From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 23

Denoting withXψthe constant vector fieldXψ≡(x, ψ), we can regardgof
Eq. (1.74) andωof Eq. (1.75) as (0,2) tensors by setting:

g(x)(Xψ,Xφ)≡g(ψ,φ), (1.87)

ω(x)(Xψ,Xφ)≡ω(ψ,φ). (1.88)

Also we can promoteJas defined in Eq. (1.76) to a (1,1) tensor field by setting:

J(x)(Xψ)=(x, Jψ), (1.89)

making, as we have already noticed before, the tensorial triple (g,J,ω) admissible
andHRa linear K ̈ahler manifold[ 36 ].
The (0,2)-tensorsgandωbeing non-degenerate, we can consider the invertible
maps ˆg,ˆω:THR→T∗HRsuch that, for anyX,Y ∈THR:g(X,Y)=ˆg(X)(Y)
andω(X,Y)=ˆω(X)(Y). Then we can define two (2,0) contravariant tensors
given respectively by a metric tensorGand a Poisson tensor Λ such that

G(ˆg(X),ˆg(Y)) =g(X,Y), Λ(ˆω(X),ωˆ(Y)) =ω(X,Y). (1.90)

We can now useGand Λ to define a Hermitean product between any twoα, β∈H∗R
(equipped with the dual complex structureJ∗):

〈α, β〉H∗R≡G(α, β)+iΛ(α, β). (1.91)

To make these structures more explicit, let us introduce an orthonormal basis
{ek}k=1,···,ninHand global coordinates (qk,pk)fork=1,···,ninHRsuch that
〈ek,x〉=(qk+ipk)(x),∀x∈H. Then, it is easy to see that:

g=dqk⊗dqk+dpk⊗dpk, (1.92)

ω=dqk⊗dpk−dpk⊗dqk, (1.93)

J=dpk⊗ ∂

∂qk

−dqk⊗ ∂

∂pk

, (1.94)

so that

G=

∂

∂qk⊗

∂

∂qk+

∂

∂pk⊗

∂

∂pk, (1.95)

Λ=

∂

∂qk

⊗

∂

∂pk

−

∂

∂pk

⊗

∂

∂qk

. (1.96)

As a convenient shorthand notation, we can introduce complex coordinates:zk≡
qk+ipk, ̄zk≡qk−ipk, so that we can write

G+i·Λ=4∂

∂zk

⊗ ∂

∂ ̄zk

, (1.97)