Direct Product vs Tensor Product
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I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product:
On page 67, it mentioned that "you can take a direct product of two j = 1/2 representations" and build representations of higher j.
On page 70, it mentioned "we can think of [the Lorentz Group] as the direct product SU(2) × SU(2)"
In each of the above, does the author mean Direct Product or Tensor Product?
group-theory
asked Dec 14 '18 at 17:30
The NotoriousThe Notorious
1033
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add a comment |
$begingroup$
I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product:
On page 67, it mentioned that "you can take a direct product of two j = 1/2 representations" and build representations of higher j.
On page 70, it mentioned "we can think of [the Lorentz Group] as the direct product SU(2) × SU(2)"
In each of the above, does the author mean Direct Product or Tensor Product?
group-theory
asked Dec 14 '18 at 17:30
The NotoriousThe Notorious
1033
$endgroup$
add a comment |
$begingroup$
I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product:
On page 67, it mentioned that "you can take a direct product of two j = 1/2 representations" and build representations of higher j.
On page 70, it mentioned "we can think of [the Lorentz Group] as the direct product SU(2) × SU(2)"
In each of the above, does the author mean Direct Product or Tensor Product?
group-theory
asked Dec 14 '18 at 17:30
The NotoriousThe Notorious
1033
$endgroup$
I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product:
On page 67, it mentioned that "you can take a direct product of two j = 1/2 representations" and build representations of higher j.
On page 70, it mentioned "we can think of [the Lorentz Group] as the direct product SU(2) × SU(2)"
In each of the above, does the author mean Direct Product or Tensor Product?
group-theory
group-theory
asked Dec 14 '18 at 17:30
The NotoriousThe Notorious
1033
asked Dec 14 '18 at 17:30
The NotoriousThe Notorious
1033
asked Dec 14 '18 at 17:30
The NotoriousThe Notorious
1033
asked Dec 14 '18 at 17:30
The NotoriousThe Notorious
1033
asked Dec 14 '18 at 17:30
The NotoriousThe Notorious
1033
1033
add a comment |
add a comment |
1 Answer
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On page $70$ the author speaks about the Lie algebras, not of the groups. So he means the direct sum $mathfrak{su}(2)oplus mathfrak{su}(2)$ of Lie algebras. In fact, $(8.14),(8.15),(8.16)$ are Lie brackets. He calls this "$SU(2)$ algebras". On the group level, $SU(2)times SU(2)$ denotes the direct product in the usual sense. For the "complexities product" he describes, see here:
Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)times SU(2)$, or their Lie algebras
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
78.2k64386
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$begingroup$
So in page 70 it's Direct Product; what about page 67? is it tensor or direct product?
$endgroup$
– The Notorious
Dec 15 '18 at 6:08
1
$begingroup$
See wikipedia, in particular the section " the $SU(2)$ case".
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– Dietrich Burde
Dec 15 '18 at 9:12
$begingroup$
Ah I see.. so the p67 is Tensor Product? (just to double check)
$endgroup$
– The Notorious
Dec 15 '18 at 9:16
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Yes, $rhootimes sigma$ denotes tensor product.
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:17
add a comment |
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1 Answer
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$begingroup$
On page $70$ the author speaks about the Lie algebras, not of the groups. So he means the direct sum $mathfrak{su}(2)oplus mathfrak{su}(2)$ of Lie algebras. In fact, $(8.14),(8.15),(8.16)$ are Lie brackets. He calls this "$SU(2)$ algebras". On the group level, $SU(2)times SU(2)$ denotes the direct product in the usual sense. For the "complexities product" he describes, see here:
Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)times SU(2)$, or their Lie algebras
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
78.2k64386
$endgroup$
$begingroup$
So in page 70 it's Direct Product; what about page 67? is it tensor or direct product?
$endgroup$
– The Notorious
Dec 15 '18 at 6:08
1
$begingroup$
See wikipedia, in particular the section " the $SU(2)$ case".
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:12
$begingroup$
Ah I see.. so the p67 is Tensor Product? (just to double check)
$endgroup$
– The Notorious
Dec 15 '18 at 9:16
$begingroup$
Yes, $rhootimes sigma$ denotes tensor product.
