Formal proof of Clebsch Gordan sum The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Conjugate Representations for $mathfraksl(2,mathbbC)$Decomposition into irreducibles of representations of semisimple Lie groups.What are the irreducible representations of a direct sum of Lie Algebras?Tensor products and decomposition of $SU(3)$ representationsAbout irreducible representations of C* algebraCompletly reducible representations : various questions (link with clebsch-gordan)Representations irreducible with respect to the tensor productDefinition of “direct sum of irreducible representations”Decomposition of the representation $chi_m(t)=e^imt$?direct sum of representation of product groups
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Formal proof of Clebsch Gordan sum
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Conjugate Representations for $mathfraksl(2,mathbbC)$Decomposition into irreducibles of representations of semisimple Lie groups.What are the irreducible representations of a direct sum of Lie Algebras?Tensor products and decomposition of $SU(3)$ representationsAbout irreducible representations of C* algebraCompletly reducible representations : various questions (link with clebsch-gordan)Representations irreducible with respect to the tensor productDefinition of “direct sum of irreducible representations”Decomposition of the representation $chi_m(t)=e^imt$?direct sum of representation of product groups
$begingroup$
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
edited Mar 24 at 15:18
Abdelmalek Abdesselam
854312
$endgroup$
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
edited Mar 24 at 15:18
Abdelmalek Abdesselam
854312
$endgroup$
$begingroup$
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
edited Mar 24 at 15:18
Abdelmalek Abdesselam
854312
$endgroup$
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
group-theory representation-theory lie-groups lie-algebras mathematical-physics
edited Mar 24 at 15:18
Abdelmalek Abdesselam
854312
edited Mar 24 at 15:18
Abdelmalek Abdesselam
854312
edited Mar 24 at 15:18
Abdelmalek Abdesselam
854312
edited Mar 24 at 15:18
Abdelmalek Abdesselam
854312
edited Mar 24 at 15:18
Abdelmalek Abdesselam
854312
854312
asked Jun 30 '14 at 17:36
user80944
$begingroup$
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 |
$begingroup$
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
$begingroup$
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
$begingroup$
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
$begingroup$
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
$begingroup$
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 |
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$begingroup$
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