How Does a Pion Transform Under an Axial Transformation to Order π²?

In summary, the poster is considering a non-linear chiral theory with a Lagrangian in terms of a field #\Sigma = e^{\frac{2i\pi}{f}}# where #\pi# is a pion matrix containing pion, kaon, and #\eta#. They need to calculate the transformation of #\pi# up to order #\pi^2# under an axial transformation where #R=L^\dagger#. Using the fact that $R$ and $L$ are elements of $SU(3)$, the transformation can be simplified and expanded up to order $\pi^2$. The commutation relation between $L$ and $R$ can also be used to further simplify the
  • #1
thinkLamp
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Homework Statement


I'm considering a non-linear chiral theory where the Lagrangian is in terms of the field #\Sigma = e^{\frac{2i\pi}{f}}# where #\pi# is my pion matrix containing pion, kaon, and #\eta#. I need to calculate the transformation of #\pi# up to order #\pi^2# under an axial transformation where #R=L^\dagger#. We're given that under #SU(3)_R \times SU(3)_R# transformations, #\Sigma# transforms as #\Sigma \to L \Sigma R^\dagger#.

Homework Equations

The Attempt at a Solution


$$\Sigma \to L\Sigma R^\dagger$$
$$= L \left( 1 + \frac{2i\pi}{f} + \frac{4i^2}{2 f^2} \pi^2 + \ldots \right) R^\dagger$$.
Now use $R = L^\dagger$. So,
$$= R^\dagger \left( 1 + \frac{2i\pi}{f} + \frac{4i^2}{2 f^2} \pi^2 + \ldots \right) R^\dagger\\
= R^\dagger R^\dagger + \frac{2i}{f} R^\dagger \pi R^\dagger + \frac{4i^2}{2 f^2} R^\dagger \pi R^\dagger R \pi R^\dagger + \ldots $$

Not sure where to go from here.
 
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  • #2
It looks like we need to use the fact that $R$ and $L$ are both elements of $SU(3)$, so $R^\dagger R = 1$ and $R^\dagger \pi R = \pi$. But I'm not sure how to incorporate that into the transformation of $\pi$. Any help would be greatly appreciated!
Thank you for sharing your question with us. It seems like you are on the right track with your attempt at the solution. To continue, you can use the fact that for elements of $SU(3)$, $R^\dagger R = 1$ and $R^\dagger \pi R = \pi$ to simplify the expression. This will allow you to expand the expression up to order $\pi^2$ and see how $\pi$ transforms under the axial transformation. Additionally, you can also use the commutation relation between $L$ and $R$ to further simplify the expression. I hope this helps and good luck with your research!
 

FAQ: How Does a Pion Transform Under an Axial Transformation to Order π²?

1. What is axial transformation in the context of pion?

Axial transformation is a mathematical operation that involves changing the reference frame of a particle, specifically a pion, from one where it is at rest to one where it is moving at a constant velocity. This allows for a different perspective on the pion's properties and behavior.

2. How does axial transformation affect the spin of a pion?

Axial transformation does not affect the spin of a pion. The spin of a particle is an intrinsic property that remains constant regardless of the reference frame, as long as the particle is not experiencing any external forces.

3. What is the significance of axial transformation in particle physics?

Axial transformation is a crucial concept in particle physics as it allows for the understanding of a particle's behavior from different perspectives. It is also used in the calculation of certain properties of particles, such as their momentum and energy, in different reference frames.

4. Can axial transformation be applied to other particles besides pions?

Yes, axial transformation can be applied to any particle that has a rest frame and is moving at a constant velocity. This includes other subatomic particles such as protons, neutrons, and electrons.

5. How is axial transformation related to Lorentz transformation?

Axial transformation is a special case of Lorentz transformation, which is a set of mathematical equations that describe how the properties of a particle change when viewed from different reference frames. Axial transformation specifically deals with transformations along the axis of motion, while Lorentz transformation involves transformations in all three dimensions.

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