Quantum Mechanics for Mathematicians

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)

p

and

J(q) =−

Re(τ)

Im(τ)

q+

(

Im(τ) +

(Re(τ))^2

Im(τ)

)

p

respectively. 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] = 1

set

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)