From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 19

Eq. (1.62) gives an identification ofHwith the fiberTψH. The latter carries a
natural linear structure, and through Δ we can give a tensorial characterization
of the linear structure of the base spaceH.
Thus, we can associate to every linear operatorAthe linear vector field:

XA:H→TH

ψ→(ψ,Aψ). (1.64)

In local coordinates, we haveXA≡Aijψj∂/∂ψi,Aij being the representative
matrix of the linear operator.
We notice that, while linear operators form an associative algebra, vector fields
do not, yielding instead a Lie algebra. Nevertheless we can recover an associative
algebra by defining now the (1,1) tensor:

TA≡Aijdψj⊗

∂

∂ψi

. (1.65)

It is easy to verify that we can recover the vector fieldXAfrom the latter and the
dilation field via:

XA=TA(Δ). (1.66)

The geometric structures just introduced allow to interpret the Schr ̈odinger
equation (1.40) as a classical evolution equation on the complex vector spaceH.
Indeed, the HamiltonianHdefines the linear vector field^13 ΓH:

ΓH:H→TH

ψ→(ψ,−(i/)Hψ). (1.67)

Thus Eq. (1.40) reads as:

LΓHψ≡

d

dtψ=−

i

Hψ. (1.68)

1.2.2.2 Riemannian and symplectic forms

Let us denote with

h:H×H→C

(φ, ψ)→h(φ, ψ)≡〈φ|ψ〉 (1.69)

the Hermitean scalar product onH.

(^13) We use here the notation ΓHinstead ofXHto stress that we include an additional factor
(−i/) in its definition.