6 From Classical Mechanics to Quantum Field Theory. A Tutorial
It is therefore evident that we will not be able to find a correspondencef→Oˆf
which fulfils all (five) requirements^5 , at least if we insist on trying to quantize the
whole space of observables.
We also remark that one is interested in the development of a technique that
is not only suited to describeT∗Q=R^2 n, but the more general case in which the
phase space is a symplectic manifold (Γ,ω).
Geometric quantizationis one of the main approaches that has been devel-
oped to deal with these questions. The idea behind this approach is to overcome
the problems described above by restricting the space of quantizable observables.
More precisely, one would like to assign to (Γ,ω) a separable Hilbert spaceHand
mappingQ:f→Oˆffrom asub-Lie algebraØ (as large as possible) of real-valued
functions on Γ into self-adjoint linear operators onH, satisfying:
(1)Q:f→Oˆfis linear;
(2)OˆidΓ=IH;
(3)Oˆ{f,g}=ı[Oˆf,Oˆg],∀f,g∈Ø;
(4) for Γ =R^2 nandω=ω 0 , the standard symplectic form, we should recover
the operators ˆqi,pˆias in (1.3);
(5) the procedure is functorial, in the sense that for any two symplectic man-
ifolds (Γ 1 ,ω 1 ), (Γ 2 ,ω 2 ) and a symplectic diffeomorphism Φ : Γ 1 →Γ 2 ,
the composition with Φ should map Ø 1 into Ø 2 and there should exist a
unitary operatorUΦ:H 1 →H 2 such thatO(1)f◦Φ=UΦ†O(2)f UΦ,∀f∈Ø 2.The solution to this problem was first given by Kostant and Souriau and goes
through two main steps, calledprequantizationandpolarization.Thetheoryof
geometric quantization has become a topic of study both in physics and in math-
ematics, but is goes beyond the scope of these lectures. We refer the interested
reader to ref.[ 2 ]for a review and an exhaustive list of references.
Instead, in these lectures we will reviewsome other methods of quantization,
which are also founded on the geometric structures of QM and represent the ones
that are more operatively used in modern theoretical physics. We will first in-
troduce coherent states and Bargmann-Fock representation, both for bosonic and
fermionic systems. Then we will move to discuss Feynman’s approach to path
integral. We will devote the last section to Weyl-Wigner formalism, to finally
introduce the so-called non-commutative *-product and discuss the quantum to
classical transitions.
(^5) This is not only because we insist on Eq. (1.4). In[ 2 ], the interested reader may found a
detailed discussions and many examples showing all existing inconsistencies among the various
assumptions we made.