Heisenberg group and Heisenberg algebra
imported>Pythagoras0님의 2012년 10월 28일 (일) 14:28 판 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로)
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=