"Heisenberg group and Heisenberg algebra"의 두 판 사이의 차이

2012년 6월 17일 (일) 09:52 판

the position operators and momentum operators satisfy the relation
\([X,P] = X P - P X = i \hbar\)

start with a Lattice \(\langle\cdot,\cdot\rangle\)
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
\([\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c\)
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

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).

@@ 38번째 줄: / 38번째 줄: @@
 <h5 style="margin: 0px; line-height: 2em;">infinite dimensional Heisenberg algebra</h5>
-*  start with a Lattice <math><\cdot,\cdot></math><br>
+*  start with a Lattice <math>\langle\cdot,\cdot\rangle</math><br>
 *  make a vector space from it<br>
 *  Construct a Loop algbera<br><math>A\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c</math><br><math>\alpha(m)=\alpha\otimes t^m</math><br>
-*  Give a bracket <br><math>[\alpha(m),\beta(n)]=m\delta_{m,-n}<\alpha,\beta>c</math><br>
+*  Give a bracket <br><math>[\alpha(m),\beta(n)]=m\delta_{m,-n}\langle\alpha,\beta\rangle c</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>