Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation (UNITEXT Book 110)

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." 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 This book pursues the accurate study of the mathematical foundations of Quantum Theories. It may be considered an introductory text on linear functional analysis with a focus on Hilbert spaces. Specific attention is given to spectral theory features that are relevant in physics. Having left the physical phenomenology in the background, it is the formal and logical aspects of the theory that are privileged.Another not lesser purpose is to collect in one place a number of useful rigorous statements on the mathematical structure of Quantum Mechanics, including some elementary, yet fundamental, results on the Algebraic Formulation of Quantum Theories.In the attempt to reach out to Master's or 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 should benefit established researchers to organise and present the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly.