Representations of SO(3)

In summary: I am more likely to receive a response. In summary, the conversation discusses the reading of Cahn's book on semi-simple lie algebras and their representations, specifically in relation to the Lie algebra of SO(3). The author is attempting to build a representation using abstract commutation relations and defines the action of T_z and T_+ on the vector v_j. However, there is no explanation of what these vectors are and how one knows they exist.
  • #1
jdstokes
523
1
Hi all, I asked this on the Quantum Physics board but didn't get a response.

I'm reading Cahn's book on semi-simple lie algebras and their representations.

http://www-physics.lbl.gov/~rncahn/book.html [Broken]

In chapter 1, he attempts to build a (2j+1)-dimensional representation [itex]T[/itex] of the Lie algebra of SO(3) starting with the abstract commutation relations

[itex][T_z,T_+] = T_+, \quad [T_z,T_-] = - T_-,\quad [T_+,T_-] = 2T_z[/itex] Eq (I.14).

He begins by defining the action of [itex]T_z,T_+[/itex] on the vector [itex]v_j[/itex] by

[itex]T_z v_j = j v_j, \quad T_+ v_j = 0[/itex]

but he does not explain what the [itex]v_j[/itex]'s are. How does one even know that such vectors exist?

Any help would be greatly appreciated.
 
Last edited by a moderator:
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  • #2
jdstokes said:
Hi all, I asked this on the Quantum Physics board but didn't get a response.

I'm reading Cahn's book on semi-simple lie algebras and their representations.

http://www-physics.lbl.gov/~rncahn/book.html [Broken]

In chapter 1, he attempts to build a (2j+1)-dimensional representation [itex]T[/itex] of the Lie algebra of SO(3) starting with the abstract commutation relations

[itex][T_z,T_+] = T_+, \quad [T_z,T_-] = - T_-,\quad [T_+,T_-] = 2T_z[/itex] Eq (I.14).

He begins by defining the action of [itex]T_z,T_+[/itex] on the vector [itex]v_j[/itex] by

[itex]T_z v_j = j v_j, \quad T_+ v_j = 0[/itex]

but he does not explain what the [itex]v_j[/itex]'s are. How does one even know that such vectors exist?

Any help would be greatly appreciated.

[EDIT]I've decided to answer on the quantum physics forum[/EDIT]
 
Last edited by a moderator:

1. What is SO(3)?

SO(3) refers to the special orthogonal group in three dimensions, which is a mathematical group that contains all possible rotations in three-dimensional space. It is often used in physics and computer graphics to describe the orientation of objects.

2. How are rotations represented in SO(3)?

In SO(3), rotations are typically represented using 3x3 matrices or unit quaternions. These representations allow for efficient and precise calculations of rotations in three-dimensional space.

3. What is the significance of SO(3) in physics?

SO(3) is a fundamental concept in physics as it describes the rotational symmetry of physical systems. It is used in various fields of physics, including classical mechanics, electromagnetism, and quantum mechanics.

4. Can SO(3) be applied to real-world situations?

Yes, SO(3) has practical applications in fields such as robotics, computer graphics, and aerospace engineering. It is used to model the movement and orientation of objects in three-dimensional space, making it a valuable tool in various industries.

5. Are there any other representations of SO(3) besides matrices and quaternions?

Yes, there are other representations of SO(3) such as Euler angles, axis-angle representations, and Cayley-Klein parameters. However, matrices and quaternions are the most commonly used representations due to their simplicity and efficiency in calculations.

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