"Heisenberg group and Heisenberg algebra"의 두 판 사이의 차이
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relation to quantum mechanics==
imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| − | <h5 style="line-height: 2em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">relation to quantum mechanics | + | <h5 style="line-height: 2em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">relation to quantum mechanics== |
* the position operators and momentum operators satisfy the relation<br><math>[X,P] = X P - P X = i \hbar</math><br> | * the position operators and momentum operators satisfy the relation<br><math>[X,P] = X P - P X = i \hbar</math><br> | ||
| 7번째 줄: | 7번째 줄: | ||
| − | ==relation to Weyl algebra | + | ==relation to Weyl algebra== |
* a quotient of the universal enveloping algebra of the Heisenberg algebra | * a quotient of the universal enveloping algebra of the Heisenberg algebra | ||
| 15번째 줄: | 15번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">finite dimensional Heisenberg algebra | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">finite dimensional Heisenberg algebra== |
* one dimensional [[central extension of groups and Lie algebras|central extension]] of abelian Lie algebra<br> | * one dimensional [[central extension of groups and Lie algebras|central extension]] of abelian Lie algebra<br> | ||
| 28번째 줄: | 28번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">differential operators | + | <h5 style="margin: 0px; line-height: 2em;">differential operators== |
* commutation relation<br><math>x</math>, <math>p=\frac{d}{dx}</math><br><math>[x,p]=1</math><br> | * commutation relation<br><math>x</math>, <math>p=\frac{d}{dx}</math><br><math>[x,p]=1</math><br> | ||
| 36번째 줄: | 36번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">infinite dimensional Heisenberg algebra | + | <h5 style="margin: 0px; line-height: 2em;">infinite dimensional Heisenberg algebra== |
* start with a Lattice <math>\langle\cdot,\cdot\rangle</math><br> | * start with a Lattice <math>\langle\cdot,\cdot\rangle</math><br> | ||
| 52번째 줄: | 52번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">highest weight module | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">highest weight module== |
* <math>\hat{A}^{+}=A\otimes\mathbb{C}[t]\oplus\mathbb{C}c</math><br> | * <math>\hat{A}^{+}=A\otimes\mathbb{C}[t]\oplus\mathbb{C}c</math><br> | ||
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| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Stone-Von Neumann theorem | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Stone-Von Neumann theorem== |
* The [[Heisenberg group and Heisenberg algebra|Heisenberg group]] has an essentially unique irreducible unitary representation on a Hilbert space H with the center acting as a given nonzero constant (the content of the Stone-von Neumann theorem). | * The [[Heisenberg group and Heisenberg algebra|Heisenberg group]] has an essentially unique irreducible unitary representation on a Hilbert space H with the center acting as a given nonzero constant (the content of the Stone-von Neumann theorem). | ||
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| − | <h5 style="line-height: 2em; margin: 0px;">Heisenberg VOA | + | <h5 style="line-height: 2em; margin: 0px;">Heisenberg VOA== |
* [[VOA associated to Heisenberg algebra]]<br> | * [[VOA associated to Heisenberg algebra]]<br> | ||
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| − | <h5 style="margin: 0px; line-height: 2em;">related items | + | <h5 style="margin: 0px; line-height: 2em;">related items== |
* [[half-integral weight modular forms|half-integral modular forms]]<br> | * [[half-integral weight modular forms|half-integral modular forms]]<br> | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">books | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">books== |
* [[2009년 books and articles|찾아볼 수학책]] | * [[2009년 books and articles|찾아볼 수학책]] | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia== |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">blogs | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">blogs== |
* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q= | * 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q= | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles== |
* [[2010년 books and articles|논문정리]] | * [[2010년 books and articles|논문정리]] | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">TeX | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">TeX == |
2012년 10월 28일 (일) 15:28 판
relation to quantum mechanics==
- the position operators and momentum operators satisfy the relation
\([X,P] = X P - P X = i \hbar\)
relation to Weyl algebra
- a quotient of the universal enveloping algebra of the Heisenberg algebra
finite dimensional Heisenberg algebra==
- one dimensional central extension of abelian Lie algebra
- \([p_i, q_j] = \delta_{ij}z\)
- \([p_i, z] = 0\)
- \([q_j, z] = 0\)
- Gannon 180p
differential operators==
- commutation relation
\(x\), \(p=\frac{d}{dx}\)
\([x,p]=1\)
infinite dimensional Heisenberg algebra==
- start with a Lattice \(\langle\cdot,\cdot\rangle\)
- make a vector space from it
- Construct a Loop algbera
\(\hat{A}=A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\)
\(\alpha(m)=\alpha\otimes t^m\)
- Give a bracket
\([\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c\)
- add a derivation \(d\)
\(d(\alpha(n))=n\alpha(n)\)
\(d(c)=0\)
- define a Lie bracket
\([d,x]=d(x)\)
- In affine Kac-Moody algebra theory, this appears as the loop algebra of Cartan subalgebra
- commutator subalgebra
- The automorphisms of the Heisenberg group (fixing its center) form the symplectic group
highest weight module==
- \(\hat{A}^{+}=A\otimes\mathbb{C}[t]\oplus\mathbb{C}c\)
- \(c.v_{h}=v_{h}\)
- for \(m>0\), \(\alpha(m)v_{h}=0\)
- \(\alpha(0)v_{h}=hv_{h}\)
Stone-Von Neumann theorem==
- The Heisenberg group has an essentially unique irreducible unitary representation on a Hilbert space H with the center acting as a given nonzero constant (the content of the Stone-von Neumann theorem).
Heisenberg VOA==
related items==
books==
encyclopedia==
blogs==
- 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
- 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
articles==
TeX ==
\([X,P] = X P - P X = i \hbar\)
- one dimensional central extension of abelian Lie algebra
- \([p_i, q_j] = \delta_{ij}z\)
- \([p_i, z] = 0\)
- \([q_j, z] = 0\)
- Gannon 180p
- commutation relation
\(x\), \(p=\frac{d}{dx}\)
\([x,p]=1\)
- start with a Lattice \(\langle\cdot,\cdot\rangle\)
- make a vector space from it
- Construct a Loop algbera
\(\hat{A}=A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\)
\(\alpha(m)=\alpha\otimes t^m\) - Give a bracket
\([\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c\) - add a derivation \(d\)
\(d(\alpha(n))=n\alpha(n)\)
\(d(c)=0\) - define a Lie bracket
\([d,x]=d(x)\) - In affine Kac-Moody algebra theory, this appears as the loop algebra of Cartan subalgebra
- commutator subalgebra
- The automorphisms of the Heisenberg group (fixing its center) form the symplectic group
- \(\hat{A}^{+}=A\otimes\mathbb{C}[t]\oplus\mathbb{C}c\)
- \(c.v_{h}=v_{h}\)
- for \(m>0\), \(\alpha(m)v_{h}=0\)
- \(\alpha(0)v_{h}=hv_{h}\)
- The Heisenberg group has an essentially unique irreducible unitary representation on a Hilbert space H with the center acting as a given nonzero constant (the content of the Stone-von Neumann theorem).
- 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
- 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=