"Heisenberg group and Heisenberg algebra"의 두 판 사이의 차이
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| − | <h5 style="line-height: 2em; margin | + | <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> |
* 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> | ||
| 50번째 줄: | 50번째 줄: | ||
* define a Lie bracket<br><math>[d,x]=d(x)</math><br> | * define a Lie bracket<br><math>[d,x]=d(x)</math><br> | ||
* In [[affine Kac-Moody algebra]] theory, this appears as the loop algebra of Cartan subalgebra<br> | * In [[affine Kac-Moody algebra]] theory, this appears as the loop algebra of Cartan subalgebra<br> | ||
| + | * commutator subalgebra<br> | ||
| 61번째 줄: | 62번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <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> |
* 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 | + | <h5 style="line-height: 2em; margin: 0px;">Heisenberg VOA</h5> |
* [[vertex algebras|Vertex Algebras and CFT]]<br> | * [[vertex algebras|Vertex Algebras and CFT]]<br> | ||
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<h5 style="margin: 0px; line-height: 2em;">related items</h5> | <h5 style="margin: 0px; line-height: 2em;">related items</h5> | ||
2010년 8월 21일 (토) 11:02 판
relation to quantum mechanics
- the position operators and momentum operators satisfy the relation
\([X,P] = X P - P X = i \hbar\)
question
- Weyl algebra
\(x\), \(p=\frac{d}{dx}\)
\([x,p]=1\)
- what's the relation with the noncommutative algebra
\(xy=qyx\) - q-series
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 \(<\cdot,\cdot>\)
- 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
\([d,x]=d(x)\) - 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
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
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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