The pair (V,J) can be thought of as givingVthe structure of a complex vector
space, withJproviding multiplication byi. Similarly, taking
v∈V→1
√
2
(v+iJv)∈VJ− (26.2)identifiesVwithVJ−, withJnow providing multiplication by−i.VJ+andVJ−
are interchanged by changing the complex structureJto−J.
For the study of quantization, the real vector space we want to choose a
complex structure on is the dual phase spaceM=M∗, since it is elements of this
space that are in a Heisenberg algebra, and taken to operators by quantization.
There will be a decomposition
M⊗C=M+J⊕M−Jand quantization will take elements ofM+J to linear combinations of creation
operators, elements ofM−J to linear combinations of annihilation operators.
The standard choice of complex structure is to takeJ=J 0 , whereJ 0 is the
linear operator that acts on coordinate basis vectorsqj,pjofMby
J 0 qj=pj, J 0 pj=−qjMaking the choice
zj=1
√
2
(qj−ipj)implies
J 0 zj=1
√
2
(pj+iqj) =izjand thezjare basis elements (over the complex numbers) ofM+J 0. The complex
conjugates
zj=1
√
2
(qj+ipj)provide basis elements ofM−J 0
With respect to the chosen basisqj,pj, the complex structure can be written
as a matrix. For the case ofJ 0 and ford= 1, on an arbitrary element ofMthe
action ofJ 0 is
J 0 (cqq+cpp) =cqp−cpq
soJ 0 in matrix form with respect to the basis (q,p) is
J 0
(
cq
cp)
=
(
0 − 1
1 0
)(
cq
cp)
=
(
−cp
cq)
(26.3)
or, the action on basis vectors is the transpose
J 0
(
q
p)
=
(
0 1
−1 0
)(
q
p)
=
(
p
−q)
(26.4)
Note that, after complexifying, three different ways to identify the original
Mwith a subspace ofM⊗Care: