Mathematics Of Open Quantum Systems, The: Dissipative And Non-unitary Representations And Quantum Measurements

Description

This book presents new developments in the open quantum systems theory with emphasis on applications to the (frequent) measurement theory. In the first part of the book, the uniqueness theorems for the solutions to the restricted Weyl commutation relations braiding unitary groups and semi-groups of contractions are discussed. The major theme involves an intrinsic characterization of the simplest symmetric operator solutions to the Heisenberg uncertainty relations, the problem posed by Jørgensen and Muhly, followed by the proof of the uniqueness theorems for the simplest solutions to the restricted Weyl commutation relations. The detailed study of unitary invariants of the corresponding dissipative and symmetric operators opens up a look at the classical Stone-von Neumann uniqueness theorem from a new angle and provides an extended version of the uniqueness result relating various realizations of a differentiation operator on the corresponding metric graphs. The second part of the book is devoted to mathematical problems of the quantum measurements under continuous monitoring. Among the topics discussed are the complementarity of the Quantum Zeno effect and Exponential Decay scenario in frequent quantum measurements, and a rigorous treatment, within continuous monitoring paradigm, of the celebrated "double-slit experiment" where the renowned exclusive and interference measurement alternatives approach in quantum theory is presented in a way that is accessible for mathematicians. One of the striking applications of the generalized (1-stable) central limit theorem is the mathematical evidence of exponential decay of unstable states of the quantum pendulum under continuous monitoring. CONTENTS Acknowledgments Preface Part 1: Representations of Operator Commutation Relations 1. Introduction 2. Preliminaries and Basic Definitions 3. The Commutation Relations and Character-Automorphic Functions 4. The Differentiation Operator on Metric Graphs 5. The Magnetic Hamiltonian 6. The Livšic Function s(D,DΘ)(z) 7. The Weyl-Titchmarsh Function M(D,DΘ)(z) 8. The Model Dissipative Operators 9. The Characteristic Function S(D,D,DΘ)(z) 10. The Transmission Coefficient and the Characteristic Function 11. Uniqueness Results 12. Dissipative Solutions to the CCR 13. Main Results 14. Unitary Dynamics on the Full Graph Part 2: Continuous Monitoring, Quantum Measurements 15. Continuous Monitoring of the Quantum Systems 15.1. Quantum Zeno Effect 15.2. Anti-Zeno Effect 15.3. The Exponential Decay 15.4. Frequent Measurements and the Time-Energy Uncertainty Principle 15.5. Presto 16. The Quantum Zeno versus Anti-zeno Effect Alternative 17. The Quantum Zeno Effect versus Exponential Decay Alternative 18. Preliminaries: Probabilities versus Amplitudes 19. Massless Particles on a Ring 20. Continuous Monitoring with Interference 20.1. Discussion 21. Continuous Monitoring with No Interference 21.1. Discussion 22. The Self-adjoint Dilation 23. General Open Quantum Systems on a Ring 23.1. Discussion 23.2. Random Phase Method 24. Operator Coupling Limit Theorems Appendix A The Characteristic Function for Rank-One Perturbations Appendix B Prime Symmetric Operators Appendix C A Functional Model of a Triple Appendix D The Spectral Analysis of the Model Dissipative Operator Appendix E Transformation Laws Appendix F The Invariance Principle Appendix G The Operator Coupling and the Multiplication Theorem Appendix H Stable Laws References Index