Spectral Theory and Quantum Mechanics

1. Front Matter

   Pages i-xxii

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2. Introduction and Mathematical Backgrounds

   • Valter Moretti

   Pages 1-38

3. Normed and Banach Spaces, Examples and Applications

   • Valter Moretti

   Pages 39-106

4. Hilbert Spaces and Bounded Operators

   • Valter Moretti

   Pages 107-196

5. Families of Compact Operators on Hilbert Spaces and Fundamental Properties

   • Valter Moretti

   Pages 197-249

6. Densely-Defined Unbounded Operators on Hilbert Spaces

   • Valter Moretti

   Pages 251-288

7. Phenomenology of Quantum Systems and Wave Mechanics: An Overview

   • Valter Moretti

   Pages 289-301

8. The First 4 Axioms of QM: Propositions, Quantum States and Observables

   • Valter Moretti

   Pages 303-392

9. Spectral Theory I: Generalities, Abstract \(C^*\)-Algebras and Operators in \({\mathfrak B}({\mathsf H})\)

   • Valter Moretti

   Pages 393-471

10. Spectral Theory II: Unbounded Operators on Hilbert Spaces

    • Valter Moretti

    Pages 473-538

11. Spectral Theory III: Applications

    • Valter Moretti

    Pages 539-593

12. Mathematical Formulation of Non-relativistic Quantum Mechanics

    • Valter Moretti

    Pages 595-664

13. Introduction to Quantum Symmetries

    • Valter Moretti

    Pages 665-792

14. Selected Advanced Topics in Quantum Mechanics

    • Valter Moretti

    Pages 793-865

15. Introduction to the Algebraic Formulation of Quantum Theories

    • Valter Moretti

    Pages 867-913

16. Back Matter

    Pages 915-950

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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."

• Functional Analysis on Hilbert Spaces
• Mathematical Physics
• Quantum Mechanics
• Spectral Theory
• Algebraic formalism

“This new edition has confirmed the important commitment to provide students and researchers with an exhaustive and self-contained presentation of the mathematical foundations of quantum mechanics, enriched by many exercises, often with solution, and an extensive literature. … The spectral theory and a due treatment of unbounded operators still provide the core of the book, which however provides the interested reader with rigorous statements and proofs for all the many different chapters of mathematics relevant to non-relativistic quantum mechanics.” (Bassano Vacchini, zbMATH 1395.81001, 2018)

• Department of Mathematics, University of Trento, Povo, Trento, Italy

  Valter Moretti

Prof. Valter Moretti is a Full Professor of Mathematical Physics at the University of Trento (Italy). His research interests include rigorous quantum field theory in curved spacetime, mathematical aspects of quantum mechanics and general relativity, the applications of operator algebras, functional analysis and global analysis to quantum field theory, and mathematical (analytic and geometric) methods for physics.

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