Fundamental Mathematical Structures of Quantum Theory

Quantum theory is built upon a remarkably rich mathematical structure, where concepts such as states, observables, probability, symmetry, and measurement acquire precise analytical and algebraic meaning. This book provides a comprehensive and rigorous introduction to that structure, presenting the mathematical foundations of quantum theory as a coherent whole. Beginning with the Hilbert-space formulation, the text develops spectral theory, operator theory, quantum logic, von Neumann and C*-algebras, symmetries, superselection rules, and the algebraic formulation of quantum theory before culminating in a modern treatment of generalized quantum measurements. Particular attention is devoted to the conceptual foundations of quantum mechanics, including contextuality, hidden-variable theories, Bell inequalities, entanglement, and the mathematical aspects of quantum probability. Extensively revised and significantly expanded, this second edition introduces two new chapters on contemporary quantum measurement theory, covering effects, POVMs, quantum operations, channels, instruments, Stinespring and Naimark dilation theorems, and related foundational issues. Throughout the book, physical motivation accompanies rigorous mathematical development, with complete proofs, numerous examples, and solved exercises. Designed for advanced Master's and PhD students as well as researchers in mathematics and theoretical physics, this volume serves both as a graduate textbook and as a lasting reference on the mathematical and conceptual foundations of quantum theory.