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

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* <math>[a_i, a_j^{\dagger}] = \delta_{ij}</math><br>
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* <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>

2009년 10월 5일 (월) 13:43 판

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).

 

introduction
  • \([p_i, q_j] = \delta_{ij}z\)
  • \([p_i, z] = 0\)
  • \([q_j, z] = 0\)

 

 

relation to quantum mechanics
  •  the position operators and momentum operators satisfy the above relation

 

 

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
    \([d,x]=d(x)\)

 

fock space representation

 

 

Representation theory

 

related items

 

books

 

 

encyclopedia

 

blogs

 

articles

 

TeX 

 

 

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