Quantum Mechanics for Mathematicians

20.3.3 The Fourier transform as an intertwining operator

For another indication of the non-trivial nature of the intertwining operators of
section 20.3.2, note that a group elementgacting by aπ 2 rotation of thed= 1
phase space interchanges the role ofqandp. It turns out that the corresponding
intertwining operatorUgis closely related to the Fourier transformF. Up to a
phase factorei
π 4
, Fourier transformation is just such an intertwining operator:
we will see in section 23.4 that, acting on wavefunctions,

Ueπ 2 L=ei

π 4

F

Squaring this gives
UeπL= (Ueπ 2 L)^2 =iF^2

and we know from the definition ofFand Fourier inversion that

F^2 ψ(q) =ψ(−q)

The non-trivial double cover here appears because

Ue 2 πL=−F^4 =− 1

which takes a wavefunctionψ(q) to−ψ(q).

20.3.4 AnRaction on thed= 1 phase space

For another sort of example ind= 1, consider the action of a subgroupR⊂
SL(2,R) ond= 1 phase space by
(
cq
cp

)

→g

(

cq

cp

)

=

(

er 0

0 e−r

)(

cq

cp

)

where

g=erL, L=

(

1 0

0 − 1

)

Now, by equation 16.7 the moment map will be

μL=

1

2

(

q p

)

(

0 − 1

− 1 0

)(

q

p

)

=−qp

which satisfies {

μL,

(

q

p

)}

=

(

q

−p

)

Quantization gives intertwining operators by

UL′ =−

i

2

(QP+PQ), Ug=erU

′L

=e−

ir 2 (QP+PQ)

These act on operatorsQandPby a simple rescaling

e−

ir 2 (QP+PQ)

(

Q

P

)

e

ir 2 (QP+PQ)

=

(

er 0

0 e−r

)(

Q

P

)