by replacing theiby an arbitrary complex numberτ. Then the condition that
q−τpbe inM+J and its conjugate inM−J is
J(q−τp) =J(q)−τJ(p) =i(q−τp)J(q−τp) =J(q)−τJ(p) =−i(q−τp)Subtracting and adding the two equations gives
J(p) =−1
Im(τ)q+
Re(τ)
Im(τ)pand
J(q) =−
Re(τ)
Im(τ)q+(
Im(τ) +
(Re(τ))^2
Im(τ))
prespectively. Generalizing 26.4, the matrix forJis
J=
(
−Re(Im(ττ)) Im(τ)(Re(τ))2
Im(τ)
−Im(^1 τ) Re(Im(ττ)))
=
1
Im(τ)(
−Re(τ) |τ|^2
− 1 Re(τ))
(26.17)
and it can easily be checked that detJ= 1, soJ∈SL(2,R) and is compatible
with Ω.
The positivity condition here is that Ω(·,J·) is positive onM, which in terms
of matrices (see 16.2) becomes the condition that the matrix
(
0 1
−1 0)
JT=
1
Im(τ)(
|τ|^2 Re(τ)
Re(τ) 1)
gives a positive quadratic form. This will be the case when Im(τ)>0. We
have thus constructed a set ofJthat are positive, compatible with Ω, and
parametrized by an elementτof the upper half-plane, withJ 0 corresponding to
τ=i.
To construct annihilation and creation operators satisfying the standard
commutation relations
[aτ,aτ] = [a†τ,a†τ] = 0, [aτ,a†τ] = 1set
aτ=1
√
2 Im(τ)(Q−τP), a†τ=1
√
2 Im(τ)(Q−τP)The Hamiltonian
Hτ=1
2
(aτa†τ+a†τaτ) =1
2 Im(τ)(Q^2 +|τ|^2 P^2 −Re(τ)(QP+PQ)) (26.18)will have eigenvaluesn+^12 forn= 0, 1 , 2 ,···. Its lowest energy state will satisfy
aτ| 0 〉τ= 0 (26.19)