Quantum Mechanics for Mathematicians

and preserves the compatibility condition since, ifJ∈Sp(2d,R), so isgJg−^1.
A complex structureJcan be characterized by the subgroup ofSp(2d,R)
that leaves it invariant, with the conditiongJg−^1 =Jequivalent to the com-
mutativity conditiongJ=Jg. For the cased= 1 andJ=J 0 this becomes
(
a b
c d

)(

0 − 1

1 0

)

=

(

0 − 1

1 0

)(

a b

c d

)

so (
b −a
d −c

)

=

(

−c −d

a b

)

which impliesb=−canda=d. The elements ofSL(2,R) that preserveJ 0
will be of the form (
a b
−b a

)

with unit determinant, soa^2 +b^2 = 1. This is theU(1) =SO(2) subgroup of
SL(2,R) of matrices of the form

(

cosθ sinθ

−sinθ cosθ

)

=eθZ

Other choices ofJwill correspond to otherU(1) subgroups ofSL(2,R), and
the space of compatible complex structures conjugate toJ 0 can be identified
with the coset spaceSL(2,R)/U(1). In higher dimensions, it turns out that
the subgroup ofSp(2d,R) that commutes withJ 0 is isomorphic to the unitary
groupU(d), and the space of compatible complex structures conjugate toJ 0 is
Sp(2d,R)/U(d).
Even before we choose a complex structureJ, we can use Ω to define an
indefinite Hermitian form onM⊗Cby:

Definition(Indefinite Hermitian form onM⊗C).Foru 1 ,u 2 ∈M⊗C,

〈u 1 ,u 2 〉=iΩ(u 1 ,u 2 ) (26.7)

is an indefinite Hermitian form onM⊗C.

This is clearly antilinear in the first variable, linear in the second, and satisfies
the Hermitian property, since

〈u 2 ,u 1 〉=iΩ(u 2 ,u 1 ) =−iΩ(u 1 ,u 2 ) =iΩ(u 1 ,u 2 ) =〈u 1 ,u 2 〉

Restricting〈·,·〉toM+J and using the identification 26.1 ofMandM+J,
〈·,·〉gives a complex-valued bilinear form onM. Anyu∈ M+J can be written
as

u=

1

√

2

(v−iJv) (26.8)