Understanding Fierz Identity Transformations

In summary, the conversation involves a calculation of amplitudes and an expression involving Fierz identity. The solution of the exercise states that the expression can be simplified to 16 times a different expression. The person is seeking guidance on how to show this. They are also recommended to read a reference for further understanding.
  • #1
Luca_Mantani
36
1
Hi,
I was calculating some amplitudes and I end up with an expression like this:
$$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L b)(\bar{d}\gamma_\mu\gamma_\nu\gamma_\rho P_L u)$$

In the solution of the exercise they say that, from the Fierz identity:
$$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L b)(\bar{d}\gamma_\mu\gamma_\nu\gamma_\rho P_Lu)=16(\bar{c}\gamma^\mu P_L b)(\bar{d}\gamma_\mu P_L u)$$

How can I show that?
 
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  • #2
Luca_Mantani said:
Hi,
I was calculating some amplitudes and I end up with an expression like this:
$$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L b)(\bar{d}\gamma_\mu\gamma_\nu\gamma_\rho P_L u)$$

In the solution of the exercise they say that, from the Fierz identity:
$$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L b)(\bar{d}\gamma_\mu\gamma_\nu\gamma_\rho P_Lu)=16(\bar{c}\gamma^\mu P_L b)(\bar{d}\gamma_\mu P_L u)$$

How can I show that?

Probably, it is useful for you to read this reference:
https://arxiv.org/pdf/hep-ph/0306087.pdf
 

1. What is the Fierz identity transformation?

The Fierz identity transformation is a mathematical expression that describes the relationship between two different representations of the same physical system. It is commonly used in particle physics and quantum field theory to connect different mathematical descriptions of particles.

2. Why is the Fierz identity transformation important?

The Fierz identity transformation is important because it allows scientists to relate different mathematical descriptions of particles and systems, making it easier to understand and analyze complex physical phenomena. It also helps to unify different theories and models in physics.

3. How is the Fierz identity transformation used in particle physics?

In particle physics, the Fierz identity transformation is used to relate different representations of particles, such as spinors and vectors, to each other. This allows scientists to study and describe particles using different mathematical frameworks and make connections between them.

4. What are some practical applications of the Fierz identity transformation?

The Fierz identity transformation has many practical applications in physics, including in the study of quantum field theory, particle interactions, and the behavior of particles in high-energy collisions. It also has applications in other fields such as condensed matter physics and astrophysics.

5. Are there any limitations to the Fierz identity transformation?

Like any mathematical tool, the Fierz identity transformation has its limitations. It is most useful in the context of quantum field theory and particle physics, and may not be applicable to other areas of physics. It also assumes certain conditions and approximations, which may not always hold true in real-world scenarios.

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