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.