From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 5

In “quantizing” a classical system, we would like to assign a set of rules that
allows us to univocally pick up a self-adjoint operatorOˆffor each classical observ-
ablef. We would like this map to preserve the algebraic structures of these two
spaces[ 2 ], i.e. to satisfy the following properties:

• the mapf→Oˆfis linear;


• iff=idT∗QthenOˆid=IH;


• ifg=Φ◦f, with Φ :R→Rfor which bothOˆfandOˆgare well defined,
  thenOˆg=OˆΦ◦f=Φ(Oˆf).

We would also like this map to include the largest class of functions as possible.
In particular, it must be possible to find the operators ˆpj,qˆj associated to the
coordinate functionspj andqj(j=1,···,n)onT∗Q, about which we put an
additional requirement. From the simple examples one can develop such as the
free particle and the harmonic oscillator, one can argue that Classical Poisson
Brackets (CPB):

{pi,qj}=δij; {qi,qj}={pi,pj}=0;{qi,id}={pi,id}= 0 (1.1)

have to be replaced by the following Canonical Commutation Relations (CCR):

[ˆqi,pˆj]=ıδij, [ˆqi,qˆj]=[ˆpi,ˆpj]=0; [ˆqi,I]=[ˆpi,I]=0. (1.2)

From the first of these commutators withi=j, one can see that at least one of
(and indeed both) the operators ˆpi,qˆihas to be unbounded[ 34 ].Atheoremby
Stone and Von-Neumann[ 34 ]then states that, up to unitary equivalence, there is
only one irreducible representation of such algebra of observables, called theWeyl
algebra,whichonH=L^2 (Q)readsas:

qˆiψ(q)=qiψ(q), pˆiψ(q)=−ı

∂

∂qiψ(q). (1.3)
Actually, knowing that in classical mechanics we can reconstruct the Poisson
bracket between any two (regular, such as polynomials) functionsf(p, q),g(p, q)
out of (1.1), we would like to find a map such that if,f→Oˆfandg→Oˆg, then:

Oˆ{f,g}=ı[Oˆf,Oˆg]. (1.4)

This is possible iff,gare both functions of only theq-orthep-coordinates or
if they are linear in them. However, we know that this does not hold already
if we consider a quadratic function such as the Hamiltonian of the 1D harmonic
oscillator. This is because, at the quantum level, we have an ordering problem due
to the fact that the operators ˆp,qˆdo not commute, contrary to the functionsp, q.