and one has all the elements needed for the standard Bargmann-Fock quantiza-
tion. Note that the Hermitian inner product here is indefinite, positive onM,
negative onM.
To understand better in a basis independent way how quantization works in
this case of a complex dual phase spaceM, one can use the identification (see
section 9.6) of polynomials with symmetric tensor products. In this case poly-
nomials in thezjget identified withS∗(M) (sincezj∈M), while polynomials
in thezjget identified withS∗(M) (sincezj∈M).
We see that the Fock spaceFdgets identified withS∗(M) and using this
identification (instead of the one with polynomials) one can ask what operator
gives the quantization of an element
u=u++u−whereu+∈ Mandu−∈M. For basis elementszj∈ Mthe operator will be
−ia†j, while forzj∈Mit will be−iaj. We will not enter here into details,
which would require more discussion of how to manipulate symmetric tensor
products (see for instance chapter 5.4 of [17]). One can show however that the
operators Γ′(u+,0),Γ′(u−,0) defined on symmetrized tensor products by (where
P+is the symmetrization operator of section 9.6)
Γ′(u+,0)P+(u 1 ⊗···⊗uN) =−i√
N+ 1P+(u+⊗u 1 ⊗···⊗uN)Γ′(u−,0)P+(u 1 ⊗···⊗uN) =−i
√
N∑N
j=1〈u−,uj〉P+(u 1 ⊗···⊗ûj⊗···⊗uN)satisfy the Heisenberg Lie algebra homomorphism relations
[Γ′(u−,0),Γ′(u+,0)] =Γ′(0,Ω(u−,u+))
=−iΩ(u−,u+) 1
= =−〈u−,u+〉 1 (26.16)when acting on elements ofSN(M) (which are given by applying the sym-
metrization operatorP+to elements of theN-fold tensor product ofM).
26.5 Complex structures ford= 1 and squeezed states
To get a better understanding of what happens for other complex structures
thanJ 0 , in this section we’ll examine the cased= 1. We can generalize the
choiceJ=J 0 , where a basis ofM+J 0 is given by
z=1
√
2
(q−ip)