182 From Classical Mechanics to Quantum Field Theory. A Tutorial
formalise, for example, field theories in curves spacetime in relationship to the
quantum phenomenology of black-hole thermodynamics.
2.4.1 Algebraicformulation....................
The algebraic formulation prescinds, anyway, from the nature of the quantum
system and may be stated for systems with finitely many degrees of freedom as
well[ 25 ]. The new viewpoint relies upon two assumptions[26; 27; 25; 29; 5].
(AA1)AphysicalsystemSis described by itsobservables,viewednowas
selfadjoint elements in a certainC∗-algebraAwith unit 11 associated toS.
(AA2)Analgebraic stateonASis a linear functionalω:AS→Csuch
that:
ω(a∗a)≥ 0 ∀a∈AS,ω(11) = 1,that is,positiveandnormalized to1.
We have to stress thatAis not seen as a concreteC∗-algebra of operators (a von
Neumann algebra for instance) on a givenHilbert space, but remains an abstract
C∗-algebra. Physically,ω(a)istheexpectation valueof the observablea∈Ain
stateω.
Remark 2.4.1.
(a)Ais usually calledthe algebra of observables ofSthough, properly speaking,
the observables are the selfadjoint elements ofAonly.
(b)Differently form the Hilbert space formulation, the algebraic approach can
be adopted to describeboth classical and quantum systems. The two cases are dis-
tinguished on the base of commutativity of the algebra of observablesAS: A commu-
tative algebra is assumed to describe a classical system whereas a non-commutative
one is supposed to be associated with a quantum systems.
(c)The notion ofspectrumof an elementaof aC∗-algebraA,withunit
element 11 , is defined analogously to the operatorial case[ 5 ]. σ(a):=C\ρ(a)
where we have introduced theresolventset:
ρ(a):={λ∈C|∃(a−λ 1 1)−^1 ∈A}.When applied to the elements ofB(H), this definition coincides with the one provi-
ously discussed for operators in view of (2) in exercise 2.2.43. It turns out that if
a∗a=aa∗,namelya∈Ais normal, then
||a||=sup
λ∈σ(a)|λ|.