# 180deg Rotation in Isospin space

• malawi_glenn
In summary, the G-parity of the pion is -1, which can be achieved by rotating in isospin space around the y-axis by angle \pi followed by C-conjugation. The rotation matrix for this transformation is given by (e.g. Sakurai, Halzen ...) and when applied to \pi^+ with isospin T = 1 and Tz = +1, it results in \pi^-. However, this is not the required -\pi^-, which can be obtained by rotating around the x-axis instead. The rotation matrix for this transformation is given by the author as a correction.
malawi_glenn
Homework Helper
Gold Member
The G-partiy of the pion is -1, which is rotation in isospin space around y-axis by angle $\pi$ followed by C-conjugation.

The rotation matrix around y-axis , with angle $\pi$, is: (e.g. Sakurai, Halzen ..)

$$\begin{pmatrix}0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$$

Thus on $\pi^+$: T = 1, Tz = +1:
$$\begin{pmatrix}0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0\\ 0 \\ 1 \end{pmatrix} = \pi^-$$

But this is not $-\pi^-$, which is required (see e.g. http://arxiv.org/PS_cache/hep-ph/pdf/9903/9903256v1.pdf page 14)

I feel really stupid now, I don't get the required minus sign for pi- and pi+, only for pi0...

The rotation matrix I use is:
$$\begin{pmatrix} \tfrac{1}{2}(1+\cos \beta ) & - \tfrac{1}{\sqrt{2}}\sin \beta & \tfrac{1}{2}(1 - \cos \beta ) \\ -\tfrac{1}{\sqrt{2}}\sin \beta& \cos \beta & - \tfrac{1}{\sqrt{2}}\sin \beta \\ \tfrac{1}{2}(1-\cos \beta ) & \tfrac{1}{\sqrt{2}}\sin \beta & \tfrac{1}{2}(1 + \cos \beta ) \end{pmatrix}$$

I found that the author has made an error, one should rotate around x-axis with that convention of C-parity.

First of all, don't feel stupid! Understanding concepts in physics can be challenging, and it's important to ask questions and seek clarification.

In this case, the issue is with the definition of the rotation matrix. The matrix you are using is for a rotation around the y-axis in a 3-dimensional space, but isospin space is a 2-dimensional space. Therefore, the rotation matrix needs to be adjusted accordingly.

The correct rotation matrix for a rotation around the y-axis in isospin space is:
\begin{pmatrix} \cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{pmatrix}

Using this matrix, we can see that for a rotation of \pi around the y-axis, the pion with isospin T=1 and Tz=+1 will become a pion with T=-1 and Tz=-1, as required.

I hope this helps clarify the issue and allows you to continue your understanding of isospin rotations. Keep asking questions and seeking understanding, it's the best way to learn!

## 1. What is 180 degree rotation in isospin space?

180 degree rotation in isospin space is a mathematical transformation that is used to describe the behavior of subatomic particles, specifically those that have a property called isospin. Isospin is a quantum number that describes the strong interactions between particles, such as protons and neutrons. 180 degree rotation in isospin space is a way to visualize and understand the relationships between these particles and their interactions.

## 2. Why is 180 degree rotation in isospin space important?

180 degree rotation in isospin space is important because it helps us understand the fundamental properties and interactions of subatomic particles. It allows us to predict and study the behavior of particles in different scenarios, and has been crucial in developing theories such as the Standard Model of particle physics.

## 3. How is 180 degree rotation in isospin space related to quantum mechanics?

180 degree rotation in isospin space is related to quantum mechanics because it is a mathematical concept that is used to describe the behavior of particles at the subatomic level. In quantum mechanics, particles are described by wave functions, and 180 degree rotation in isospin space is a transformation that can be applied to these wave functions to better understand the behavior of particles with isospin.

## 4. Can 180 degree rotation in isospin space be observed experimentally?

No, 180 degree rotation in isospin space cannot be directly observed experimentally. It is a mathematical concept that is used to describe the behavior of particles, but it cannot be physically observed. However, its effects can be observed through the interactions and behavior of particles in experiments.

## 5. Are there any practical applications of 180 degree rotation in isospin space?

Yes, there are practical applications of 180 degree rotation in isospin space. It is used in nuclear physics to describe the behavior of nuclei and in particle physics to understand the interactions between particles. It has also been applied in other fields such as quantum computing and cryptography.

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