From Classical Mechanics to Quantum Field Theory

22 From Classical Mechanics to Quantum Field Theory. A Tutorial

We have also already mentionedthat every equivalence class [|ψ〉] can be iden-
tified with the rank-one projector:ρψ=|ψ〉〈ψ|,where,asusual,thevector|ψ〉is
supposed to be normalized. The space of rank-one projectors is usually denoted
asD 11 (H) and it is then clear that in this way we can identify it withPH.
More explicitly, points in CPn are equivalence classes of vectors Z =
(Z^0 ,Z^1 ,···,Zn)∈Cn+1w.r.t. the equivalence relationZ≈λZ;λ∈C−{ 0 }.
The spaceCPnis endowed with:

(i) the Fubini-Study metric[ 3 ], whose pull-back toCn+1is given by:

gFS=

1

(Z·Z ̄)^2

[

(Z·Z ̄)dZ⊗SdZ ̄−(dZ·Z ̄)⊗S(Z·dZ ̄)

]

, (1.84)

whereZ·Z ̄=ZaZ ̄a,dZ·Z ̄=dZaZ ̄a,dZ⊗SdZ ̄=dZadZ ̄a+dZ ̄adZa;

(ii) the compatible Kostant Kirillov Souriau symplectic form[ 23 ]:

ωFS= i

(Z·Z ̄)^2

[

(Z·Z ̄)dZ∧dZ ̄−(dZ·Z ̄)∧(Z·dZ ̄)

]

=dθFS, (1.85)

where:

θFS=^1

2 i

ZdZ−ZdZ

Z·Z

. (1.86)

This shows thatPHis intrinsically a K ̈ahler manifold. A more detailed discussion
of theses structures, together with the ones that will be defined in the next sections,
may be found in[ 14 ].
We end this example with a remark. The careful reader might have noticed that
it appears that the most natural setting for QM is not primarily the Hilbert space
itself but ratherPH=D 11 (H), which is not a vector space. On the other hand, we
know that the superposition rule is the key ingredient to interpret all interference
phenomena that yields one of the fundamental building blocks of QM. In[ 27 ],itis
shown that a superposition of rank-one projectors which yields another rank-one
projector is possible, but requires the arbitrary choice of a fiducial projectorρ 0.
This procedure is equivalent to the introduction of a connection on the bundle,
usually called the Pancharatnam connection[ 29 ].

1.2.2.3 Geometric structures on the Hilbert space

In the following, points inHRwill be denoted by Latin letters (x=u+ıv,···)
while we will use Greek letters (ψ,···) for tangent vectors.