# Compute 3-Point Function QFT Homework with Fermions

• nikosbak
In summary, the conversation discusses the computation of a three-point function involving fermion currents and the source functional for fermions. The specific form of the current and source functional is mentioned, along with the need to insert the gammas between the derivatives in order to prove the correspondence between appearances of the fermion field and the derivatives of the source functional. The speaker is seeking help or an example on how to compute this three-point function.
nikosbak

## Homework Statement

I'm working on path integrals for fermions and I came across an exercise that ask to compute the three point functions , one of that is the:
$$<0|J^{\mu}(x_1)J^{\nu}(x_2)J^{\rho}(x_3)|0>$$
where $$J^{\mu}$$ is the current $$J^{\mu}=\bar{\psi}\gamma^{\mu}\psi$$.

***Can you give me an idea or an example on how to compute this things?***

Because I'm trying to use the usual logic about I don't see what I can do about the gammas isnside the correlation.

The sourse functional for fermions is :
$$Z[\eta,\bar{\eta}]=\exp\{-i\int dx\;dy\; \bar{\eta}(x)S(x-y)\eta(y)\}$$
where $$S(x)=(i\gamma^{\mu}{\partial_{\mu}-m)^{-1}}$$.

The correspondence between appearances of ##\psi(x)## and derivatives ##\delta/\delta \eta## is derived from the form of the source functional before the fermion field has been integrated out. You should be able to prove that the ##\gamma##s have to be inserted between the derivatives.

## 1. What is a 3-point function in quantum field theory (QFT)?

A 3-point function in QFT is a mathematical quantity that describes the interaction between three particles in a quantum field. It is calculated using Feynman diagrams and is important in understanding the behavior of particles in a quantum field.

## 2. What are fermions in QFT?

Fermions are a type of particle in QFT that follow Fermi-Dirac statistics. They have half-integer spin and obey the Pauli exclusion principle, meaning that no two fermions can occupy the same quantum state. Examples of fermions include electrons, protons, and neutrons.

## 3. How do you compute a 3-point function in QFT with fermions?

The computation of a 3-point function in QFT with fermions involves using Feynman diagrams and perturbation theory. First, the Feynman rules for fermions are used to assign mathematical expressions to each vertex in the diagram. Then, the diagrams are evaluated using Feynman integrals and the results are combined to obtain the final 3-point function.

## 4. What is the significance of computing a 3-point function with fermions in QFT?

The 3-point function with fermions in QFT is important because it provides insights into the interactions between particles in a quantum field. It can be used to calculate scattering amplitudes and cross-sections, which are essential in predicting the behavior of particles in experiments. It also helps in understanding the fundamental forces in nature, such as the strong and weak nuclear forces.

## 5. Are there any limitations or challenges in computing a 3-point function with fermions in QFT?

Yes, there are some limitations and challenges in computing a 3-point function with fermions in QFT. One of the main challenges is the complexity of the calculations involved, as they often require advanced mathematical techniques and can be time-consuming. Additionally, the results may be non-perturbative, meaning they cannot be obtained using perturbation theory, which can make the calculations more difficult. There may also be limitations in the accuracy of the results, as higher-order Feynman diagrams may need to be included to improve the precision of the calculation.

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