# Understanding the Use of Trace in QFT Lagrangian for Chiral Symmetry in QCD

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• Phis
In summary, the conversation revolves around the use of trace in QFT, specifically in the lagrangian for chiral symmetry in QCD. The individual asking the question is struggling to understand the concept and has put the question on multiple forums without receiving a satisfactory answer. The individual references a specific article and offers a few possible interpretations for the use of trace, including the application of Jacobi's formula and the properties of hermitian unitary matrices. The main question is why the lagrangian is written as a trace and the reason behind it.
Phis
I have a question about the use of trace in QFT in general - more specifically the use of trace in the lagrangian in the effective theory concerning chiral symmetry in QCD. I am slowly trying to get a hang of everything, and most things i am able to calculate, but i still have som very specific areas where i lack a deeper physical intuition/understanding. The biggest hole in my learning is the use of trace.

Bear with me if you think this is extremely trivial. I have put this question on another forum before, but people seem to think that this is so trivial, that it doesn't deserve an answer. I haven't had QFT courses nor advanced particle physics, so a lot of this is new to me. Everybody seems to think this is trivial, but at the moment it confuses me a lot! I feel that a lot of other smaller things will fall into place after i understand this tiny bit.

To get a context or some form of reference, I am using the article https://arxiv.org/pdf/hep-ph/9806303v1.pdf and it's specifically the jump from the lagrangian in 4.18 to the one in 4.25 on page 40 and 41, that confuses me.

I have a couple of interpretations myself...

Has it something to do with Jacobi’s formula and the fact that ##det(e^{A})=e^{Tr(a)}##
Where we have ## U(\phi) = exp(i \frac{\Phi}{f}) ## where ## \Phi(x) \equiv \lambda \phi = \sqrt{2} \cdot [3x3 matrix] ## and the 3x3 matrix is the one containing the pions, kaons and eta's. Because U is a hermitian unitary matrix belonging to SU(3) we know that ## det(U) = 1## and therefore ## e^{Tr(U)} = 1 ## which is the same as saying that U is traceless and of course the Gell-mann matrices are traceless.

Or is it something much simpler with calculating/summing the eigenvalues of a linear algebra problem where we have a diagonalized matrix and seek our observables.

Or is it maybe that the order of matrices in our lagrangian is put in a way that we have ## [1x4] \cdot [4x4] \cdot \ldots \cdot [4x4] \cdot [4x1]## and we end up with a [1x1] matrix. Because the trace of a scalar is just the scalar it self, we can just as well write our lagrangian as a trace because we then may make use of the cyclic properties of the trace which helps a great deal when making transformations.

Or is it... ?

I can definitely see why we WANT to use it - mostly due to its cyclic properties, and the fact that we can pull all constants outside - but i can not see WHY we are allowed to write our lagrangian as a trace all of a sudden? I get that the lagrangian needs to be a scalar, since it is just ## L=T-V ## but why exactly make it a trace, what is the reason for the only interesting thing happening only in the diagonal, and why a sum, and not say a product or...?

I hope someone can give an in depth and hopefully simple explanation. Thank you very much!

Phis said:
I have a question about the use of trace in QFT in general - more specifically the use of trace in the lagrangian in the effective theory concerning chiral symmetry in QCD. I am slowly trying to get a hang of everything, and most things i am able to calculate, but i still have som very specific areas where i lack a deeper physical intuition/understanding. The biggest hole in my learning is the use of trace.

Bear with me if you think this is extremely trivial. I have put this question on another forum before, but people seem to think that this is so trivial, that it doesn't deserve an answer. I haven't had QFT courses nor advanced particle physics, so a lot of this is new to me. Everybody seems to think this is trivial, but at the moment it confuses me a lot! I feel that a lot of other smaller things will fall into place after i understand this tiny bit.

To get a context or some form of reference, I am using the article https://arxiv.org/pdf/hep-ph/9806303v1.pdf and it's specifically the jump from the lagrangian in 4.18 to the one in 4.25 on page 40 and 41, that confuses me.

I have a couple of interpretations myself...

