From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 85

find the known form of the Schr ̈odinger equation,

i

dψt

dt

=−

^2

2 m

Δψt+Uψt,

whenψτ ∈S(R^3 )forτ varying in a neighborhood oft(this requirement may
be relaxed). Actually the meaning of the derivative on the left-hand side should
be specified. We only say here that it is computed with respect to the natural
topology ofL^2 (R^3 ,d^3 x).

2.2 Observables in Infinite Dimensional Hilbert Spaces:
Spectral Theory

The main goal of this section is to present a suitable mathematical theory, suf-
ficient to extend to the infinite dimensional case the mathematical formalism of
QM introduced in the previous section. As seen in Sect. 2.1.5, the main issue
concerns the fact that, in the infinite dimensional case, there are operators rep-
resenting observables which do not have proper eigenvalues and eigenvectors, like
XandP. So, naive expansions as (2.4) cannot be literally extended to the gen-
eral case. These expansions, together withthe interpretation of the eigenvalues
as values attained by the observable associated with a selfadjoint operator, play
a crucial rˆole in the mathematical interpretation of the quantum phenomenol-
ogy introduced in Sect. 2.1.1 and mathematically discussed in Sect. 2.1.2. In
particular, we need a precise definition of selfadjoint operator and a result re-
garding a spectral decomposition in the infinite dimensional case. These tools
are basic elements of the so called spectral theory in Hilbert spaces, literally
invented by von Neumann in his famous book [ 7 ]to give a rigorous form to
Quantum Mechanics and successively developed by various authors towards many
different directions of pure and applied mathematics. The same notion of ab-
stract Hilbert space, as nowadays known, was born in the second chapter of that
book, joining and elaborating previous mathematical constructions by Hilbert and
Riesz. The remaining part of this section is devoted to introduce the reader to
some basic elements of that formalism. Reference books are, e.g., [8; 5; 6; 9;
10 ].

2.2.1 Hilbert spaces

AHermitian scalar productover the complex Hilbert spaceHis a map

〈·,·〉:H×H→C

such that, fora, b∈Candx, y, z∈H,