68 From Classical Mechanics to Quantum Field Theory. A Tutorial
Within Section 3, thecorpusofthelectures,wepasstoanalysethemathemati-
cal structure of QM from a finer and advanced viewpoint, adopting the framework
based on orthomodular lattices’ theory. This approach permits one to justify
some basic assumptions of QM, like the mathematical nature of the observables
represented by selfadjoint operators andthe quantum states viewed as trace class
operators. QM is essentially a probability measure on the non-Boolean lattice
L(H) of elementary observables. A key tool of that analysis is the theorem by
Gleason characterising the notion probability measure onL(H) in terms of certain
trace class operators. We also discuss the structure of the algebra of observables
in the presence of superselection rules after having introduced the mathematical
notion of von Neumann algebra. The subsequent part of the third section is de-
voted to present the idea of quantum symmetry, illustrated in terms of Wigner
and Kadison theorems. Some basic mathematical facts about groups of quantum
symmetries are introduced and discussed, especially in relation with the prob-
lem of their unitarisation. Bargmann’s condition is stated. The particular case
of a strongly continuous one-parameter unitary group will be analysed in detail,
mentioning von Neumann’s theorem and the celebrated Stone theorem, remarking
its use to describe the time evolution of quantum systems. A quantum formu-
lation of Noether theorem ends this part. The last part of Section 3 aims to
introduce some elementary results about continuous unitary representations of Lie
groups, discussing in particular a theorem by Nelson which proposes sufficient con-
ditions for lifting a (anti)selfadjoint representation of a Lie algebra to a unitary
representation of the unique simply connected Lie group associated to that Lie
algebra.
The last section closes the paper focussing on elementary ideas and results
of the so called algebraic formulation of quantum theories. Many examples and
exercises (with solutions) accompany the theoretical text at every step.
2.1.1 Physical facts about quantum mechanics
Let us quickly review the most relevant and common features of quantum sys-
tems. Next, we will present a first elementary mathematical formulation which
will be improved in the rest of the lectures, introducing a suitable mathematical
technology.
2.1.1.1 When a physical system is quantum
Loosely speaking, Quantum Mechanics is the physics of microscopic world (ele-
mentary particles, atoms, molecules). That realm is characterized by a universal
physical constant denoted byhand calledPlanck constant. A related constant