From Classical Mechanics to Quantum Field Theory

128 From Classical Mechanics to Quantum Field Theory. A Tutorial

elementary property of the physical system,μ(E) denotes the probability that the
propertyEis true for the system at timet.

Remark 2.3.1. The evolution equation ofμ, in statistical mechanics is given
by the well-known Liouville’s equationassociate with the Hamiltonian flow. In
that case,μis proportional to the natural symplectic volume measure ofΓ,Ω=
ω∧···∧ω(n-times, where 2 n=dim(Γ)). In fact, we haveμ=ρΩ,wherethe
non-negative functionρis the so-calledLiouville densitysatisfying the famous
Liouville’s equation. In symplectic local coordinates that equation reads

∂ρ(t, q, p)

∂t

+

∑n

k=1

(

∂ρ

∂qk

∂H

∂pk

− ∂ρ

∂pk

∂H

∂qk

)

=0.

We shall not deal any further with this equation in this section.

More complicated classical quantities of the system can be described byBorel
measurablefunctionsf:Γ→R. Measurability is a good requirement as it permits
one to perform physical operations like computing, for instance, theexpectation
value(at a given time) when the state isμ:

〈f〉μ=

∫

Γ

fμ.

Also elementary properties can be pictured by measurable functions, in fact they
are one-to-one identified with all the Borel measurable functionsg:Γ→{ 0 , 1 }.
The Borel setEgassociated togisg−^1 ({ 1 })andinfactg=χEg.
A generic physical quantity, a measurable functionf:Γ→R, is completely
determined by the class of Borel sets (elementary properties)E(Bf) :=f−^1 (B)

whereB∈B(R). The meaning ofE(Bf)is

E(Bf)= “the value offbelongs toB”. (2.73)

It is possible to prove[ 5 ]that the mapB(R)B→EB(f)permits one to reconstruct
the functionf.ThesetsEB(f):=f−^1 (B)formaσ-algebra as well and the class of

setsEB(f)satisfies the following elementary properties whenBranges inB(R).

(Fi)E(Rf)=Γ;

(Fii)EB(f)∩EC(f)=E(Bf∩)C;

(Fiii)IfN⊂Nand{Bk}k∈N⊂B(R) satisfiesBj∩Bk=∅ifk=j,then

∪j∈NEB(fj)=E(∪fj)∈NBj.

These conditions just say thatB(R)B→EB(f)is ahomomorpism ofσ-
algebras.