72 From Classical Mechanics to Quantum Field Theory. A Tutorial
vector space equipped with a Hermitian scalar product , henceforth denoted by
〈·,·〉, where the linear entry is the second one. WithHas above,L(H) will de-
note the complex algebra of operatorsA:H→H. We remind the reader that,
ifA∈L(H) withHfinite dimensional, theadjoint operator,A†∈L(H), is the
unique linear operator such that
〈A†x, y〉=〈x, Ay〉 for allx, y∈H. (2.2)Ais said to beselfadjointifA=A†, so that, in particular
〈Ax, y〉=〈x, Ay〉 for allx, y∈H. (2.3)Since〈·,·〉is linear in the second entry and antilinear in the first entry, we imme-
diately have that all eigenvalues of a selfadjoint operatorAare real.
Our assumptions on the mathematical description of quantum systems are the
following ones in agreement with the discussion in the first part:
- A quantum mechanical systemSis always associated to a complex vec-
tor spaceH(here finite dimensional) equipped with a Hermitian scalar
product〈·,·〉; - observables are pictured in terms ofselfadjointoperatorsAonH;
- states are equivalence classes ofunitvectorsψ ∈H,whereψ ∼ψ′iff
ψ=eiaψ′for somea∈R.
Remark 2.1.2.
(a) It is clear that states are therefore one-to-one represented by all of the
elements of the complex projective spacePH. The states we are considering within
this introductory section are calledpurestates. A more general notion of state,
already introduced in the first part, will be discussed later.
(b)His an elementary version of complex Hilbert space since it is automati-
cally complete, it being finite dimensional.
(c)Sincedim(H)<+∞, every selfadjoint operatorA∈L(H)admits a spectral
decomposition
A=∑
a∈σ(A)aPa(A), (2.4)whereσ(A)is thefiniteset of eigenvalues – which must berealasAis selfadjoint –
andPa(A)is the orthogonal projector onto the eigenspace associated toa.Notice
thatPaPa′=0ifa=a′as eigenvectors with different eigenvalue are orthogonal.
Let us show how the mathematical assumptions (1)-(3) permit us to set the phys-
ical properties of quantum systems (1)-(3) into a mathematically nice form.