This book discusses the mathematical foundations of quantum theories. It offers an introductory text on linear functional analysis with a focus on Hilbert spaces, highlighting the spectral theory features that are relevant in physics. After exploring physical phenomenology, it then turns its attention to the formal and logical aspects of the theory.
Further, this Second Edition collects in one volume a number of useful rigorous results on the mathematical structure of quantum mechanics focusing in particular on von Neumann algebras, Superselection rules, the various notions of Quantum Symmetry and Symmetry Groups, and including a number of fundamental results on the algebraic formulation of quantum theories.
Intended for Master's and PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book also benefits established researchers by organizing and presenting the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly."
Author(s): Valter Moretti
Series: UNITEXT 110 / La Matematica per il 3+2
Edition: 2nd
Publisher: Springer
Year: 2018
Language: English
Pages: 950
Tags: Physics, Quantum
Front Matter ....Pages i-xxii
Introduction and Mathematical Backgrounds (Valter Moretti)....Pages 1-38
Normed and Banach Spaces, Examples and Applications (Valter Moretti)....Pages 39-106
Hilbert Spaces and Bounded Operators (Valter Moretti)....Pages 107-196
Families of Compact Operators on Hilbert Spaces and Fundamental Properties (Valter Moretti)....Pages 197-249
Densely-Defined Unbounded Operators on Hilbert Spaces (Valter Moretti)....Pages 251-288
Phenomenology of Quantum Systems and Wave Mechanics: An Overview (Valter Moretti)....Pages 289-301
The First 4 Axioms of QM: Propositions, Quantum States and Observables (Valter Moretti)....Pages 303-392
Spectral Theory I: Generalities, Abstract \(C^*\)-Algebras and Operators in \({\mathfrak B}({\mathsf H})\) (Valter Moretti)....Pages 393-471
Spectral Theory II: Unbounded Operators on Hilbert Spaces (Valter Moretti)....Pages 473-538
Spectral Theory III: Applications (Valter Moretti)....Pages 539-593
Mathematical Formulation of Non-relativistic Quantum Mechanics (Valter Moretti)....Pages 595-664
Introduction to Quantum Symmetries (Valter Moretti)....Pages 665-792
Selected Advanced Topics in Quantum Mechanics (Valter Moretti)....Pages 793-865
Introduction to the Algebraic Formulation of Quantum Theories (Valter Moretti)....Pages 867-913
Back Matter ....Pages 915-950
