Quantum Mechanics for Mathematicians

for some non-zerov∈M, so

〈u 1 ,u 2 〉=iΩ(u 1 ,u 2 )

=i

1

2

Ω(v 1 +iJv 1 ,v 2 −iJv 2 )

=

1

2

(−Ω(Jv 1 ,v 2 ) + Ω(v 1 ,Jv 2 )) +

i

2

(Ω(v 1 ,v 2 ) + Ω(Jv 1 ,Jv 2 ))

=Ω(v 1 ,Jv 2 ) +iΩ(v 1 ,v 2 ) (26.9)

where we have used compatibility ofJandJ^2 =−1 to get

Ω(Jv 1 ,v 2 ) = Ω(J^2 v 1 ,Jv 2 ) =−Ω(v 1 ,Jv 2 )

We thus can recover Ω onMas the imaginary part of the form〈·,·〉.
This form〈·,·〉is not positive or negative-definite onM ⊗C. One can
however restrict attention to thoseJthat give a positive-definite form onM+J:

Definition(Positive compatible complex structures). A complex structureJ
onMis said to be positive and compatible withΩif it satisfies the compatibility
condition 26.5 (i.e., is inSp(2d,R)) and one of the equivalent (by equation
26.9) positivity conditions

〈u,u〉=iΩ(u,u)> 0 (26.10)

for non-zerou∈M+J. or
Ω(v,Jv)> 0 (26.11)

for non-zerov∈M.

For such aJ,〈·,·〉restricted toM−J will be negative-definite since

Ω(u,u) =−Ω(u,u)

and complex conjugation interchangesM+J andM−J. The standard complex
structureJ 0 is positive since

〈zj,zk〉=iΩ(zj,zk) =i{zj,zk) =δjk

and〈·,·〉is thus the standard Hermitian form onM+J 0 for which thezj are
orthonormal.

26.3 Complex structures and quantization

Recall that the Heisenberg Lie algebra is the Lie algebra of linear and constant
functions onM, so can be thought of as

h 2 d+1=M⊕R