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This book explores the Hilbert space formulation of quantum mechanics and its measurement theory, tracing the evolution of the Mathematical Foundations of Quantum Mechanics since von Neumann’s classic work. It delves into fundamental non-classical features of quantum mechanics—such as indeterminacy, incompatibility of observables, unavoidable measurement disturbance, entanglement, and nonlocality—using operational quantum theory. The content is organized into four parts: 1. **Mathematics** systematically presents the Hilbert space and operator theoretic tools, along with relevant measure and integration theory, culminating in the Naimark and Stinespring dilation theorems. 2. **Elements** develops key concepts of quantum mechanics and measurement theory, emphasizing approximate joint measurability. 3. **Realisations** offers detailed studies of fundamental observables in quantum mechanics and their measurement implementations. 4. **Foundations** addresses foundational topics such as quantum-classical contrast, Bell nonlocality, measurement limitations, the measurement problem, and operational axioms from a measurement theoretic viewpoint. The book is intended for physicists, mathematicians, and philosophers of physics interested in the mathematical and conceptual foundations of quantum physics, particularly through the lens of measurement theory.

The amazing accuracy in verifying quantum effects experimentally has recently renewed interest in quantum mechanical measurement theory. In this book the authors give within the Hilbert space formulation of quantum mechanics a systematic exposition of the quantum theory of measurement. Their approach includes the concepts of unsharp objectification and of nonunitary transformations needed for a unifying description of various detailed investigations. The book addresses advanced students and researchers in physics and philosophy of science. In this second edition Chaps. II-IV have been substantially rewritten. In particular, an insolubility theorem for the objectification problem has been formulated in full generality, which includes unsharp object observables as well as unsharp pointers.