This is a book about the Hilbert space formulation of quantum mechanics and its measurement theory. It contains a synopsis of what became of the Mathematical Foundations of Quantum Mechanics since von Neumann’s classic treatise with this title. Fundamental non-classical features of quantum mechanics—indeterminacy and incompatibility of observables, unavoidable measurement disturbance, entanglement, nonlocality—are explicated and analysed using the tools of operational quantum theory.

The book is divided into four parts: 1. Mathematics provides a systematic exposition of the Hilbert space and operator theoretic tools and relevant measure and integration theory leading to the Naimark and Stinespring dilation theorems; 2. Elements develops the basic concepts of quantum mechanics and measurement theory with a focus on the notion of approximate joint measurability; 3. Realisations offers in-depth studies of the fundamental observables of quantum mechanics and some of their measurement implementations; and 4. Foundations discusses a selection of foundational topics (quantum-classical contrast, Bell nonlocality, measurement limitations, measurement problem, operational axioms) from a measurement theoretic perspective.

The book is addressed to physicists, mathematicians and philosophers of physics with an interest in the mathematical and conceptual foundations of quantum physics, specifically from the perspective of measurement theory.

Table of Contents

1 Introduction

Part I Mathematics

2 Rudiments of Hilbert Space Theory

3 Classes of Compact Operators

4 Operator Integrals and Spectral Representations: The Bounded Case

5 Operator Integrals and Spectral Representations: The Unbounded Case

6 Miscellaneous Algebraic and Functional Analytic Techniques

7 Dilation Theory

8 Positive Operator Measures: Examples

Part II Elements

9 States, Effects and Observables

10 Measurement

11 Joint Measurability

12 Preparation Uncertainty

13 Measurement Uncertainty

Part III Realisations

14 Qubits

15 Position and Momentum

16 Number and Phase

17 Time and Energy

18 State Reconstruction

19 Measurement Implementations

Part IV Foundations

20 Bell Inequalities and Incompatibility

21 Measurement Limitations Due to Conservation Laws

22 Measurement Problem

23 Axioms for Quantum Mechanics

Review

“This book is addressed to students of physics or mathematics who like mathematically rigorous formulations and deductions. It may as well serve lecturers or researchers as a useful reference text. The reader gets well introduced into the physical problems and the mathematical tools used for their solution. … The content of this book is highly rich. … This book is best recommended.” (K.-E. Hellwig, zbMath 1416.81001, 2019)

“The book is constructed as a series of theorems, proofs, propositions, lemmas and corollaries … . Each chapter throughout the book is supplemented by a detailed set of references for further reading. Consequently, this monograph will be very useful as a reference work. … this is a volume that is really intended for mathematicians.” (Stephen J. Blundell, Contemporary Physics, Vol. 58 (4), September, 2017)