The case of the groupSO(2)⊂Sp(4,R) can be generalized to a larger
subgroup, the groupGL(2,R) of all invertible linear transformations ofR^2 ,
performed simultaneously on position and momentum space. Replacing the
matrixLby (
A 0
0 A
)
forAany real 2 by 2 matrix
A=
(
a 11 a 12
a 21 a 22)
we get an action of the groupGL(2,R)⊂Sp(4,R) onM, and after quantization
a Lie algebra representation
UA′ =i(
Q 1 Q 2
)
(
a 11 a 12
a 21 a 22)(
P 1
P 2
)
which will satisfy
[
UA′,
(
Q 1
Q 2
)]
=−A
(
Q 1
Q 2
)
,
[
UA′,
(
P 1
P 2
)]
=AT
(
P 1
P 2
)
Note that the action ofAon the momentum operators is the dual of the action
on the position operators. Only in the case of an orthogonal action (theSO(2)
earlier) are these the same, withAT=−A.
20.3.2 AnSO(2) action on thed= 1 phase space
Another sort ofSO(2) action on phase space provides ad= 1 example that
mixes position and momentum coordinates. This will lead to quite non-trivial
intertwining operators, with an action on wavefunctions that does not come
about as an induced action from a group action on position space. This ex-
ample will be studied in much greater detail when we get to the theory of the
quantum harmonic oscillator, beginning with chapter 22. Such a physical sys-
tem is periodic in time, so the usual groupRof time translations becomes this
SO(2), with the corresponding intertwining operators giving the time evolution
of the quantum states.
In this cased= 1 and one has elementsg∈SO(2)⊂Sp(2,R) acting on
cqq+cpp∈Mby
(
cq
cp)
→g(
cq
cp)
=
(
cosθ sinθ
−sinθ cosθ)(
cq
cp)
so
g=eθL
where
L=