From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 21

can reconstruct the Hermitean structurehwhen any two of the tensorsg,ω,Jare
given. This is so because:

h(φ, ψ)=ω(Jφ,ψ)+ıω(φ, ψ)=g(φ, ψ)−ıg(Jφ,ψ). (1.77)

Notice also that:

ω(Jφ,Jψ)=ω(φ, ψ)aswellasg(Jφ,Jψ)=g(φ, ψ). (1.78)

We can summarize what has been proven up to now by saying thatH is a K ̈ahler
manifold[ 22 ], with Hermitean metrich,ωbeing the fundamental two-form andg
the K ̈ahler metric.
As a final remark, we observe that a vector field ΓHof the form (1.67) is such
that:

(iΓHω)(ψ)=ω

(

−

i



Hφ,ψ

)

=

1

2 

[〈Hφ|ψ〉+〈ψ|Hφ〉]. (1.79)

If we define now the quadratic function

fH(φ)=

1

2 

〈φ|Hφ〉 (1.80)

and its differential as the one-form

dfH(φ)=

1

2

[〈·|Hφ〉+〈φ|H·〉]=

1

2

[〈·|Hφ〉+〈Hφ|·〉] (1.81)

(the last passage following fromHbeing self-adjoint), then it is easy to prove that:
(iΓHω)(ψ)=dfH(φ)(ψ)∀ψ. Hence we have:

iΓHω=dfH, (1.82)

i.e. ΓHisHamiltonianw.r.t. the symplectic structure, withfHas quadratic
Hamiltonian.

Example 1.2.10.The projective Hilbert space.
We have already seen that a physical state is not identified with a unique vector
in some Hilbert space, but rather with a “ray”, i.e. an equivalence class of vectors
differing by multiplication through a nonzero complex number: even fixing the
normalization, an overall phase remains unobservable. Quotienting with respect
to these identifications, we get the following double fibration:

R+ ↪→H 0 =H−{ 0 }

↓

U(1)↪→ S^2 n−^1

↓

P(H)

(1.83)

whose final result is the projective Hilbert spacePHCPn−^1.