Quantum Mechanics for Mathematicians

24.4SU(1,1) and Bogoliubov transformations

Changing bases in complexified phase space fromq,ptoz,zchanges the group of
linear transformations preserving the Poisson bracket from the groupSL(2,R)
of real 2 by 2 matrices of determinant one to an isomorphic group of complex 2
by 2 matrices. We have

Theorem.The groupSL(2,R)is isomorphic to the groupSU(1,1)of complex
2 by 2 matrices (
α β
β α

)

such that
|α|^2 −|β|^2 = 1

Proof.The equations forz,zin terms ofq,pimply that the change of basis
between these two bases is
(
z
z

)

=

1

√

2

(

1 i

1 −i

)(

q

p

)

The matrix for this transformation has inverse

1

√

2

(

1 1

−i i

)

Conjugating by this change of basis matrix, one finds

1

√

2

(

1 1

−i i

)(

α β

β α

)

1

√

2

(

1 i

1 −i

)

=

(

Re(α+β) −Im(α−β)

Im(α+β) Re(α−β)

)

(24.11)

The right hand side is a real matrix, with determinant one, since conjugation
doesn’t change the determinant.

Note that the change of basis 24.11 is reflected in equations 24.3 and 24.8, where
the matrices on the right hand side are the matrixZandGrespectively, but
transformed to thez,zbasis by 24.11.
Another equivalent characterization of the groupSU(1,1) is as the group
of linear transformations ofC^2 , with determinant one, preserving the indefinite
Hermitian inner product
〈(
c 1
c 2

)

,

(

c′ 1

c′ 2

)〉

1 , 1

=c 1 c′ 1 −c 2 c′ 2

One finds that
〈(
α β
γ δ

)(

c 1

c 2

)

,

(

α β

γ δ

)(

c′ 1

c′ 2

)〉

1 , 1

=

〈(

c 1

c 2

)

,

(

c′ 1

c′ 2

)〉

1 , 1

whenγ=β,δ=αand|α|^2 −|β|^2 = 1.