identities only be used with the effective action (as opposed to the generating functional for connected diagrams)?

My question is about the derivation of Ward identities. I will sketch it here in the case of an O(N) symmetric model and point out what it bothering me when I am done. I am being very sloppy with the ...
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41 views

Charge renormalization point in massive abelian gauge theory

Let's assume massless QED. The Ward identities hold. Through this identity we determine the finite part of the counterterm when fix the value of electric charge to be the experimentally observed one: ...
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85 views

Why do Lorentz invariance and Ward identities imply the structure $\Pi(k^2)(k^2\eta^{\mu\nu}-k^{\mu}k^{\nu})$ for the photon self energy?

In the first page of this link we can read that Ward identities and Lorentz invariance make the form if the photon self-energy be $$\Sigma^{\mu\nu}=\Pi(k^2)(k^2\eta^{\mu\nu}-k^{\mu}k^{\nu}).$$ Why?
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Propagation speed of photons when taking higher-order QFT corrections into account

In our group of experimental physicist who have nothing to do with and know very little about quantum field theory, we recently had a question concerning the propagation speed of photons in vacuum: ...
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1answer
138 views

Gauge invariance (QED)

In his book, the author says that according to the Feynman diagrams of this process in QED $$e^+ e^- \rightarrow \gamma \gamma,$$ gauge invariance requires that $$k_{1\nu}(A^{\mu\nu} + ...
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Diagrammatics of Slavnov Taylor Identity

Is there a reference other than the original paper of 't-Hooft and Veltman, where I can get a pedagogical introduction to the diagrammatic approach to understanding the BRST-Ward or Slavnov-Taylor ...
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562 views

Ward identity derived from global symmetry and SDE, different from that derived from gauge symmetry?

In QED, according to Schwinger-Dyson equation $^{[1]}$, $$\left(\eta^{\mu\nu}(\partial ^2)-(1-\frac{1}{\xi})\partial^{\mu}\partial^{\nu}\right)\langle 0|\mathcal{T}A_{\nu}(x)...|0\rangle = e\,\langle ...
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79 views

Ward identities and operator product expansions

Polchinski's (2.3.11) gives the Ward Identity $$i\epsilon[Res_{z\rightarrow z_0}j(z)\mathcal A(z_0,\bar z_0)+\bar {Res}_{\bar z\rightarrow \bar z_0}\tilde j(\bar z)\mathcal A(z_0,\bar ...
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1answer
141 views

Ward Identity in CFT

This is about Polchinski's eq(2.3.11). It says that $$Res_{z\rightarrow z_0}j(z)\mathcal A(z_0,\bar z_0)+\bar{Res}_{z\rightarrow z_0}\tilde j(\bar z)\mathcal A(z_0,\bar z_0)=\frac1{i\epsilon}\delta ...
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105 views

Can you gauge a $U(1)_L$ symmetry?

I recently calculating the one loop correction for the propagator of a gauge boson, $\hspace{5cm}$ I assumed arbitrary left and right couplings, $ g _L $ and $ g _R $. I found that the one loop ...
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240 views

Chiral Anomaly in Massless QED

Classical massless QED has axial current conservation. When quantizing the theory, we expect that suddenly $\partial_\mu \hat{j}^{\mu5}\neq0$ (as an operator equality). I have two questions regarding ...
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The Ward identity for EM amplitude with massive vector boson

Let's have theory of massive vector boson interacting with EM field: $$ L = |D_{\mu}W_{\nu} - D_{\nu}W_{\mu}|^{2} + m^{2}|W|^{2}, \quad D_{\mu} = \partial_{\mu} - ieA_{\mu}. $$ The question: how to ...
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187 views

Constructing Ward identity associated with conserved currents

Consider constructing the Ward identity associated with Lorentz invariance. It is possible to find a 3rd rank tensor $B^{\rho \mu \nu}$ antisymmetric in the first two indices, then the stress-energy ...
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1answer
232 views

Correlation functions and connection to ward identities

I have the following definition of a general correlation function $$ \langle \Phi(x_1)\dots \Phi(x_n)\rangle = \frac{1}{Z} \int [d\Phi] \Phi(x_1)\dots\Phi(x_n)e^{-S[\Phi]} $$ I have only just ...
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1answer
269 views

Question on derivation of Ward identity

I'm currently reading these notes about the Ward identity (pages 259 - 261). I will repeat some of the steps to make the question self-contained. Let us consider a local transformation on the field ...
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2answers
402 views

Peskin's book page 334 proof of $Z_1=Z_2$ to all orders in QED perturbation theory

Peskin in his QFT page 334 argued that $Z_1=Z_2$ to all orders in QED perturbation theory, but I couldn't understand his argument: ... With a generalization of the argument given there (section ...
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415 views

gauge invariance of the Feynman amplitudes

When we calculate the photon polarization sums over amplitudes, $$X=\sum\limits_{r=1}^{2}|\mathcal M_r|^2=\mathcal M_\alpha\mathcal M_\beta^*\sum\limits_{r=1}^{2}\epsilon_r^\alpha\epsilon_r^\beta$$ ...
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164 views

Some fundamental results in QFTs [closed]

In quantum theory we have some principles that guides us, e.g. Pauli's principle. What I am after in this question is a list of fundamental results, be it equation or identities that must hold in a ...
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1answer
609 views

Traceless of stress-energy tensor in $d=2$

This is a question regarding Francesco, section 4.3.3. In this section, he considers the two-point function $$ S_{\mu\nu\rho\sigma}(x) = \left< T_{\mu\nu}(x) T_{\rho\sigma}(0)\right> $$ He then ...
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1answer
777 views

How does the Ward-Takahashi Identity imply that non-transverse photons are unphysical in QED?

Peskin and Schroeder say that the Ward Identity of QED proves that non-transverse photon polarizations can be consistently ignored, but I'm confused about the details. Setup One starts by ...
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3answers
3k views

Is it really proper to say Ward identity is a consequence of gauge invariance?

Many (if not all) of the materials I've read claim Ward identity is a consequence of gauge invariance of the theory, while actually their derivations only make use of current conservation ...
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1answer
463 views

Some questions about Ward-Takahashi Identity

I'm a learner of Peskin and Schroeder's textbook of quantum field theory. I have proceeded to Ward-Takahashi identity and have one question when I look for Wikipedia for reference. The following is ...
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86 views

CP-symmetry and Ward identities and finite temperature

I have a few questions about Ward-identities which I summarize here. For each I am very greateful for answers and references to literature. Wikipedia states about Ward-identities: The ...
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1answer
90 views

Application of Ward identities for OPE under scaling and rotations

I think this is a very straightforward question but I don't see it right now. In Tong's notes on String theory (http://www.damtp.cam.ac.uk/user/tong/string/four.pdf) section 4.2.3 he defines the ...
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281 views

How does the conformal Ward identity guarantee a vanishing 3-point function in this case?

I was looking through some conformal Ward identity related things when I noticed that this paper (arXiv:1212.3788) writes in their equation (33), a 3-point function between a conserved current and two ...