From Classical Mechanics to Quantum Field Theory

214 From Classical Mechanics to Quantum Field Theory. A Tutorial

and satisfy

Nˆ|f 1 ,f 2 〉=2|f 1 ,f 2 〉.

Then-particle states can be identified with

|f 1 ,f 2 ,···,fn〉=√^1

n!

a(f 1 )†a(f 2 )†···a(fn)†| 0 〉,

and the number operator satisfies

Nˆ|f 1 ,f 2 ,···,fn〉=n|f 1 ,f 2 ,···,fn〉.
Now because of the commutation properties of the bosonic field operators, the
space of states withn-particles is not the tensor productH⊗nofnHilbert spaces of
one-particle statesH. Instead it can be identified with the subspace of symmetric
states involvingnparticles,
sH⊗n⊂H⊗H⊗···⊗Hn =H⊗n,

in the space of quantum states ofndistinguishable particles.
In this sense, the bosonic nature of the commutation relations implies the
bosonic statistics of the corresponding particles. To some extent this example
illustrates the existence of a link between the spin of the fields and the statistics
of the corresponding particles. In general, for any field theory the spin-statistics
connection follows from fundamental principles (spin-statistics theorem)[ 25 ].
The Fock space is the Hilbert space of all multiparticle bosonic states^9 ,

F=

⊕∞

n=0

sH⊗n.

Fis the Hilbert space of the (bosonic) quantum field theory. In the free theory
the HamiltonianHˆ and the number of particles operatorNˆcommute. Thus, all
energy levels have a definite number of particles. However, in the presence of
interactions this is not longer true, e.g. for

V(φ)=

1

2 m

(^2) φ (^2) +λ
4!φ
4
we have that
[N,Vˆ ]=0,
which means that the number of particles might change by time evolution. This is
one of the novel characteristics of quantum field theory. New process like, decaying
of particles, pair creation and photon emission in atoms can occur in the theory.
(^9) See the first part of this volume.