Relation between Poisson bracket and commutator.
$begingroup$
In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b in mathbb{C}_q[T].$ It seems that we have
$$
[a, b]=(q-1){a,b}+o((q-1)^2).
$$
Is this true? When $qto 1$, we have $[a,b]/(q-1)to {a,b}$.
representation-theory classical-mechanics quantum-mechanics quantum-groups poisson-geometry
asked Jan 26 '15 at 5:19
LJRLJR
6,65141850
$endgroup$
add a comment |
$begingroup$
In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b in mathbb{C}_q[T].$ It seems that we have
$$
[a, b]=(q-1){a,b}+o((q-1)^2).
$$
Is this true? When $qto 1$, we have $[a,b]/(q-1)to {a,b}$.
representation-theory classical-mechanics quantum-mechanics quantum-groups poisson-geometry
asked Jan 26 '15 at 5:19
LJRLJR
6,65141850
$endgroup$
3
$begingroup$
What is $mathbb{C}_q[T]$?
$endgroup$
– Qiaochu Yuan
Jan 26 '15 at 5:56
add a comment |
$begingroup$
In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b in mathbb{C}_q[T].$ It seems that we have
$$
[a, b]=(q-1){a,b}+o((q-1)^2).
$$
Is this true? When $qto 1$, we have $[a,b]/(q-1)to {a,b}$.
representation-theory classical-mechanics quantum-mechanics quantum-groups poisson-geometry
asked Jan 26 '15 at 5:19
LJRLJR
6,65141850
$endgroup$
In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b in mathbb{C}_q[T].$ It seems that we have
$$
[a, b]=(q-1){a,b}+o((q-1)^2).
$$
Is this true? When $qto 1$, we have $[a,b]/(q-1)to {a,b}$.
representation-theory classical-mechanics quantum-mechanics quantum-groups poisson-geometry
representation-theory classical-mechanics quantum-mechanics quantum-groups poisson-geometry
asked Jan 26 '15 at 5:19
LJRLJR
6,65141850
asked Jan 26 '15 at 5:19
LJRLJR
6,65141850
edited Jan 27 '15 at 4:50
LJR
asked Jan 26 '15 at 5:19
LJRLJR
6,65141850
asked Jan 26 '15 at 5:19
LJRLJR
6,65141850
asked Jan 26 '15 at 5:19
LJRLJR
6,65141850
6,65141850
3
$begingroup$
What is $mathbb{C}_q[T]$?
$endgroup$
– Qiaochu Yuan
Jan 26 '15 at 5:56
add a comment |
3
$begingroup$
What is $mathbb{C}_q[T]$?
$endgroup$
– Qiaochu Yuan
Jan 26 '15 at 5:56
3
3
$begingroup$
What is $mathbb{C}_q[T]$?
$endgroup$
– Qiaochu Yuan
Jan 26 '15 at 5:56
$begingroup$
What is $mathbb{C}_q[T]$?
$endgroup$
– Qiaochu Yuan
Jan 26 '15 at 5:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your notation entails undefined terms and peculiar notation. I will, instead, use the conventional ones.
Using the invertible Wigner map from linear operators Φ in Hilbert space to functions in phase space f(x,p), one has
$$
f(x,p)= 2 int_{-infty}^infty text{d}y~e^{-2ipy/hbar}~ langle x+y| Phi [f] |x-y rangle.
$$
In phase space, one may now compare apples with apples, and indeed, the commutator of two operator maps to the bilinear of the corresponding two functions f and g,
$$
f star g - g star f = f , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , g - g , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , f \
= 2i~~ f sin left ( {frac{hbar }{2} (overset{leftarrow}{partial_x}
overset{rightarrow}{partial_p}-overset{leftarrow}{partial_p}overset{rightarrow}{partial_x})} right )
g =ihbar {f,g} + O(hbar^3),
$$
with the classical $hbarto 0$ limit you are evoking.
answered Dec 31 '18 at 21:06
Cosmas ZachosCosmas Zachos
1,810522
$endgroup$
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%2f1119968%2frelation-between-poisson-bracket-and-commutator%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
$begingroup$
Your notation entails undefined terms and peculiar notation. I will, instead, use the conventional ones.
Using the invertible Wigner map from linear operators Φ in Hilbert space to functions in phase space f(x,p), one has
$$
f(x,p)= 2 int_{-infty}^infty text{d}y~e^{-2ipy/hbar}~ langle x+y| Phi [f] |x-y rangle.
$$
In phase space, one may now compare apples with apples, and indeed, the commutator of two operator maps to the bilinear of the corresponding two functions f and g,
$$
f star g - g star f = f , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , g - g , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , f \
= 2i~~ f sin left ( {frac{hbar }{2} (overset{leftarrow}{partial_x}
overset{rightarrow}{partial_p}-overset{leftarrow}{partial_p}overset{rightarrow}{partial_x})} right )
g =ihbar {f,g} + O(hbar^3),
$$
with the classical $hbarto 0$ limit you are evoking.
answered Dec 31 '18 at 21:06
Cosmas ZachosCosmas Zachos
1,810522
$endgroup$
add a comment |
$begingroup$
Your notation entails undefined terms and peculiar notation. I will, instead, use the conventional ones.
Using the invertible Wigner map from linear operators Φ in Hilbert space to functions in phase space f(x,p), one has
$$
f(x,p)= 2 int_{-infty}^infty text{d}y~e^{-2ipy/hbar}~ langle x+y| Phi [f] |x-y rangle.
$$
In phase space, one may now compare apples with apples, and indeed, the commutator of two operator maps to the bilinear of the corresponding two functions f and g,
$$
f star g - g star f = f , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , g - g , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , f \
= 2i~~ f sin left ( {frac{hbar }{2} (overset{leftarrow}{partial_x}
overset{rightarrow}{partial_p}-overset{leftarrow}{partial_p}overset{rightarrow}{partial_x})} right )
g =ihbar {f,g} + O(hbar^3),
$$
with the classical $hbarto 0$ limit you are evoking.
answered Dec 31 '18 at 21:06
Cosmas ZachosCosmas Zachos
1,810522
$endgroup$
add a comment |
$begingroup$
Your notation entails undefined terms and peculiar notation. I will, instead, use the conventional ones.
Using the invertible Wigner map from linear operators Φ in Hilbert space to functions in phase space f(x,p), one has
$$
f(x,p)= 2 int_{-infty}^infty text{d}y~e^{-2ipy/hbar}~ langle x+y| Phi [f] |x-y rangle.
$$
In phase space, one may now compare apples with apples, and indeed, the commutator of two operator maps to the bilinear of the corresponding two functions f and g,
$$
f star g - g star f = f , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , g - g , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , f \
= 2i~~ f sin left ( {frac{hbar }{2} (overset{leftarrow}{partial_x}
overset{rightarrow}{partial_p}-overset{leftarrow}{partial_p}overset{rightarrow}{partial_x})} right )
g =ihbar {f,g} + O(hbar^3),
$$
with the classical $hbarto 0$ limit you are evoking.
answered Dec 31 '18 at 21:06
Cosmas ZachosCosmas Zachos
1,810522
$endgroup$
Your notation entails undefined terms and peculiar notation. I will, instead, use the conventional ones.
Using the invertible Wigner map from linear operators Φ in Hilbert space to functions in phase space f(x,p), one has
$$
f(x,p)= 2 int_{-infty}^infty text{d}y~e^{-2ipy/hbar}~ langle x+y| Phi [f] |x-y rangle.
$$
In phase space, one may now compare apples with apples, and indeed, the commutator of two operator maps to the bilinear of the corresponding two functions f and g,
$$
f star g - g star f = f , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , g - g , exp{left( frac{i hbar}{2} left(overleftarrow {partial }_x
overrightarrow{partial }_p-overleftarrow{partial }_p overrightarrow {partial }_x right) right)} , f \
= 2i~~ f sin left ( {frac{hbar }{2} (overset{leftarrow}{partial_x}
overset{rightarrow}{partial_p}-overset{leftarrow}{partial_p}overset{rightarrow}{partial_x})} right )
g =ihbar {f,g} + O(hbar^3),
$$
with the classical $hbarto 0$ limit you are evoking.
answered Dec 31 '18 at 21:06
Cosmas ZachosCosmas Zachos
1,810522
answered Dec 31 '18 at 21:06
Cosmas ZachosCosmas Zachos
1,810522
answered Dec 31 '18 at 21:06
Cosmas ZachosCosmas Zachos
1,810522
answered Dec 31 '18 at 21:06
Cosmas ZachosCosmas Zachos
1,810522
1,810522
add a comment |
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%2f1119968%2frelation-between-poisson-bracket-and-commutator%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
3
$begingroup$
What is $mathbb{C}_q[T]$?
$endgroup$
– Qiaochu Yuan
Jan 26 '15 at 5:56