"Heisenberg group and Heisenberg algebra"의 두 판 사이의 차이
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| − | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">relation to quantum mechanics</h5> | |
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| + | * the position operators and momentum operators satisfy the relation<br><math>[X,P] = X P - P X = i \hbar</math><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%;">finite dimensional Heisenberg algebra</h5> | |
| − | < | + | * one dimensional [[central extension of groups and Lie algebras|central extension]] of abelian Lie algebra<br> |
* <math>[p_i, q_j] = \delta_{ij}z</math><br> | * <math>[p_i, q_j] = \delta_{ij}z</math><br> | ||
* <math>[p_i, z] = 0</math><br> | * <math>[p_i, z] = 0</math><br> | ||
* <math>[q_j, z] = 0</math><br> | * <math>[q_j, z] = 0</math><br> | ||
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* Gannon 180p<br> | * Gannon 180p<br> | ||
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* add a derivation <math>d</math><br><math>d(\alpha(n))=n\alpha(n)</math><br><math>d(c)=0</math><br> | * add a derivation <math>d</math><br><math>d(\alpha(n))=n\alpha(n)</math><br><math>d(c)=0</math><br> | ||
* define a Lie bracket<br><math>[d,x]=d(x)</math><br> | * define a Lie bracket<br><math>[d,x]=d(x)</math><br> | ||
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| + | The automorphisms of the Heisenberg group (fixing its center) form the symplectic group | ||
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| + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">Stone-Von Neumann theorem</h5> | ||
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| + | * 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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2010년 3월 26일 (금) 08:56 판
relation to quantum mechanics
- the position operators and momentum operators satisfy the relation
\([X,P] = X P - P X = i \hbar\)
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
infinite dimensional Heisenberg algebra
- start with a Lattice \(<\cdot,\cdot>\)
- make a vector space from it
- Construct a Loop algbera
\(A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\)
\(\alpha(m)=\alpha\otimes t^m\) - Give a bracket
\([\alpha(m),\beta(n)]=m\delta_{m,-n}<\alpha,\beta>c\) - add a derivation \(d\)
\(d(\alpha(n))=n\alpha(n)\)
\(d(c)=0\) - define a Lie bracket
\([d,x]=d(x)\)
The automorphisms of the Heisenberg group (fixing its center) form the symplectic group
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).
fock space representation
Representation theory
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=
TeX