Questions aboug Special Groups SO(n) and SU(n)

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In summary: DUsually, in Quantum Mechanics, SO(3) and SU(2) are utilized to describe, first, rotations in space, and SU(2) to describe and calculate rotations on an abstract space. My question is on about the description that SO(3) and SU(2) can do on earth... if SU(2) has "less information" or is less descriptive than SO(3) or are isomorphic. And this leaves to 2nd answer...:D This surjective homomorphism, means that are equal?Until you can accurately state what the 'information content' is in mathematical terms then we really can't help.
  • #1
Plott029
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Dear Friends,

I have many questions about the special Orthogonal Group SO(n) and the Special Unitary Group SU(n). The first, SO(n) has [tex]\frac {n (n-1)}{n}[/tex] parameters or degrees of freedom, and the second, SU(n) has [tex] n^2 -1[/tex].

If I take for example the group SO(3), this has 3 degrees of freedom, and SU(2) has too 3 degrees of freedom, and there's a relation 2->1 from SO to SU. The question is, if SU(n) can describe the same that SO(n) or if there's a lose of information in using SU(n).

Best Reggards.
 
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  • #2
Plott029 said:
Dear Friends,

I have many questions about the special Orthogonal Group SO(n) and the Special Unitary Group SU(n). The first, SO(n) has [tex]\frac {n (n-1)}{n}[/tex] parameters or degrees of freedom, and the second, SU(n) has [tex] n^2 -1[/tex].

If I take for example the group SO(3), this has 3 degrees of freedom, and SU(2) has too 3 degrees of freedom, and there's a relation 2->1 from SO to SU. The question is, if SU(n) can describe the same that SO(n) or if there's a lose of information in using SU(n).

Best Reggards.

Excuse me? Didn't you just say that SO(3) has[itex]\frac{3(3-1)}{3}= 2[/itex] degrees of freedom? And that SU(3) has 32- 1= 8 degrees of freedom?
 
  • #3
HallsofIvy said:
Excuse me? Didn't you just say that SO(3) has[itex]\frac{3(3-1)}{3}= 2[/itex] degrees of freedom? And that SU(3) has 32- 1= 8 degrees of freedom?

There's a mistake. SO(n) has [n(n-1)/2] and SU(n) [n2-1], thus, S(3) has 3 degrees of freedom and SU(2) has 3 degrees of freedom. The question is if SU(2) has the same information that SO(3).

Best reggards.
 
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  • #4
What on Earth does 'information' mean?
 
  • #5
Plott029 said:
There's a mistake. SO(n) has [n(n-1)/2] and SU(n) [n2-1], thus, S(3) has 3 degrees of freedom and SU(2) has 3 degrees of freedom. The question is if SU(2) has the same information that SO(3).

Best reggards.

Ah, I misread SU(2) as SU(3)! SU(2) and SO(3) both have dimension (degrees of freedom) 3(3-1)/2= 22- 1= 3. That does not mean that the are isomorphic in which "knowing one tells us everything about the other". There exist a surjective homomorphism from SU(3) to SO(2) with kernel {I, -I}.
 
  • #6
matt grime said:
What on Earth does 'information' mean?

:D

Usually, in Quantum Mechanics, SO(3) and SU(2) are utilized to describe, first, rotations in space, and SU(2) to describe and calculate rotations on an abstract space. My question is on about the description that SO(3) and SU(2) can do on earth... if SU(2) has "less information" or is less descriptive than SO(3) or are isomorphic. And this leaves to 2nd answer...


HallsofIvy said:
Ah, I misread SU(2) as SU(3)! SU(2) and SO(3) both have dimension (degrees of freedom) 3(3-1)/2= 22- 1= 3. That does not mean that the are isomorphic in which "knowing one tells us everything about the other". There exist a surjective homomorphism from SU(3) to SO(2) with kernel {I, -I}.

This surjective homomorphism, means that are equal?
 
  • #7
Until you can accurately state what the 'information content' is in mathematical terms then we really can't help.And equal is not the same as isomorphic. No SU group is equal to any SO group. One is a complex lie group the other is a real lie group. They are never equal, though some of them might be isomorphic (as groups).
 

1. What is the difference between SO(n) and SU(n)?

SO(n) and SU(n) are two different types of special groups in mathematics. SO(n) stands for Special Orthogonal group, while SU(n) stands for Special Unitary group. The main difference between these two groups is their underlying spaces. SO(n) operates on n-dimensional Euclidean space, while SU(n) operates on n-dimensional complex space.

2. What are the applications of SO(n) and SU(n)?

SO(n) and SU(n) have various applications in physics, engineering, and computer graphics. In physics, SO(n) and SU(n) are used to describe the symmetries of physical systems, such as rotations and transformations in quantum mechanics. In engineering, these groups are used in the study of rigid body motion and control systems. In computer graphics, they are used to represent 3D rotations and transformations.

3. How are SO(n) and SU(n) related to Lie groups?

SO(n) and SU(n) are both examples of Lie groups, which are groups that are also smooth manifolds. Lie groups are used to study continuous symmetries in mathematics and physics. SO(n) and SU(n) are also special cases of compact Lie groups, which have applications in representation theory and harmonic analysis.

4. Can you explain the structure of SO(n) and SU(n)?

SO(n) and SU(n) have a rich mathematical structure that is closely related to their underlying spaces. Both groups are non-abelian and have a center, which is the identity element. They also have a Lie algebra, which describes the tangent space at the identity element and the group's infinitesimal structure. Additionally, both groups are connected and simply connected, meaning that any two elements can be continuously connected by a path.

5. How are SO(n) and SU(n) represented mathematically?

SO(n) and SU(n) can be represented using matrices. SO(n) is represented by n × n orthogonal matrices with determinant 1, while SU(n) is represented by n × n unitary matrices with determinant 1. These matrices are known as the defining representations of the groups. Other representations of these groups can also be constructed using the representation theory of Lie groups.

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