CONTENTS -- Preface -- Memorial -- Part A Introductions -- The Mathematical Basis for Deterministic Quantum Mechanics G. 't Hooft -- 1. Introduction -- 2. Variables, Beables and Changeables -- 3. The Harmonic Oscillator -- 4. Two (or more) Harmonic Oscillators -- 5. Energy and Hamiltonian -- 6. Interactions -- 7. Limit Cycles -- 8. Discussion -- Appendix A. -- 9. The random deterministic model -- References -- What Did We Learn from Quantum Gravity? A. Ashtekar -- 1. Introduction -- 2. Quantum Riemannian Geometry -- 3. Application: Homogeneous Isotropic Cosmology -- 4. Discussion -- Acknowledgments -- References -- Bose-Einstein Condensates and EPR Quantum Non-Locality F. Lalo e -- 1. EPR Argument and Its Refutation by Bohr -- 2. Detecting the Transverse Direction of Spins -- calculation -- 3. Physical Discussion -- 3.1. Role of the Integral -- 3.2. Small and Big Condensates -- Ampli cation During Measurement -- 3.3. EPR Non-locality with Fock States -- 4. Possible Objections -- 5. Conclusion -- Acknowledgments -- References -- The Quantum Measurement Process: Lessons from an Exactly Solvable Model A. E. Allahverdyan, R. Balian and Th. M. Nieuwenhuizen -- 1. Introduction -- 2. The model and its solution -- 2.1. The Hamiltonian -- 2.2. Disappearance of Schr odinger cats -- 2.2.1. Dephasing -- 2.2.2. Decoherence -- 2.3. Registration of the measurement -- 3. Microscopic versus macroscopic aspects of quantum measurement -- 4. Measurement, a process of quantum statistical mechanics -- 5. Irreversibility -- 6. Meaning of von Neumann's reduction and of Born's rule -- 7. Relation to the pre-measurement -- 8. Statistical interpretation of quantum mechanics -- Acknowledgement -- References -- Part B Quantum Mechanics and Quantum Information -- POVMs: A Small but Important Step Beyond Standard Quantum Mechanics W. M. de Muynck -- 1. Introduction.

2. Complementarity -- 2.1. The Summhammer, Rauch, Tuppinger experiment -- 2.2. The Martens inequality -- 2.3. Martens inequality versus Heisenberg uncertainty relation -- 3. Bell Inequalities -- 3.1. Generalized Aspect experiment -- 3.2. Complementarity and nonlocality as alternative explanations of violation of the Bell inequalities -- References -- State Reduction by Measurements with a Null Result G. Nienhuis -- 1. Introduction -- 2. Macroscopic Fluorescence Switching -- 2.1. Evolution of density matrix -- 2.2. Intensity correlation and waiting-time distribution -- 2.3. Single histories -- 3. Spontaneous Decay as a Quantum Trajectory -- 4. Discussion -- References -- Solving Open Questions in the Bose{Einstein Condensation of an Ideal Gas via a Hybrid Mixture of Laser and Statistical Physics M. Kim, A. Svidzinsky and M. O. Scully -- 1. Introduction -- 2. Laser Master Equation Analysis of BEC Statistics -- 2.1. Low temperature approximation -- 2.2. Quasithermal approximation -- 3. A Very Accurate Hybrid Approach -- 3.1. Cumulants of BEC uctuation for an ideal Bose gas -- 3.2. Hybrid approach to condensate uctuations -- Acknowledgments -- Appendix A. Mean number of condensate particles: cusp vs smooth crossover -- References -- Twin-Photon Light Scattering and Causality G. Puentes, A. Aiello and J. P. Woerdman -- 1. Introduction -- 2. Our experiments -- 3. Causality condition -- 4. Scattering processes as trace-preserving and non-trace-preserving quantum maps -- 5. Non-trace-preserving maps and the causality condition -- 6. Conclusion -- Acknowledgments -- References -- Simultaneous Measurement of Non-Commuting Observables G. Aquino and B. Mehmani -- 1. Introduction -- 2. Statement of the Problem -- 3. Spin-1 Assistant in a Known Pure State -- 4. Assistant System as a Coherent State of Light -- 5. Conclusions -- Acknowledgments -- References.

Quantum Decoherence and Gravitational Waves M. T. Jaekel, B. Lamine, A. Lambrecht, S. Reynaud and P. Maia Neto -- 1. Introduction -- 2. Gravitational Environment -- 3. Quantum Decoherence of Atomic Interferometers -- 4. Quantum Decoherence of Planetary Systems -- 5. Gravitational Quantum Decoherence -- References -- Role of Various Entropies in the Black Hole Information Loss Problem Th. M. Nieuwenhuizen and I. V. Volovich -- 1. Introduction -- 2. Different Kinds of Entropies -- 3. On the Black Hole Information Paradox -- 4. Information Loss in Gases -- 5. Information Loss in Quantum Gravity -- 6. Conclusions -- Acknowledgments -- References -- Quantum and Super-Quantum Correlations G. S. Jaeger -- 1. Introduction -- 2. Quantum Correlations -- 3. Super{Quantum Correlations -- 4. The Collapse of Super{Quantum Complexity Theory -- 5. Beyond the Quantum -- 6. The Simulation of Quantum Correlations -- References -- Part C Long Distance Correlations and Bell Inequalities -- Understanding Long{Distance Quantum Correlations L. Marchildon -- 1. Introduction -- 2. Long-Distance Correlations -- 3. Copenhagen and Related Views -- 4. Bohmian Mechanics -- 5. The Transactional Interpretation -- 6. Extended Parameter Space -- 7. Summary and Conclusion -- Acknowledgments -- References -- Connection of Probability Models to EPR Experiments: Probability Spaces and Bell's Theorem K. Hess and W. Philipp -- 1. Introduction -- 2. Random Variables, EPR Experiments and the Theorem of Bell -- 3. Connection of Quantum Mechanics to Experiments -- 4. Connection of Kolmogorov Type Models to Experiments -- 5. Connecting Quantum Mechanics and Kolmogorov's Probability Theory -- 6. Probability Spaces, Time Evolution, Non-Commuting Operators, Closed Loops -- 6.1. Different probability spaces and unitary time evolution -- 6.2. Probability spaces, joint distributions and commutativity.

6.3. Probability spaces and EPR experiments -- 7. Probability Spaces and Relativity: Physics and the Sample Space -- 8. Discussion and Conclusions -- References -- Fair Sampling vs No-Signalling Principle in EPR Experiments G. Adenier and A. Yu. Khrennikov -- 1. Introduction -- 2. Channel efficiencies -- 3. Normalizing using the single counts -- 4. Experimental Results -- 5. Interpretation -- 6. Conclusion -- Acknowledgments -- References -- Part D Mathematical Foundations -- Where the Mathematical Structure of Quantum Mechanics Comes From G. M. D'Ariano -- 1. Introduction -- 2. The Postulates -- 3. The Statistical and Dynamical Structure -- 4. Banach Structure -- 5. Observables -- 6. Dimensionality Theorems -- 7. The Complex Hilbert Space Structure for Finite Dimensions -- 8. Infinite Dimension: the C -Algebra of Transformations -- References -- Phase Space Description of Quantum Mechanics and Non-Commutative Geometry: Wigner{Moyal and Bohm in a Wider Context B. J. Hiley -- 1. Introduction -- 2. The Moyal Approach -- 3. Relation to the Bohm Approach -- 4. The Star-Product and Its Properties -- 5. Time Dependent Equations in the Moyal Approach -- 6. The Operator Formalism -- 7. Conclusion -- References -- Quantum Mechanics as Simple Algorithm for Approximation of Classical Integrals A. Yu. Khrennikov -- 1. Introduction -- 2. Finite Dimensional Quantum Mechanics -- 2.1. Quantum mechanics over real numbers -- 2.2. Quantum mechanics over complex numbers -- 3. Illustration for the von Neumann Trace{Class Formula in the One Dimensional Case -- 4. Finite Dimensional Quantum Mechanics Over Reals as an Approximative Theory -- 5. Prequantum Phase Space { the Two Dimensional Case -- 6. Prequantum Phase Space for Finite Dimensional Quantum Mechanics -- References.

Noncommutative Quantum Mechanics Viewed from Feynman Formalism J. Lages, A. Berard, H. Mohrbach, Y. Grandati and P. Gosselin -- 1. Introduction -- 2. Mathematical Foundations of Feynman's Formalism -- 3. Maxwell's Equations -- 4. SO(3) Algebra and Poincar e Momentum -- 5. Noncommutative Quantum Mechanics -- 5.1. Jacobi Identities -- 5.2. Position Transformation -- 5.3. Field Properties -- 5.4. Angular Momentum -- 5.5. Physical Realization -- 6. Conclusion -- References -- Beyond the Quantum in Snyder Space J. F. S. van Huele and M. K. Transtrum -- 1. Why Go Beyond the Quantum? -- 2. Go Beyond What Quantum? -- 3. Snyder's Goes Beyond the Quantum -- 4. Snyder Space Essentials -- 5. Snyder Space Solutions -- 6. Snyder Space Overlapping Beyond the Quantum -- 7. Lessons Learned from Snyder Space Beyond the Quantum -- References -- Part E Stochastic Electrodynamics -- Some Quantum Experiments from the Point of View of Stochastic Electrodynamics V. Spi cka, J. J. Mare s, P. Hub k and J. Kri sto k -- 1. Introduction -- 1.1. Contents -- 2. Development of concepts: QED and SED -- 2.1. Notes to history of quantum mechanics, QED and SED -- 2.1.1. Essential problems of QED: in nities and the structure of vacuum -- 2.1.2. Summary of the early development of quantum mechanics and QED -- 2.1.3. Randomness and discussion of Earnshaw's theorem: a way to SED -- 2.2. Stochastic approach to quantum phenomena -- 2.2.1. Stochastic quantum mechanics -- 2.2.2. Stochastic electrodynamics -- 2.2.3. Quantum phenomena explained by SED -- 3. Black-body radiation spectrum and stochastic electrodynamics -- 3.1. Black-body radiation spectrum and quantum mechanics -- 3.2. Zero-point radiation -- 3.2.1. Lorentz invariance of ZPR spectral distribution -- 3.2.2. Divergence of ZPR spectrum and natural cut-off -- 3.2.3. Black-body radiation spectrum in SED.

3.3. Measurement of black-body radiation and ZPR spectrum.