Mathematical Foundations of Quantum Mechanics 83Since〈ψ,ψ〉=||ψ||^2 = 1 by hypotheses, (2.22) is proved. Obviously the open
problem is to justify the validity of (2.5) and (2.8) also in the infinite dimensional
case.
Another philosophically important consequence of the CCR (2.21) is that they
resemble theclassical canonical commutation relationsof the Hamiltonian vari-
ablesqhandpk, referring to the standardPoisson brackets{·,·}P,
{qh,pk}P=δhk, {qh,qk}P=0, {ph,pk}P=0. (2.23)as soon as one identifies (i)−^1 [·,·] with{·,·}P. This fact, initially noticed by
Dirac[ 3 ], leads to the idea of “quantization” of a classical Hamiltonian theory (see
the first part of this book).
One starts from a classical system described on a symplectic manifold (Γ,ω), for
instance Γ =R^2 nequipped with the standard symplectic form asωand considers
the (real) Lie algebra (C∞(Γ,R),{·,·}P). To “quantize” the system one looks for
a map associating classical observablesf∈C∞(Γ,R) to quantum observablesÔf,
i.e. selfadjoint operators restricted^4 to a common invariant domainSof a certain
Hilbert spaceH.(IncaseΓ=T∗Q,Hcan be chosen asL^2 (Q, dμ)whereμis some
natural measure.) The mapf→Ôf is expected to satisfy a set of constraints.
The most important are listed here:
(1)R-linearity;
(2)Ôid=I|S;
(3)Ô{f,g}P=−i[Ôf,Ôg];
(4) If (Γ,ω)isR^2 nequipped with the standard symplectic form, they must
holdÔxk=Xk|SandÔpk=Pk|S,k=1, 2 ,...,n.The penultimate requirement says that the mapf→Ôftransforms the real Lie
algebra (C∞(Γ,R),{·,·}P) into a real Lie algebra of operators whose Lie bracket
isi[Ôf,Ôg]. A map fulfilling these constraints, in particular the third one, is
possible iff,gare both functions of only theqor thepcoordinates separately or
if they are linear in them. But it is false already if we consider elementary physical
systems as some discussed in the first part of the book. The ultimate reason of
this obstructions is due to the fact that the operatorsPk,Xkdo not commute,
(^4) The restriction should be such that it admits a unique selfadjoint extension. A sufficient
requirement onSis that everyÔfisessentially selfadjointthereon, notion we shall discuss in
the next section.