From Classical Mechanics to Quantum Field Theory

20 From Classical Mechanics to Quantum Field Theory. A Tutorial

More properly,hshould be seen as a (0,2) (constant) tensor field evaluated at
the pointφ∈Hand withφ, ψ∈TφH. Thus we should write, in a more precise
way:

h(φ)(Γφ(φ),Γψ(φ)) =〈φ|ψ〉. (1.70)

Since the r.h.s. of this equation does not depend onφ, from Eq. (1.40), we obtain
the chain of equalities:

0=LΓH(h(φ, ψ)) = (LΓHh)(φ, ψ)+h(LΓHφ, ψ)+h(φ,LΓHψ)

=(LΓHh)(φ, ψ)+

i



{〈Hφ|ψ〉−〈φ|Hψ〉}, (1.71)

which implies in turn, asHis self-adjoint, that:

LΓHh=0. (1.72)

This means that the Hermitean structurehis invariant^14 under the unitary flow
of ΓH. Vice versa, if the Hermitean structure is not invariant thenHwill not be
self-adjoint w.r.t. the given Hermitean structure.
Let us now decompose the Hermitean structure into its real and imaginary
parts:

h(·,·)=g(·,·)+iω(·,·), (1.73)

so defining the two (0,2) tensors

g(φ, ψ)≡〈φ|ψ〉+〈ψ|φ〉

2

, (1.74)

ω(φ, ψ)≡〈φ|ψ〉−〈ψ|φ〉

2 ı

. (1.75)

Clearlygis symmetric andωis skew-symmetric, while Eq. (1.72) implies that
both tensors are separately invariant under ΓH,sinceω(φ, iψ)=g(φ, ψ). Also,ω
is closed because it is represented by a constant (and unitarily invariant) matrix.
Insummary,wehaveprovedthatωis asymplectic formandgis a (Riemannian)
metric form.
Let us recall also thatHis endowed with a natural complex structureJ,the
(1,1) tensor defined by:

J:H→H

φ→−ıφ

with J^2 =−I. (1.76)

Sinceω(Jφ,ψ)=g(φ, ψ), the complex structureJis said to becompatible with
the pair (g,ω) and the triple (g,ω,J) is calledadmissible. This means that we

(^14) This means that ΓHis a Killing vector field for the Hermitean structure.