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:17
add a comment |
$begingroup$
On page $70$ the author speaks about the Lie algebras, not of the groups. So he means the direct sum $mathfrak{su}(2)oplus mathfrak{su}(2)$ of Lie algebras. In fact, $(8.14),(8.15),(8.16)$ are Lie brackets. He calls this "$SU(2)$ algebras". On the group level, $SU(2)times SU(2)$ denotes the direct product in the usual sense. For the "complexities product" he describes, see here:
Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)times SU(2)$, or their Lie algebras
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
78.2k64386
$endgroup$
$begingroup$
So in page 70 it's Direct Product; what about page 67? is it tensor or direct product?
$endgroup$
– The Notorious
Dec 15 '18 at 6:08
1
$begingroup$
See wikipedia, in particular the section " the $SU(2)$ case".
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:12
$begingroup$
Ah I see.. so the p67 is Tensor Product? (just to double check)
$endgroup$
– The Notorious
Dec 15 '18 at 9:16
$begingroup$
Yes, $rhootimes sigma$ denotes tensor product.
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:17
add a comment |
$begingroup$
On page $70$ the author speaks about the Lie algebras, not of the groups. So he means the direct sum $mathfrak{su}(2)oplus mathfrak{su}(2)$ of Lie algebras. In fact, $(8.14),(8.15),(8.16)$ are Lie brackets. He calls this "$SU(2)$ algebras". On the group level, $SU(2)times SU(2)$ denotes the direct product in the usual sense. For the "complexities product" he describes, see here:
Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)times SU(2)$, or their Lie algebras
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
78.2k64386
$endgroup$
On page $70$ the author speaks about the Lie algebras, not of the groups. So he means the direct sum $mathfrak{su}(2)oplus mathfrak{su}(2)$ of Lie algebras. In fact, $(8.14),(8.15),(8.16)$ are Lie brackets. He calls this "$SU(2)$ algebras". On the group level, $SU(2)times SU(2)$ denotes the direct product in the usual sense. For the "complexities product" he describes, see here:
Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)times SU(2)$, or their Lie algebras
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
78.2k64386
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
78.2k64386
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
78.2k64386
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
78.2k64386
78.2k64386
$begingroup$
So in page 70 it's Direct Product; what about page 67? is it tensor or direct product?
$endgroup$
– The Notorious
Dec 15 '18 at 6:08
1
$begingroup$
See wikipedia, in particular the section " the $SU(2)$ case".
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:12
$begingroup$
Ah I see.. so the p67 is Tensor Product? (just to double check)
$endgroup$
– The Notorious
Dec 15 '18 at 9:16
$begingroup$
Yes, $rhootimes sigma$ denotes tensor product.
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:17
add a comment |
$begingroup$
So in page 70 it's Direct Product; what about page 67? is it tensor or direct product?
$endgroup$
– The Notorious
Dec 15 '18 at 6:08
1
$begingroup$
See wikipedia, in particular the section " the $SU(2)$ case".
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:12
$begingroup$
Ah I see.. so the p67 is Tensor Product? (just to double check)
$endgroup$
– The Notorious
Dec 15 '18 at 9:16
$begingroup$
Yes, $rhootimes sigma$ denotes tensor product.
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:17
$begingroup$
So in page 70 it's Direct Product; what about page 67? is it tensor or direct product?
$endgroup$
– The Notorious
Dec 15 '18 at 6:08
$begingroup$
So in page 70 it's Direct Product; what about page 67? is it tensor or direct product?
$endgroup$
– The Notorious
Dec 15 '18 at 6:08
1
1
$begingroup$
See wikipedia, in particular the section " the $SU(2)$ case".
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:12
$begingroup$
See wikipedia, in particular the section " the $SU(2)$ case".
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:12
$begingroup$
Ah I see.. so the p67 is Tensor Product? (just to double check)
$endgroup$
– The Notorious
Dec 15 '18 at 9:16
$begingroup$
Ah I see.. so the p67 is Tensor Product? (just to double check)
$endgroup$
– The Notorious
Dec 15 '18 at 9:16
$begingroup$
Yes, $rhootimes sigma$ denotes tensor product.
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:17
$begingroup$
Yes, $rhootimes sigma$ denotes tensor product.
$endgroup$
– Dietrich Burde
Dec 15 '18 at 9:17
add a comment |
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