From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 127

q^1 ,...,qn,p 1 ,...,pn. The state of the system at timetis a points∈Γ, in local
coordinatess≡(q^1 ,...,qn,p 1 ,...,pn), whose evolutionRt→s(t)isasolution
of theHamiltonian equationof motion. Always in local symplectic coordinates,
they read

dqk

dt

=

∂H(t, q, p)

∂pk

,k=1,...,n, (2.71)

dpk

dt

=−

∂H(t, q, p)

∂qk

,k=1,...,n, (2.72)

Hbeing the Hamiltonian function of the system, depending on the (inertial) refer-
ence frame. Every physicalelementary property,E, that the system may possess
at a certain timet, i.e., which can be true or false at that time, can be identified
with a subsetE⊂Γ. The property is true ifs∈Eand it is not ifs∈E.Fromthis
point of view, the standard set theory operations∩,∪,⊂,¬(where¬E:= Γ\E
from now on is thecomplement operation) have a logical interpretation:

(i)E∩Fcorresponds to the property “EANDF”;

(ii)E∪Fcorresponds to the property “EORF”;

(iii)¬Ecorresponds to the property “NOTF”;

(iv)E⊂Fmeans “EIMPLIESF”.

In this context:

(v) Γ is the property which is always true;

(vi)∅is the property which is always false.

This identification is possible because, as is well known, the logical operations have
the same algebraic structure of the set theory operations.
As soon as we admit the possibility to construct statements includingcountably
infinite number of disjunctions or conjunctions, we can enlarge our interpretation
towards the abstract measure theory, interpreting the states asprobability Dirac
measuressupported on a single point. To this end, we first restrict the class of pos-
sible elementary properties to the Borelσ-algebra of Γ,B(Γ). For various reasons,
this class of sets seems to be sufficiently large to describe physics (in particular
B(Γ) includes the preimages of measurable sets under continuous functions). A
state at timet,s∈Γ,canbeviewedasaDiracmeasure,δs, supported onsitself.
IfE∈B(Γ),δs(E)=0ifs∈Eorδs(E)=1ifs∈E.
If we do not have a perfect knowledge of the system, as for instance it happens
instatistical mechanics, the state at timet,μ, is a proper probability measure
onB(Γ) which now, is allowed to attain all values of [0,1]. IfE∈B(Γ) is an