"Heisenberg group and Heisenberg algebra"의 두 판 사이의 차이
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imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | + | ==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> | ||
| 15번째 줄: | 15번째 줄: | ||
| − | + | ==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: | ||
| − | + | ==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: | ||
| − | + | ==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: | ||
| − | + | ==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> | ||
| 61번째 줄: | 61번째 줄: | ||
| − | + | ==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). | ||
| 69번째 줄: | 69번째 줄: | ||
| − | + | ==Heisenberg VOA== | |
* [[VOA associated to Heisenberg algebra]]<br> | * [[VOA associated to Heisenberg algebra]]<br> | ||
| 77번째 줄: | 77번째 줄: | ||
| − | + | ==related items== | |
* [[half-integral weight modular forms|half-integral modular forms]]<br> | * [[half-integral weight modular forms|half-integral modular forms]]<br> | ||
| 88번째 줄: | 88번째 줄: | ||
| − | + | ==books== | |
* [[2009년 books and articles|찾아볼 수학책]] | * [[2009년 books and articles|찾아볼 수학책]] | ||
| 101번째 줄: | 101번째 줄: | ||
| − | + | ==encyclopedia== | |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 113번째 줄: | 113번째 줄: | ||
| − | + | ==blogs== | |
* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q= | * 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q= | ||
| 120번째 줄: | 120번째 줄: | ||
| − | + | ==articles== | |
* [[2010년 books and articles|논문정리]] | * [[2010년 books and articles|논문정리]] | ||
| 134번째 줄: | 134번째 줄: | ||
| − | + | ==TeX == | |
2012년 10월 28일 (일) 17:27 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
- half-integral modular forms
- Kac-Moody algebras
- central extension of semisimple lie algebra
- Weyl algebra
books
- 찾아볼 수학책
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Heisenberg_algebra
- http://en.wikipedia.org/wiki/Weyl_algebra
- http://en.wikipedia.org/wiki/Stone–von_Neumann_theorem
- http://en.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(첨부파일로 올릴것)
blogs
- 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
- 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
articles
- 논문정리
- The Cover of June/July 2003 Volume 50 Issue 6
- Notices of AMS
- http://www.zentralblatt-math.org/zmath/en/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
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