(Note that for such phase space rotations, we are making the opposite choice
for convention of the positive direction of rotation, clockwise instead of counter-
clockwise).
To find the intertwining operators, we first find the quadratic functionμL
inq,pthat satisfies
{
μL,

(

q

p

)}

=LT

(

q

p

)

=

(

−p

q

)

By equation 16.7 this is

μL=

1

2

(

q p

)

(

1 0

0 1

)(

q

p

)

=

1

2

(q^2 +p^2 )

QuantizingμLusing the Schr ̈odinger representation Γ′S, one has a unitary
Lie algebra representationU′ofso(2) with

UL′=−

i

2

(Q^2 +P^2 )

satisfying [

UL′,

(

Q

P

)]

=

(

−P

Q

)

(20.6)

and intertwining operators

Ug=eθU

L′

=e−i

θ 2 (Q (^2) +P (^2) )
These give a representation ofSO(2) only up to a sign, for reasons mentioned
in section 17.1 that will be discussed in more detail in chapter 24.
Conjugating the Heisenberg Lie algebra representation operators by the uni-
tary operatorsUgintertwines the representations corresponding to rotations of
the phase space plane by an angleθ
e−i
θ 2 (Q (^2) +P (^2) )

(

Q

P

)

ei

θ 2 (Q (^2) +P (^2) )

(

cosθ −sinθ

sinθ cosθ

)(

Q

P

)

(20.7)

Note that this is a different calculation than in the spin case where we also
constructed a double cover ofSO(2). Despite the different context (SO(2)
acting on an infinite dimensional state space), again one sees an aspect of the
double cover here, as eitherUgor−Ugwill give the sameSO(2) rotation action
on the operatorsQ,P(while each having a different action on the states, to be
worked out in chapter 24).
In our discussion here we have blithely assumed that the operatorUL′ can
be exponentiated, but doing so turns out to be quite non-trivial. As remarked
earlier, this representation on wavefunctions does not arise as the induced action
from an action on position space. UL′ is (up to a factor ofi) the Hamiltonian
operator for a quantum system that is not translation invariant. It involves
quadratic operators in bothQandP, so neither the position space nor momen-
tum space version of the Schr ̈odinger representation can be used to make the
operator a multiplication operator. Further details of the construction of the
needed exponentiated operators will be given in section 23.4.