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.
From the reviews:
“The present text is an extended translation of its original Italian version published 2010 and gives a comprehensive introduction to the mathematical tools used in quantum mechanics. … The text is well-written and gives a self-contained and concise introduction to these topics. Many exercises make it suitable as a basis for various courses in this area.” (G. Teschl, Monatshefte für Mathematik, 2013)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.