Direct Product vs Tensor Product
$begingroup$
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
$endgroup$
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
1
active
oldest
votes
$begingroup$
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
79.1k647103
$endgroup$
$begingroup$
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 |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039680%2fdirect-product-vs-tensor-product%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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
79.1k647103
$endgroup$
$begingroup$
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
79.1k647103
$endgroup$
$begingroup$
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
79.1k647103
$endgroup$
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
79.1k647103
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
79.1k647103
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
79.1k647103
answered Dec 14 '18 at 19:15
Dietrich BurdeDietrich Burde
79.1k647103
79.1k647103
$begingroup$
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 |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039680%2fdirect-product-vs-tensor-product%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
