we introduced complex coordinates on phase space, making the choice
zj=1
√
2
(qj−ipj), zj=1
√
2
(qj+ipj)Thezjwere then quantized using creation operatorsa†j, thezjusing annihilation
operatorsaj. In the Bargmann-Fock representation, where the state space is a
space of functions of complex variableszj, we have
aj=∂
∂zj, a†j=zjand there is a distinguished state, the constant function, which is annihilated
by all theaj.
In this section we’ll introduce the notion of a complex structure on a real vec-
tor space, with such structures characterizing the possible ways of introducing
complex coordinateszj,zjand thus annihilation and creation operators. The
abstract notion of a complex structure can be formalized as follows. Given any
real vector spaceV=Rn, we have seen that taking complex linear combinations
of vectors inV gives a complex vector spaceV⊗C, the complexification ofV,
and this can be identified withCn, a real vector space of twice the dimension.
Whenn= 2dis even there is another way to turnV =R^2 dinto a complex
vector space, by using the following additional piece of information:
Definition(Complex structure).A complex structure on a real vector spaceV
is a linear operator
J:V→V
such that
J^2 =− 1
Given such a pair (V=R^2 d,J) complex linear combinations of vectors inV
can be decomposed into those on whichJacts asiand those on which it acts
as−i(sinceJ^2 =− 1 , its eigenvalues must be±i), so we have
V⊗C=VJ+⊕VJ−whereVJ+is the +ieigenspace of the operatorJonV⊗CandVJ−is the−i
eigenspace. Note that we have extended the action ofJonV to an action on
V⊗Cusing complex linearity. Complex conjugation takes elements ofVJ+to
VJ−and vice-versa. The choice ofJhas thus given us two complex vector spaces
of complex dimensiond,VJ+andVJ−, related by this complex conjugation.
Since
J(v−iJv) =i(v−iJv)
for anyv∈V, the real vector spaceVcan by identified with the complex vector
spaceVJ+by the map
v∈V→1
√
2
(v−iJv)∈VJ+ (26.1)