From Classical Mechanics to Quantum Field Theory

84 From Classical Mechanics to Quantum Field Theory. A Tutorial

contrary to the functionspk,qkwhich do. The problem can be solved, in the
paradigm of the so-calledGeometric Quantization(see the first part of the book),
replacing (C∞(Γ,R),{·,·}P) with a sub-Lie algebra (as large as possible). There
are other remarkable procedures of “quantization” in the literature, we shall not
insist on them any further here (see the first part of the book).

Example 2.1.15.
(1)The full Hilbert space of an electron is therefore given by the tensor product
L^2 (R^3 ,d^3 x)⊗Hs⊗Hc.
(2)Consider a particle in 3Dwith massm, whose potential energy is a bounded-
below real functionU∈C∞(R^3 ) with polynomial growth. Classically, its Hamil-
tonian function reads

h:=

∑^3

k=1

p^2 k

2 m

+U(x).

A brute force quantization procedure inL^2 (R^3 ,d^3 x) consists of replacing every
classical object with corresponding operators. It may make sense at most when
there are no ordering ambiguities in translating functions likep^2 x, since classically
p^2 x=pxp=xp^2 , but these identities are false at quantum level. In our case,
these problems do not arise so that

H:=

∑^3

k=1

Pk^2

2 m+U, (2.24)

where (Uψ)(x):=U(x)ψ(x), could be accepted as first quantum model of the
Hamiltonian function of our system. The written operator is at least defined on
S(R^3 ), where it satisfies〈Hψ,φ〉=〈ψ,Hφ〉. The existence of selfadjoint extensions
is a delicate issue[ 5 ]we shall not address here. Taking (2.19) into account, always
onS(R^3 ), one immediately finds

H:=−

^2

2 m

Δ+U,

where Δ is the standard Laplace operator inR^3. If we assume that the equation
describing the evolution of the quantum system is again^5 (2.14), in our case we

(^5) Afactorhas to be added in front of the left-hand side of (2.14) if we deal with a unit system
where=1.