# General form of electromagnetic vertex function in QFT

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• Wrichik Basu
In summary, the authors discuss electromagnetic form factors in a chapter on QFT and provide a general form for the vertex function ##\Gamma##. This form includes all possible combinations of ##\gamma## and ##\gamma_5## matrices and takes into account Lorentz invariance.
Wrichik Basu
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TL;DR Summary
How did the authors write the general form of the electromagnetic vertex function out of nowhere?
I am studying a beginner's book on QFT.

In a chapter on electromagnetic form factors, the authors have written, using normalized states,
$$\begin{eqnarray} \langle \vec{p'}, s'| j_\mu (x) |\vec{p}, s \rangle \ = \ \exp(-i \ q \cdot x) \langle \vec{p'}, s'| j_\mu (0) |\vec{p}, s \rangle \nonumber \\ \Rightarrow \ \langle \vec{p'}, s'| j_\mu (x) |\vec{p}, s \rangle \ = \ \dfrac{\exp(-i \ q \cdot x)}{\sqrt{2 E_p V} \sqrt{2 E_{p'} V}} \bar{u}_{s'} (\vec{p'}) e \Gamma_\mu(p, p') u_s(\vec{p}) \nonumber \end{eqnarray}$$
where ##q = p - p'##, ##E_p = p^0##, ##E_{p'} = p'^0##, ##\Gamma## is the vertex function, ##u_s## is the plane wave solution of the Dirac Equation, ##\bar{u}_s## is the Dirac conjugate of ##u_s##, and other symbols have their usual meanings.

After this, the authors have said that the most general form of ##\Gamma## is $$\Gamma_\mu \ = \ \gamma_\mu(F_1 + \tilde{F}_1 \gamma_5) \ + (\ i F_2 + \tilde{F}_2 \gamma_5) \sigma_{\mu \nu} q^\nu \ + \tilde{F}_3 q_\mu \not\!q\gamma_5 \ + \ q_\mu (F_4 + \tilde{F}_4\gamma_5),$$ where ##\sigma_{\mu \nu} \ = \ \frac{i}{2} \left[\gamma_\mu, \ \gamma_\nu\right]## and ##\gamma_5 \ = \ \frac{i}{4!} \epsilon_{\mu\nu\lambda\rho} \gamma^\mu \gamma^\nu \gamma^\lambda \gamma^\rho##, all the ##F##'s are the electromagnetic form factors and other symbols have their usual meanings.

I understand that the book is for beginners, but how did the authors, out of nowhere, write down the general form for ##\Gamma##?

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The products of one or more ##\gamma## and ##\gamma_5##, forms a basis for all matrices of dimension 4. ##\Gamma## can thus be written in a basis of them. You need then also to take into account e.g. Lorentz invariance in order to arrive at the expression for ##\Gamma##.

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Wrichik Basu

## 1. What is the general form of an electromagnetic vertex function in quantum field theory (QFT)?

The general form of an electromagnetic vertex function in QFT is a mathematical expression that describes the interaction between a charged particle and a photon. It is a fundamental concept in QFT and is used to calculate the probability of a particle emitting or absorbing a photon.

## 2. How is the general form of an electromagnetic vertex function derived?

The general form of an electromagnetic vertex function is derived from the principles of quantum mechanics and special relativity. It takes into account the properties of the particles involved, such as their charge and spin, as well as the dynamics of their interaction with the photon.

## 3. What is the significance of the general form of an electromagnetic vertex function in QFT?

The general form of an electromagnetic vertex function is significant because it allows us to make predictions about the behavior of particles at the quantum level. By using this mathematical expression, we can calculate the probability of different interactions between particles and photons, which helps us understand the fundamental forces of nature.

## 4. Can the general form of an electromagnetic vertex function be applied to all types of particles?

Yes, the general form of an electromagnetic vertex function can be applied to all types of particles, as long as they have an electric charge and can interact with photons. This includes both fundamental particles, such as electrons and quarks, and composite particles, such as protons and neutrons.

## 5. How does the general form of an electromagnetic vertex function relate to experimental data?

The general form of an electromagnetic vertex function is used to make predictions about the behavior of particles, which can then be compared to experimental data. If the predictions match the experimental results, it provides evidence for the validity of the theory. If there is a discrepancy, it may indicate the need for further refinement of the theory.

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