- Mis identified withM+J 0 by equation 26.1, with basis elementqjgoing
tozj, andpjtoizj. - Mis identified withM−J 0 by equation 26.2, with basis elementqjgoing
tozj, andpjto−izj. - Mis identified with elements ofM+J 0 ⊕ M−J 0 that are invariant under
conjugation, with basis elementqjgoing to√^12 (zj+zj) andpjto√i 2 (zj−
zj).
26.2 Compatible complex structures and posi-
tivity
Our interest is in vector spacesMthat come with a symplectic structure Ω,
a non-degenerate antisymmetric bilinear form. To successfully use a complex
structureJfor quantization, it will turn out that it must be compatible with Ω
in the following sense:
Definition(Compatible complex structure).A complex structure onMis said
to be compatible withΩif
Ω(Jv 1 ,Jv 2 ) = Ω(v 1 ,v 2 ) (26.5)Equivalently,J∈Sp(2d,R), the group of linear transformations ofMpreserv-
ingΩ.
The standard complex structureJ=J 0 is compatible with Ω, since (treating
thed= 1 case, which generalizes easily, and using equations 16.2 and 26.3)
Ω(J 0 (cqq+cpp),J 0 (c′qq+c′pp))=
((
0 − 1
1 0
)(
cq
cp))T(
0 1
−1 0
)((
0 − 1
1 0
)(
c′q
c′p))
=
(
cq cp)
(
0 1
−1 0
)(
0 1
−1 0
)(
0 − 1
1 0
)(
c′q
c′p)
=
(
cq cp)
(
0 1
−1 0
)(
c′q
c′p)
=Ω(cqq+cpp,c′qq+c′pp)More simply, the matrix forJ 0 is obviously inSL(2,R) =Sp(2,R).
Note that elementsgof the groupSp(2d,R) act on the set of compatible
complex structures by
J→gJg−^1 (26.6)
This takes complex structures to complex structures since
(gJg−^1 )(gJg−^1 ) =gJ^2 g−^1 =− 1