Has it something to do with Jacobi’s formula and the fact that ##det(e^{A})=e^{Tr(a)}##
Where we have ## U(\phi) = exp(i \frac{\Phi}{f}) ## where ## \Phi(x) \equiv \lambda \phi = \sqrt{2} \cdot [3x3 matrix] ## and the 3x3 matrix is the one containing the pions, kaons and eta's. Because U is a hermitian unitary matrix belonging to SU(3) we know that ## det(U) = 1## and therefore ## e^{Tr(U)} = 1 ## which is the same as saying that U is traceless and of course the Gell-mann matrices are traceless.

Or is it something much simpler with calculating/summing the eigenvalues of a linear algebra problem where we have a diagonalized matrix and seek our observables.

Or is it maybe that the order of matrices in our lagrangian is put in a way that we have ## [1x4] \cdot [4x4] \cdot \ldots \cdot [4x4] \cdot [4x1]## and we end up with a [1x1] matrix. Because the trace of a scalar is just the scalar it self, we can just as well write our lagrangian as a trace because we then may make use of the cyclic properties of the trace which helps a great deal when making transformations.

Or is it... ?

I can definitely see why we WANT to use it - mostly due to its cyclic properties, and the fact that we can pull all constants outside - but i can not see WHY we are allowed to write our lagrangian as a trace all of a sudden? I get that the lagrangian needs to be a scalar, since it is just ## L=T-V ## but why exactly make it a trace, what is the reason for the only interesting thing happening only in the diagonal, and why a sum, and not say a product or...?

I hope someone can give an in depth and hopefully simple explanation. Thank you very much!

The key point about using traces is that it gives quantities that are invariant under the chiral transformation. If you have a quantity that transforms as
$U \rightarrow A U B$ (with $A A^\dagger = B B^\dagger =1$ ) then the quantity $D_\mu U D^\mu U^\dagger$ for example transforms as
$$D_\mu U D^\mu U^\dagger \Rightarrow A D_\mu U D^\mu U^\dagger A^\dagger$$
so it is not invariant but the trace of that is invariant.

## 1. What is QFT Lagrangian and how is it used in Chiral Symmetry in QCD?

The Quantum Field Theory (QFT) Lagrangian is a mathematical framework used to describe the interactions between particles in quantum mechanics. It is used in Chiral Symmetry in Quantum Chromodynamics (QCD) to study the behavior of quarks and gluons, the fundamental particles that make up protons and neutrons. Chiral Symmetry is a fundamental symmetry in QCD that relates the left and right-handed components of a particle's spin. The QFT Lagrangian is used to understand how this symmetry is broken and how it affects the behavior of quarks and gluons.

## 2. What is the significance of trace in QFT Lagrangian for Chiral Symmetry in QCD?

The trace in QFT Lagrangian is a mathematical operation that allows for the calculation of the symmetry-breaking effects in Chiral Symmetry. It is a key factor in understanding how quarks and gluons interact with each other and how the symmetry is broken. The trace is also used to calculate the mass of the particles and how they interact with the strong nuclear force.

## 3. How does understanding the use of trace in QFT Lagrangian contribute to our understanding of QCD?

By understanding the use of trace in QFT Lagrangian, we can gain a deeper understanding of the behavior of quarks and gluons and how they interact with each other. This, in turn, helps us understand the strong nuclear force and how it binds protons and neutrons together to form atomic nuclei. It also allows us to make predictions about the properties of particles and their interactions, which can be tested through experiments.

## 4. What are some potential applications of understanding the use of trace in QFT Lagrangian for Chiral Symmetry in QCD?

Understanding the use of trace in QFT Lagrangian can have various applications in theoretical physics, such as in the study of particle physics and cosmology. It can also have practical applications in the development of new technologies, such as in the construction of more efficient nuclear reactors or the creation of new materials with unique properties.

## 5. Are there any challenges or limitations in using trace in QFT Lagrangian for Chiral Symmetry in QCD?

One of the challenges in using trace in QFT Lagrangian is that it requires complex mathematical calculations, which can be difficult to perform and interpret. Additionally, there may be limitations in using this approach to understand certain aspects of QCD, such as the confinement of quarks within protons and neutrons. However, ongoing research and advancements in technology may help overcome these challenges and limitations in the future.

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