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

2010년 3월 26일 (금) 09:28 판

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

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

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).

@@ 17번째 줄: / 17번째 줄: @@
+<h5 style="margin: 0px; line-height: 2em;"> </h5>
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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>
 *  define a Lie bracket<br><math>[d,x]=d(x)</math><br>
-*  In [[affine Kac-Moody algebra]] theory, this appears as the loop <br>
+*  In [[affine Kac-Moody algebra]] theory, this appears as the loop algebra of Cartan subalgebra<br>