Spectral Theory and Quantum Mechanics

With an Introduction to the Algebraic Formulation
2013
ISBN: 978-88-470-2834-0
Verlag: Springer Italia

Buch, Englisch, 728 Seiten, Book, Format (B × H): 155 mm x 235 mm

Reihe: La Matematica per il 3+2

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.

Zielgruppe

Graduate

Autoren/Hrsg.

Moretti, Valter

Inhaltsverzeichnis

Introduction and mathematical backgrounds.- Normed and Banach spaces, examples and applications.- Hilbert spaces and bounded operators.- Families of compact operators on Hilbert spaces and fundamental properties.- Densely-defined unbounded operators on Hilbert spaces.- Phenomenology of quantum systems and Wave Mechanics: an overview.- The first 4 axioms of QM: propositions, quantum states and observables.- Spectral Theory I: generalities, abstract C -algebras and operators in B(H).- Spectral theory II: unbounded operators on Hilbert spaces.- Spectral Theory III: applications.- Mathematical formulation of non-relativistic Quantum Mechanics.- Introduction to Quantum Symmetries.- Selected advanced topics in Quantum Mechanics.- Introduction to the Algebraic Formulation of Quantum Theories.- Order relations and groups.- Elements of differential geometry.

Über Autor(innen)

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