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Formal proof of Clebsch Gordan sum

0

$begingroup$

physicist here.

When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be have values like $j = 0, 1/2, 1, 3/2, ...$. Let $D_j$ denote the representation of $so(3)$ with index j.

If one has two Hilbert spaces $H_1, H_2$ which both carry a representation $D_j_1, D_j_2$, the tensorspace of $H_1$ and $H_2$ carries a representation $D_j_1 otimes D_j_2$.

In my script for quantum mechanics it says that this tensor representation can be decomposed into a direct sum of irreducible representations in the following way:

$D_j_1 + j_2 oplus D_j_1 + j_2 -1 oplus ... oplus D_$

One can easily check that if a representation $D_j$ has dimension $2j+1$ then the dimension of the direct sum is the same as the dimension of the tensorrepresentation, which is basically the whole proof given in the script.

For my good conscience I'd like to know a little bit more about this:

1.) What theorem tells me that the tensor representation can be decomposed into a direct sum of irreps?

2.) Which theorem tells me that the addends of the direct sum are the already found irreducible representations? Why can't it be a completely different function?

3.) Why does the sum go from $j_1 + j_2$ to $|j_1 - j_2|$? Intuitively this makes sense if one looks at the 3 component of the angular momentum operator since this can be $j_1, j_2$ maximally, but I don't understand the mathematical argumentation behind it.

I hope my questions are clear and that someone can point me in the right direction.

Cheers!

group-theory representation-theory lie-groups lie-algebras mathematical-physics

share|cite|improve this question

edited Mar 24 at 15:18

Abdelmalek Abdesselam

854312

asked Jun 30 '14 at 17:36

user80944

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  1) This is basic Lie theory, but the answer is a chapter or two in a book. 2) You are right that with other groups you actually get different reps in there (me thinks). It is just that $SO(3)$ (or $SL_2$) is a special case. This is another non-trivial result but not as long as the answer to 1. 3) The dimensions were used up, so no further summands fit in there. The "true" explanation comes from rep theory (for my money), but it is hardly a coincidence that it matches with our intuition about addition of angular momenta.
  $endgroup$
  – Jyrki Lahtonen
  Jun 30 '14 at 20:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But it really looks like your question may be a bit broad. I am not voting to close yet, as somebody may want to try to give you an overview. The textbook I would recommend (Humphreys) is from Lie algebra side, and Lie group people may be able to recommend something that gives you a decent understanding of what's going on without needing to wade through quite a bit of advanced linear algebra and proving those basics.
  $endgroup$
  – Jyrki Lahtonen
  Jun 30 '14 at 20:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hey thanks a lot for the indepth answer. I guess I'll just accept it for now and will return to the proper understanding once I'm done with the exam.
  $endgroup$
  – user80944
  Jul 1 '14 at 15:55
  

  
  
  
  

add a comment | 

0

$begingroup$

physicist here.

When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be have values like $j = 0, 1/2, 1, 3/2, ...$. Let $D_j$ denote the representation of $so(3)$ with index j.

If one has two Hilbert spaces $H_1, H_2$ which both carry a representation $D_j_1, D_j_2$, the tensorspace of $H_1$ and $H_2$ carries a representation $D_j_1 otimes D_j_2$.

In my script for quantum mechanics it says that this tensor representation can be decomposed into a direct sum of irreducible representations in the following way:

$D_j_1 + j_2 oplus D_j_1 + j_2 -1 oplus ... oplus D_$

One can easily check that if a representation $D_j$ has dimension $2j+1$ then the dimension of the direct sum is the same as the dimension of the tensorrepresentation, which is basically the whole proof given in the script.

For my good conscience I'd like to know a little bit more about this:

1.) What theorem tells me that the tensor representation can be decomposed into a direct sum of irreps?

2.) Which theorem tells me that the addends of the direct sum are the already found irreducible representations? Why can't it be a completely different function?

3.) Why does the sum go from $j_1 + j_2$ to $|j_1 - j_2|$? Intuitively this makes sense if one looks at the 3 component of the angular momentum operator since this can be $j_1, j_2$ maximally, but I don't understand the mathematical argumentation behind it.

I hope my questions are clear and that someone can point me in the right direction.

Cheers!

group-theory representation-theory lie-groups lie-algebras mathematical-physics

share|cite|improve this question

edited Mar 24 at 15:18

Abdelmalek Abdesselam

854312

asked Jun 30 '14 at 17:36

user80944

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  1) This is basic Lie theory, but the answer is a chapter or two in a book. 2) You are right that with other groups you actually get different reps in there (me thinks). It is just that $SO(3)$ (or $SL_2$) is a special case. This is another non-trivial result but not as long as the answer to 1. 3) The dimensions were used up, so no further summands fit in there. The "true" explanation comes from rep theory (for my money), but it is hardly a coincidence that it matches with our intuition about addition of angular momenta.
  $endgroup$
  – Jyrki Lahtonen
  Jun 30 '14 at 20:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But it really looks like your question may be a bit broad. I am not voting to close yet, as somebody may want to try to give you an overview. The textbook I would recommend (Humphreys) is from Lie algebra side, and Lie group people may be able to recommend something that gives you a decent understanding of what's going on without needing to wade through quite a bit of advanced linear algebra and proving those basics.
  $endgroup$
  – Jyrki Lahtonen
  Jun 30 '14 at 20:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hey thanks a lot for the indepth answer. I guess I'll just accept it for now and will return to the proper understanding once I'm done with the exam.
  $endgroup$
  – user80944
  Jul 1 '14 at 15:55
  

  
  
  
  

add a comment | 

$begingroup$

physicist here.

When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be have values like $j = 0, 1/2, 1, 3/2, ...$. Let $D_j$ denote the representation of $so(3)$ with index j.

If one has two Hilbert spaces $H_1, H_2$ which both carry a representation $D_j_1, D_j_2$, the tensorspace of $H_1$ and $H_2$ carries a representation $D_j_1 otimes D_j_2$.

In my script for quantum mechanics it says that this tensor representation can be decomposed into a direct sum of irreducible representations in the following way:

$D_j_1 + j_2 oplus D_j_1 + j_2 -1 oplus ... oplus D_$

One can easily check that if a representation $D_j$ has dimension $2j+1$ then the dimension of the direct sum is the same as the dimension of the tensorrepresentation, which is basically the whole proof given in the script.

For my good conscience I'd like to know a little bit more about this:

1.) What theorem tells me that the tensor representation can be decomposed into a direct sum of irreps?

2.) Which theorem tells me that the addends of the direct sum are the already found irreducible representations? Why can't it be a completely different function?

3.) Why does the sum go from $j_1 + j_2$ to $|j_1 - j_2|$? Intuitively this makes sense if one looks at the 3 component of the angular momentum operator since this can be $j_1, j_2$ maximally, but I don't understand the mathematical argumentation behind it.

I hope my questions are clear and that someone can point me in the right direction.

Cheers!

group-theory representation-theory lie-groups lie-algebras mathematical-physics

share|cite|improve this question

edited Mar 24 at 15:18

Abdelmalek Abdesselam

854312

asked Jun 30 '14 at 17:36

user80944

$endgroup$

share|cite|improve this question

edited Mar 24 at 15:18

Abdelmalek Abdesselam

854312

asked Jun 30 '14 at 17:36

user80944

share|cite|improve this question

edited Mar 24 at 15:18

Abdelmalek Abdesselam

854312

asked Jun 30 '14 at 17:36

user80944

edited Mar 24 at 15:18

Abdelmalek Abdesselam

854312

edited Mar 24 at 15:18

Abdelmalek Abdesselam

854312