From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 103

Fourier transform (2.33) with inverse (2.34). Using these integral expressions we
easily obtain

(Pmψ)(x)=(Fˆ†KmFˆψ)(x)=−i

∂

∂xm

ψ(x) (2.37)

because inS(Rn), which is invariant under the Fourier (and inverse Fourier) inte-
gral transformation,
∫

Rn

eik·xkm(Fψ)(k)dnk=−i

∂

∂xm

∫

Rn

e−ik·x(Fψ)(k)dnk.

This way leads us to consider the operatorsPm′ andPm′′inL^2 (Rn,dnx) with

D(Pm′)=C∞ 0 (Rn;C),D(Pm′′)=S(Rn)

and, forx∈Rnandψ,φin the respective domains,

(Pm′ψ)(x):=−i ∂

∂xm

ψ(x), (Pm′′φ)(x):=−i ∂

∂xm

φ(x).

Both operators are symmetric as one can easily prove by integrating by parts,
but not selfadjoint. They admit selfadjoint extensions because they commute
with the conjugation (Cψ)(x)=ψ(−x) (see remark 2.2.34). It is furthermore
possible to prove that both operators are essentially selfadjoint by direct use of
Proposition 2.2.33[ 5 ]. However we already know thatPm′′is essentially selfadjoint
as it coincides with the essentially selfadjoint operatorFˆ†Km′′FˆbeacauseS(Rn)
is invariant underFˆ.
The unique selfadjoint extension of both operators turns out to bePm.We
conclude thatC∞ 0 (Rn;C)andS(Rn)arecoresfor them-axis momentum operator.
With the given definitions of selfadjoint operators,XkandPk,S(Rn) turn out
to be an invariant domain and thereon the CCR (2.21) hold rigorously.
As a final remark to conclude, we say that, ifn=1,D(P) coincides to the
already introduced domain (2.17). In that domain,P is nothing but the weak
derivative times the factor−i.
(3)The most elementary example of application of Nelson’s criterion is in
L^2 ([0,1],dx). ConsiderA=−d

2
dx^2 with dense domainD(A) given by the functions
in C∞([0,1];C) such thatψ(0) = ψ(1) and dψdx(0) = dψdx(1). Ais symmetric
thereon as it arises immediately by integration by parts, in particular its domain
is dense since it includes the Hilbert basis of exponentialsei^2 πnx,n∈Z,whichare
eigenvectors ofA.ThusAis also essentially selfadjoint on the above domain.
A more interesting case is the Hamiltonian operator of the harmonic
oscillator, H(see the first part) obtained as follows. One starts by

H 0 =−^1

2 m

d^2

dx^2

+mω

2

2

x^2