Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation

ISBN-13: 9783319707051 / Angielski / Miękka / 2018 / 950 str.

Valter Moretti

Opis

This mathematically rigorousm self-contained book explains the foundations of Quantum Mechanics and Quantum Theories in general. Includes a primer on general topology and abstract measure theory, along with material in the appendices on differential geometry.

Recenzje

"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)

Spis treści

1 Introduction and Mathematical Backgrounds.- 2 Normed and Banach Spaces, Examples and Applications.- 3 Hilbert Spaces and Bounded Operators.- 4 Families of Compact Operators on Hilbert Spaces and Fundamental Properties.- 5 Densely-Defined Unbounded Operators on Hilbert Spaces.- 6 Phenomenology of Quantum Systems and Wave Mechanics: an Overview.- 7 The First 4 Axioms of QM: Propositions, Quantum States and Observables.- 8 Spectral Theory I: Generalities, Abstract C -algebras and Operators in B(H).- 9 Spectral theory II: Unbounded Operators on Hilbert Spaces.- 10 Spectral Theory III: Applications.- 11 Mathematical Formulation of Non-Relativistic Quantum Mechanics.- 12 Introduction to Quantum Symmetries.- 13 Selected Advanced Topics in Quantum Mechanics.- 14 Introduction to the Algebraic Formulation of Quantum Theories.- 15 Appendix A: Order Relations and Groups.- 16 Appendix B: Elements of Differential Geometry.

O autorze

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.

Blurb

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.