Chapter 2
Mathematical Foundations of Quantum
Mechanics: An Advanced Short Course
Valter Moretti
Department of Mathematics of the University of Trento and INFN-TIFPA,
via Sommarive 14, I-38122 Povo (Trento), Italy.
email: [email protected]Within these lectures, I review the formulation of Quantum Mechanics, and quantum theories
in general, from a mathematically advanced viewpoint essentially based on the orthomodular
lattice of elementary propositions, discussing some fundamental ideas, mathematical tools and
theorems also related to the representation of physical symmetries. The final step consists of an
elementary introduction of the so-called (C*-) algebraic formulation of quantum theories.
2.1 Introduction: Summary of Elementary Facts of QM
A concise account of the basic structure of quantum mechanics andquantization
procedureshas already been extensively presentedinthefirstpartofthisbook,with
several crucial examples. In the rest of Section 1, we quickly review again some
elementary facts and properties, either of physical or mathematical nature, related
to Quantum Mechanics, without fully entering into the mathematical details.
Section 2 is instead devoted to present some technical definitions and results of
spectral analysis in complex Hilbert spaces, especially the basic elements of spec-
tral theory, including the classic theorem about spectral decomposition of (gen-
erally unbounded) selfadjoint operators and the so called measurable functional
calculus. A brief presentation of the three most important operator topologies for
applications in QM closes Section 2.
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