Quantum Mechanics for Mathematicians

what happens to this state under the Heisenberg group action.
Elements of the complexified Heisenberg Lie algebrah 3 ⊗Ccan be written
as
iαz+βz+γ

forα,β,γinC(this choice ofαsimplifies later formulas). The Lie algebrah 3
is the subspace of real functions, which will be those of the form

iαz−iαz+γ

forα∈Candγ∈R. The Lie algebra structure is given by the Poisson bracket

{iα 1 z−iα 1 z+γ 1 ,iα 2 z−iα 2 z+γ 2 }= 2Im(α 1 α 2 )

Hereh 3 is identified withC⊕R, and elements can be written as pairs (α,γ),
with the Lie bracket

[(α 1 ,γ 1 ),(α 2 ,γ 2 )] = (0,2Im(α 1 α 2 ))

This is just a variation on the labeling ofh 3 elements discussed in chapter 13, and
one can again use exponential coordinates and write elements of the Heisenberg
groupH 3 also as such pairs, with group law

(α 1 ,γ 1 )(α 2 ,γ 2 ) = (α 1 +α 2 ,γ 1 +γ 2 + Im(α 1 α 2 ))

Quantizing using equation 22.5, one has a Lie algebra representation Γ′, with
operators for elements ofh 3

Γ′(α,γ) = Γ′(iαz−iαz+γ) =αa†−αa−iγ 1 (23.1)

and exponentiating these will give the unitary representation

Γ(α,γ) =eαa

†−αa−iγ

We define operators
D(α) =eαa

†−αa

which satisfy (using Baker-Campbell-Hausdorff)

D(α 1 )D(α 2 ) =D(α 1 +α 2 )e−iIm(α^1 α^2 )

Then
Γ(α,γ) =D(α)e−iγ

and the operators Γ(α,γ) give a representation, since they satisfy

Γ(α 1 ,γ 1 )Γ(α 2 ,γ 2 ) =D(α 1 +α 2 )e−i(γ^1 +γ^2 +Im(α^1 α^2 ))= Γ((α 1 ,γ 1 )(α 2 ,γ 2 ))

Acting on| 0 〉withD(α) gives: