Foundations and Interpretation of Quantum Mechanics

The following sections are included:

• SCHEME OF THE BOOK

• FOREWORD

• Contents

The following sections are included:

• Historical Premise

• Aims and Character of the Book

• Some Preliminary Epistemological Considerations

• Presentation of the Content

• Methodological Principles Followed in this Book

• Technical Instructions

• Acknowledgments

In that which follows we give a short review of some classical concepts.

Spaces For each physical system of n degrees of freedom we distinguish a coordinate configuration space ℜⁿ {q₁,q₂,…,q_n} and a momentum configuration space ℜⁿ {p₁,p₂,…,p_n}, where the q_j's (j = 1,…,n) are the generalized coordinates and the p_j's (j = 1,…,n) the generalized momenta, the variables pertinent to a description of a classical system. For a system Ѕ with n degrees of freedom the phase space Г is the set ℜ²ⁿ {q₁,q₂,…,q_n,p₁,p₂,…,p_n}…

• This part begins with an historical examination of the first steps of the theory [chapter 2].

• Chapter 3 starts with the interpretative difficulties regarding the new discoveries (waves or particles?) and with the classical two-slit experiment, and in a short presentation of the historical development from Heisenberg's article of 1925 until the formalisms of groups and propagators, synthesizes the mathematical basis of the theory (which we term Basic-QM) — which will be used in the following. Obviously, these results are basic only relative to later developments and not in themselves (for their time they were of foundational type).

• Chapter 4 is dedicated to the relativistic developments of the theory (in a very short synthesis) and particularly to the localization problem — from which important contributions of QM stochastic theories were later developed.

• Chapter 5 is devoted to quantum optics, which, in the last twenty years, has acquired an enormous weight in the testing of the theory.

The following sections are included:

• The Quantum Postulate
  • Exposition of QP1
  • Exposition of QP2

• Physical consequences of the Quantum Postulate
  • Classical Physical Description
  • Classical Measurement Problem and Continuity
  • Quantum Mechanics' Specific Problem

• Examination of the Quantum Postulate

The following sections are included:

• Antecedents and First Experimental Results

• First Attempts at an Independent Formalism

• Matrix Mechanics and Observables
  • A Non-Commutative Multiplication Rule
  • Observables as Operators

• Wave Mechanics
  • Waves
  • Dirac's algebra
  • Hilbert Spaces
  • Equivalence Between Matrix Mechanics and Wave Mechanics
  • Values and Observables
  • Superposition Principle

• States, Density Matrices and Projectors
  • Density Operator
  • Entangled States
  • Projectors
  • Spectral Theorem and Consequences

• Unitary Transformations
  • Beginnings
  • Time Evolution of the Wave function and states
  • Schrödinger, Heisenberg and Dirac Pictures
  • Different Representations

• Distribution and Characteristic Functions
  • Expectation Value
  • (Un)correlations
  • Characteristic Function and Variance

• Particles Statistics and Pauli Exclusion Principle

• Spin Theory

• Symmetries and Groups
  • Symmetries and Transformations
  • Groups
  • Galilei Group

• Scattering matrix and Propagators
  • Green Function
  • Path Integrals
  • Scattering Matrix

The following sections are included:

• Poincaré Group

• Klein/Gordon and Dirac Equations
  • Klein/Gordon Equation
  • Dirac Equation
  • Hole Theory

• Field Formalism
  • Klein/Gordon Field
  • Dirac Field

• Relativistic Scattering

• Localization
  • The Problem of Renormalization
  • The Problem of Sharp Localization

The following sections are included:

• General Formalism
  • States
  • Intensity and Visibility
  • Interaction Between Electromagnetic and Electron Wave fields
  • Photon Count
  • Coherent States

• Quantum Optics in Phase Space
  • Q-function
  • P-function
  • Characteristic Function
  • Wigner Function
  • Other Problems of Quantum Mechanics in Phase Space
  • Weyl Group
  • Interference in Phase Space

• Q-Optics Experimental Apparatus and Methods
  • Interferometry
  • Downconversions

This part is centered on the first, and the most important, global interpretation of the theory, the Copenhagen interpretation.

• In chapter 6, we first discuss Schrödinger's and Heisenberg's first attempts; then we examine the Ignorance interpretation (from Einstein promoted) and the Statistical interpretation.

• Then, in chapter 7, we examine the fundamental Uncertainty Principle: as we shall see, the principle's place in the theory and its character have been clarified in the last years by overcoming some traditional interpretations based on a supposed centrality of the disturbance or of the standard deviation.

• The part ends with the Complementarity Principle [chapter 8], different aspects of which are shown, and which are then partly discussed on the basis of recent experiments.

The following sections are included:

• Schrödinger's Interpretation
  • Exposition of Schrödinger's Interpretation
  • Examination of Schrödinger's Interpretation

• Heisenberg's First Interpretation

• Some Observations Concerning the Structure of the Theory

• Born's Probabilistic Interpretation

• The Ignorance and Statistical Interpretations
  • Ignorance Interpretation
  • Statistical Interpretation

• Discussion of the Ignorance and Statistical Interpretations
  • Examination of the Ignorance Interpretation
  • Examination of Statistical Interpretation
  • Concluding Remarks

The following sections are included:

• Heisenberg's formulation
  • The Origins of the Problem
  • General formalism and UR1
  • Other Uncertainty Relations
  • Formulation of the Uncertainty Principle and First Problems

• Bohr's Interpretation of the Principle
  • First discussion with Einstein
  • Second discussion

• Heisenberg's Interpretation: The Microscope Experiment

• Statistical Formulations of the Uncertainty Principle

• Some Problems
  • The Problem of Operators
  • The Problem of the Standard Deviation

• A New More General Formulation

The following sections are included:

• Definition of Complementarity

• Complementarity Between Conjugate Observables
  • Analysis
  • Some Consequences

• Analysis of the Wave/Particle Dualism
  • First Formulation
  • Second Formulation

• Complementarity Between Localization and Dynamic Laws
  • Formulation of the Principle
  • An Experiment

• Complementarity Between Separability and Phenomenon
  • General Features
  • Other Experiments
  • Classical Concepts and Apparatus

• Conclusions on the Copenhagen Interpretation

The aim of this part is to better develop the foundations of the formalism treated in the first two parts.

Four problems are considered:

• General axiomatic of the theory [chapter 9], where different attempts are shown to reduce the complexity of Basic-QM, particularly the ‘dualism’ between states (vectors) and observables (operators on vectors).

• The structure of Hilbert space [chapter 10], where particular attention is devoted to the problem of the observables which in Basic-QM are not represented by operators (time, energy, angle and phase), or which pose some problems in Relativistic-QM (position).

• The quantum probability [chapter 11], where particular attention is dedicated to the important Gleason theorem.

• Finally the geometric phase [chapter 12], which is a new — very abstract but also very promising — domain of the theory, particularly suitable for explaining phenomena such as the Aharonov/Bohm effect.

The following sections are included:

• Boolean Algebra and Some Other Technical Notions

• QM Axiomatic and Quantum Logic
  • Traditional Approach
  • Quantum Logic
  • Conclusions on Quantum Logic

• Other Approaches
  • Algebraic Approach
  • Convexity Approach
  • States/Observables Approach

• Concluding Remarks on the Axiomatic
  • Relation Between State and Observables
  • Models and Contributions

• QM Logical Calculus

The following sections are included:

• Is a Complex Hilbert Space Necessary?
  • Real and Quaternionic Hilbert Spaces
  • Deductions of Complex Hilbert Space
  • Wootters' Proposal
  • Superselection Rules

• Time and Energy Operators
  • General Remarks on the Relationship between Operators and Observables
  • A Short History of the Problem
  • Time-of-Arrival Operator
  • Another Proposal
  • Interfering Times

• Position Operator

• Angle Operator*

The following sections are included:

• Introduction to the Concept of Probability
  • Three Interpretations
  • Other Proposals
  • Some Lessons

• Classical Probability

• QM Probability

• Generalization to S–Dimensional Probabilities

• Gleason Theorem
  • Introduction
  • Probability Measure
  • Gleason Theorem
  • Proof of the Gleason Theorem

The following sections are included:

• Antecedents

• Berry's Contribution

• Generalization with Fiber Bundles*

• Experiments and Conclusions

The fourth part handles the first thematic area of the book and probably the most important part of the theory: the measurement problem.

• First we examine some basic concepts [chapter 13].

• Then we reconstruct von Neumann's original proposal and its refinements [chapter 14].

Von Neumann was the first to assume a dichotomy between two different types of evolutions: the unitary one and the measurement process. On this matter, different positions have been developed:

• there are those who have accepted such a dichotomy and tried to assign a fundamental function to the consciousness (after von Neumann: London, Bauer, Wigner): this is the content of chapter 14;

• some have tried to eliminate the jump-like features of the measurement by rejecting the projection postulate (Margenau, Ballentine, Everett, deWitt); this is the content of chapter 15;

• others have tried to explain measurement in terms of intervention of macrosystems: either as a limiting case of a thermodynamical evolution (Green, van Hove and others) or using SSRs for obtaining the required classical behaviour of the pointer: this is the issue in chapter 16;

• there are those who have tried to find some ad hoc parameters in order to explain the reduction (Ghirardi/Rimini/Weber): this will be treated later in chapter 23, because the proposal will account more generally for spontaneous transitions to localisation, and hence it is more like a theory of the relation between microphysics and macrophysics;

• others have developed a form of non-unitary evolution by using the specific nature of open systems: this is the theory of decoherence (Zurek, Cini, Machida/Namiki, among others), which is treated in chapter 17.

It is quit astonishing that measurement theory has led to so many different positions and that, consequently, for fifty years has been a problematic area of QM. The reasons will be briefly discussed in subsection 14.1.1.

The last three chapters of this part are devoted to new developments which partly extend and integrate the theory of decoherence.

• The operational approach [chapter 18] represents a formal generalization of the quantum theory of open systems.

• Chapter 19 is dedicated to the Quantum Non-demolition Measurement.

The last two chapters contain results which are generally valid and neutral with respect to a particular interpretation. Anyway, their importance for QM (and partly also for some interpretational problems) justifies the place they occupy here.

• Finally we investigate a new field of research which has very interesting interpretational implications and a lot of technological consequences: the so-called interaction-free measurements [chapter 20].

In the following we shall use the word ‘Measurement’ to mean the measurement process in general, while the word ‘measurement’ always indicates a particular measurement process.

The following sections are included:

• Measurement's Definition

• Elements of the Measurement

• Steps of the Measurement

• Premeasurement

• Types of Measurement

The following sections are included:

• Von Neumann's Original Exposition
  • Introduction
  • General Features of the Problem
  • Von Neumann's Solution
  • Poincaré-Sphere Representation of the Measurement
  • Necessity and Definition of the Apparatus
  • Discussion of the Copenhagen Interpretation and von Neumann's Interpretation

• Criticisms and Problems
  • The Observer
  • Limits of the Projection Postulate
  • Measurement and Relativity
  • Measurement and Conservation Laws

• Value Reproducibility and Objectification
  • Objectification Problem
  • State Transformer and Reduced Final States
  • Pointer Objectification
  • Value Objectification
  • Examination of the Objectification Condition

The following sections are included:

• Presentation of the Interpretation
  • Antecedents of the Interpretation
  • Everett's Formulation
  • DeWitt's Formulation

• Difficulties of the Many-World Interpretation

• Concluding Remarks

The following sections are included:

• Classicality as Superselection Rules
  • Classicality of the Pointer
  • Models of Measurement based on Superselection Rules
  • A Difficulty with the Classicality of the Pointer

• Semiclassical Apparatus
  • Unstable Apparatus
  • Van Hove's Master Equation*
  • Daneri/Loinger/Prosperi's Model
  • Conclusion

The following sections are included:

• A Recent and Yet Rich History
  • Hepp's Algebraic Approach*
  • Scully/Shea/McCullen's Reduced Density Matrix

• Zurek's Solution
  • General Features
  • General Formalism
  • Characters of the Measurement Process
  • Detailed Analysis of the Decoherence Process
  • The Action of a Large Environment

• Machida-Namiki Model

• Cini's Model

• An Easy Computer Model

• Concluding Remarks on Decoherence

The following sections are included:

• Fundamentals of Operational Stochastic Quantum Mechanics
  • Operations and Effects
  • Measurement Theory

• Stochastic Analysis of Non-commutativity and Complementarity
  • Uncertainty, Commutativity, and Coexistence
  • A Probabilistic Formulation of the Complementarity

• Covariance, Conservation Laws and Measurement
  • Davies Theorem
  • Noise Operators*

• Sharpness and Unsharpness: Optimal Measurements

• Self-Adjointness and Symmetry

• Informational Completeness and Measurement
  • Informational Completeness

• Statistical Treatment of Definition and Observation*
  • Decision Theory
  • Estimation Theory

• Weak Measurement
  • Generals Remarks on Weak Measurement
  • Weak Measurement and Time Order
  • An Example
  • A Generalization of the Example
  • Concluding Remarks on the Weak Measurement

• Discrimination Between Non-Orthogonal States*

• Stochastic Analysis of Spin*
  • Sharp Treatment
  • Unsharp Treatment
  • Proper Screen Observable
  • Determinative Measurement of Spin

The following sections are included:

• Brief History

• More on the Measurement Process

• Indirect Measurement

• Principles of Non-Demolition Measurement
  • Non-Demolition Measurement
  • Non-Demolition Observables

• Evaluation of the Minimal Measurement Error
  • An Example
  • Concluding Remarks

• QND-Measurement and the Uncertainty Principle
  • Linear Case
  • Nonlinear Case
  • Classical Forces

• Experimental Context

• Zeno Paradox
  • General Features
  • Watchdog Effect
  • Collapse or Not?
  • Zeno Effect and Decoherence*
  • Zeno Effect of a Single System

• Boundaries Between Apparatus and Object

The following sections are included:

• Renninger's Experiment

• Interaction-Free Interferometry

• More Recent Experiments

• Discussion

The following sections are included:

• Introduction to Part V
  • Schrödinger's Analysis
  • A Short Discussion
  • Content of Part V

The following sections are included:

• Heat Baths and Master Equation
  • Different Master Equations: General Formal Discussion
  • Indistinguishability of States

• Baths of Harmonic Oscillators
  • First Model of Decoherent Measurement
  • Walls-Milburn Model
  • A Non-Demolition Model of Decoherence

• Damping
  • The Model
  • High-Temperature Case
  • Low-Temperature Case
  • Physical Interpretation

• Ignorance and Dynamics: Baths and Damping

• Unruh/Zurek Model*

• Another Model

• Measure of Decoherence
  • General Measure of Purity and Decoherence
  • Decoherence Time-Scale After Zurek's Model
  • Decoherence Parameter After Machida/Namiki's Model
  • Coherence Range
  • Decoherence and Probability

• Conclusion

The following sections are included:

• General Theory
  • Photon Counting for ‘Classical’ Fields
  • Photoelectron Counting for Quantized Fields
  • Quantum Trajectories
  • Wave Function Approach

• Interpretation of Quantum Jumps and Trajectories

• Models and Experiments: Jumps as Telegraph Signals

The following sections are included:

• Classical Stochastic Theories
  • Hydrodynamics and QM
  • The Fokker Equation
  • A Development of the Probabilistic Interpretation
  • Nelson's Proposal
  • General Criticism

• Ghirardi/Rimini/Weber
  • Exposition of the Model
  • Discussion and Criticisms

• Stochastic Non-Linear Jump-Theories
  • Brief Exposition
  • Difficulties

The following sections are included:

• How to Construct a Schrödinger Cat

• Short History
  • Non-Linear Media Arrangements
  • Amplification Methods
  • Microcavity Experiments
  • Other Proposals

• Detailed Exposition of a Performed Experiment
  • Preparation
  • Generation of a Schrödinger Cat

• Schrödinger Cats and Decoherence
  • Schrödinger Cat
  • A Decoherence Experiment

• Conclusion
  • No Jump-Like Transition between Microphysics and Macrophysics
  • Persistence of Quantum Effects
  • Diagonalization, Dissipation and Reductionism
  • Self-Referentiality

As we have seen [chapter 21], there can be spontaneous decoherence, such that the interference terms fall to zero in a relatively short time. It is natural to ask: Is it possible to generalize that model in order to build a microtheory in which the interference is lost and a deterministic history is gained? This is the aim of the consistent or decoherent- histories approach. The question is very interesting on the cosmological level, because, if the answer is affirmative, then we can describe quantum mechanically a consistent history of our universe.

Another problem is the following: von Neumann assumed two types of evolution, an unitary reversible evolution and the reduction of the wave-packet. In part V we have showed how is possible to overcome the gap between micro- and macro-world postulated by von Neumann. But a problem remains: the basic equations of QM seem reversible, while decoherence seems to presuppose some type of reversibility. How to solve such a problem? Summing up:

• In chapter 25 we analyse different proposals to build cosmological decoherent wave functions.

• Then the ‘delayed choice’ issue is analysed theoretically and experimentally [chapter 26], because of its enormous consequence on the theory, and in particular on the interpretation of QM.

• While in chapter 26 time is treated of as an external parameter, in chapter 27 we discuss the problem of the reversibility and irreversibility of quantum systems, i.e. treating time as a dynamical observable of the system itself, following the discussion of the time operator [see subsection 10.2.3].

Even if the theme of this part has only taken a central place in scientific debate since the seventies, its connection with the preceding part and its importance for the successive two parts, justify its treatment here.

The following sections are included:

• Consistent Histories
  • General Theory
  • Criticism
  • Omnès' Proposal

• Gell-Mann/Hartle's Model
  • Wheeler/DeWitt Equation
  • General Formalism
  • A Symmetric Proposal
  • Discussion

• Yamada/Takagi Theorem

• Conclusions

The following sections are included:

• The Theoretical Problem

• Proposed Delayed Choice Experiments
  • Another Two-Slit Experiment
  • Again on the Heisenberg Microscope
  • Beam Splitting
  • Energy and Time
  • Other Examples

• A Delayed Choice Performed Experiment
  • The Experiment
  • Conclusion

• Lessons of the Delayed Choice
  • Definition and Measurement
  • The Present Determines the ‘Past’
  • Can the Universe be Treated as a Closed System?
  • Participation
  • Universality of Quantum Mechanics
  • Epistemological Aspects

The following sections are included:

• A Spin Experiment

• Haunted Measurements

• Schrödinger Cats and Revival Behaviour

• Montecarlo Methods
  • First Attempts
  • A ‘Unitarily Reversible’ Measurement
  • Is New Information Gained?
  • Formal Generalization of Unitarily Reversible Measurements

• Concluding Remarks

We have touched many times upon the problem of WP-Dualism. Now we shall analyse the problem, by starting with what is chronologically the first discussion about it. An organic answer to the problem came from de Broglie, but, for many reasons yet to be analysed in greater depth, it was not accepted by the majority of physicists. This theory was later included in the theoretical program of the Hidden-Variable theory by David Bohm, so that one normally speaks of the ‘de Broglie-Bohm Pilot wave’. However, de Broglie's proposal was directed more toward the problem of WP-Dualism as such, while Bohm's proposal was directed more toward the construction of a deterministic theory by contesting the completeness of QM — and here the point of departure is more Einstein's Ignorance interpretation [see proposition 6.3: p. 106] and the historical article of Einstein/Podolsky/Rosen [EINSTEIN et al. 1935]. As a consequence of the Hidden Variable theory the problem of non-locality was posed.

The problem of the WP-Dualism and that of Hidden Variable theories (and non-locality) are theoretically different and hence they require different examinations. Therefore we divide all matter into three different parts: in this part we shall discuss de Broglie's theory and the problem of WP-Dualism; in part VIII, the problem of completeness and determinism (Bohm's Hidden Variable theory); in part IX the problem of separability and non locality.

While part VIII, and especially chapter 33, is devoted to an interpretation of QM particles, we discuss in particular here what quantum waves are; if they are classical entities, if and how they can be ontologically interpreted, and what a quantum state is. We shall confirm our main distinction between states and observables.

• We begin [in chapter 28] with the classical discussion stimulated by de Broglie with his proposal of the double solution theory — a proposal following which both particles and waves always exist together. Then more recent proposals concerning the nature of quantum waves and of quantum states are discussed (non-analytical solutions, empty waves, ontological interpretation of waves based on protective measurements).

• Three-valued logic [chapter 29] represents to a certain extent the opposite point of view relative to de Broglie's: the microentities are not determined in themselves (neither corpuscular nor wave-like), and, only under very specific circumstances (measurements), can the truth or falsehood of propositions about the values of observables be decided. Even if three-valued logic is chronological precedent to the matter of the precedent chapter, for systematical reasons it seemed better to discuss it after the latter.

• Then in chapter 30 the relationship between which-path information and superposition is discussed, and very recent (proposed and performed) experiments are reported; finally we begin to discuss an ontological interpretation.

The following sections are included:

• De Broglie's Pilot Wave
  • Pilot Wave and Double Solution
  • De Broglie's Theory
  • First Critical Positions
  • Further Examination of de Broglie's Theory

• Non-Linear Solutions
  • Non-Linear Theory
  • Experimental Proofs
  • Concluding Remarks

• Empty Waves
  • Croca's and Hardy's Proposed Experiment
  • Other Proposals for Detecting the Energy of the Empty Waves

• Meaning of the Wave Function
  • Protective Measurement applied to a Spatial Wave Function
  • Reversibility and Measurement of the Density Operator of a Single System
  • Discussion of Aharonov's Proposal

• Measurement of the Density Matrix
  • General Abstract Formalism
  • Concrete Measurement Proposals in Q-Optics
  • Zeno Effect and Indetermination of the Initial Density Matrix*

• Classical Waves or Quantum Waves?

• Conclusion

The following sections are included:

• Three-valued Logic
  • Exposition of Three-valued Logic
  • An analysis of Three-valued Logic
  • A Possibility of an Infinite-Valued Logic

• Ontological Aspects
  • Propensity
  • Heisenberg's Later Interpretation

The following sections are included:

• Wootters/Zurek Gedankenexperiment

• Inequality Between Predictability and Visibility
  • Greenberger/Yasin Inequality
  • Englert's Model
  • Other Experiments

• Stochastic Complementarity Visibility/Predictability
  • Mittelstaedt/Prieur/Schieder Model
  • de Muynck/Martens/Stoffels Model

• Beginning of an Ontological Interpretation

The eighth part is devoted to the Hidden-Variables theories, an attempt to interpret QM in a deterministic way.

• The starting point of the discussion is most surely represented by the experiment proposed by Einstein/Podolsky/Rosen (= EPR), which aims to show the incompleteness of the theory [chapter 31]. As we have seen, from its beginning, the ‘strangeness’ of QM was intended by a number of physicists and philosophers as evidence of the incompleteness of the theory and of the necessity of finding a more adequate account of microphenomena. In particular, one tried to interpret the statistical aspect of the theory, i.e. the Probabilistic assumption [postulate 3.5: p. 38], as an approximation to a more basic theory with which QM statistics would have the same relationship as classical statistical mechanics has with classical mechanics. It is not by chance that Einstein, the proponent of the Ignorance interpretation [proposition 6.3: p. 106], was also the proponent of an argument which sought to demonstrate the incompleteness of QM.

In the fifties, many proposals aimed to prove that Hidden-Variable corrections to QM were not possible. In the middle of the sixties, Bell proposed a theorem with a related inequality, by means of which he could quantify the difference between the predictions of Hidden-Variable theories and those of QM, allowing the possibility of experimentally testing such predictions. But, as we shall see, Bell Inequality has consequences which go much further than the problem of Hidden-Variable theories because they pose the problem of non-locality. Although dependent on Hidden-Variable theory, the latter problem presents a lot of new and positive questions. Hence we postpone this aspect until part IX.

• In this part, we limit ourselves to the analysis of the more abstract (logical) confutations of Bohm's proposal [chapter 32]. In the seventh part we already examined a certain spread in the position of quantum entities due to their wave-like nature. Here we analyse the complementary problem which is more concerned with their corpuscular nature: is there a contradiction if we add to QM an assumption such as the Determined Value Assumption [postulate 6.3: p. 105]?

Then we discuss the interpretational issues of Hidden-Variable theories. Bohm began with the proposal of EPR and tried to accomplish this by introducing some variables underlying the quantum observables able to account for latter ones in order to reduce QM to a deterministic theory. As we shall see, the proposal of Bohm, especially his physical interpretation, can also be understood as a partial continuation of de Broglie's theory of double solution.

• Finally, in chapter 33, a radical (undeterministic) counterproposal is presented: stochastic theories. The problem of renormalization and that of the position operator is also discussed again.

The following sections are included:

• Basic Definitions
  • Theory and reality
  • Correctness
  • Completeness
  • Separability
  • Reality
  • Counterfactuality

• The Argument
  • Structure
  • The Argument Itself

• Bohm's Reformulation
  • Theory
  • Preparing a Singlet State

• Wave Function and Density Matrix

• Concluding Remarks

The following sections are included:

• Basic Definitions of the Hidden Variable Problem

• First Proofs
  • Von Neumann's Proof Concerning Hidden Variables
  • Jauch/Piron Theorem
  • Gudder Theorem
  • Informational Completeness and Hidden Variable Theories

• A Corollary of the Gleason Theorem
  • General Features
  • Some Lemmas
  • The First Bell Theorem and the Proof of the Corollary
  • Brief Discussion

• The Kochen/Specker Theorem
  • First Exposition
  • Brown/Svetlichny Theorem*

• Other Proofs
  • Non-local Measurement
  • Mermin's Generalization

• Interpretation of Hidden-Variable Theories
  • Bohm's Basic Theory
  • Bohm's Later Interpretation
  • Modal Interpretation
  • Cini's Proposal

• Concluding Remarks

The following sections are included:

• General Statement of the Problem
  • Again Concerning Jumps
  • Uncertainty and Extended Particles
  • Complementary Observables

• Stochastic Quantum Mechanics in Phase Space
  • Non-Relativistic Treatment
  • Relativistic Treatment*
  • Statistical Mechanics on Stochastic Phase Spaces*

• Conclusion

The ninth part is devoted to a problem originally developed from the HV problematic, but which in previous years has gone much further than the original discussion: the non-locality problem.

• In chapter 34 we discuss the initial historical criticisms of the EPR argument.

• Bell inequalities are then discussed [chapter 35].

• In [chapter 36] the type of locality which is violated by QM is examined.

• Chapter 37 is devoted to different theorems in order to determine more precise values which are violated by QM or classical theories.

• In the following chapters different generalizations of the subject are examined. Firstly, we ask if mixtures violate Bell inequalities [chapter 38].

• Then a generalization on a probabilistic level is provided and several probabilistic formulations of Bell inequalities are discussed [chapter 39].

• Thereafter, Bell inequalities for observables different from the spin [chapter 40] are examined.

• Finally, other non-local effects such as entanglements from different sources, AB-Effect and tunnelling time [chapter 41] are discussed.

The theoretical and mathematical foundations of the non-local features of QM are especially due to Bell's article of 1964. However, this area was mostly developed in the eighties, especially as the new techniques developed in the Q-Optics domain allowed for the first time an experimental verification and development of the theory.

The following sections are included:

• Schrödinger's Answer

• Bohr's Answer

• Bub's Formulation

The following sections are included:

• The Second Bell Theorem

• Refinements of Bell Theorem
  • Clauser/Horne/Shimony/Holt Inequality
  • Bell in 1971
  • Clauser/Horne Inequality
  • Holt's Proof
  • Generalization with an Arbitrary Number of Quanta

• Experimental Tests
  • A General Presentation of Different Tests
  • From the Theory to the Experiment
  • Other Atomic Cascade Experiments
  • Low Energy Proton-Proton Scattering
  • Chained Bell Inequalities

• Loopholes
  • Aspect/Dalibard/Grangier/Roger Experiment
  • Spontaneous Parametric Downconversion Experiments
  • Detection Loophole
  • Radial Correlation
  • Duration of the Pump Pulse

• Fine's Theorems

• Hidden-Variable Theories and Classicality

The following sections are included:

• Stapp Theorem
  • First Requirement
  • Second Requirement
  • Third Requirement
  • Proof of the Stapp Theorem

• The Problem of Counterfactuals

• The Relationship Between Stapp Theorem and Bell Theorem

• Other Proofs on the Same General Lines as Stapp's Proof
  • Greenberger/Horne/Zeilinger Proof
  • Greenberger/Horne/Shimony/Zeilinger Proof

• Eberhard Theorem

• Hidden Variables and Lorentz Invariance
  • Hardy's Proof of the Non-separability of Quantum Mechanics
  • Hardy's Theorem

• Context-Dependent Entanglement

The following sections are included:

• Introduction to the Problem

• Tsirelson Theorem
  • The Theorem and the Proofs
  • Khalfin/Tsirelson Inequality

• Landau Theorem

• Hillery/Yurke Theorem

• Conditions for Separability

The following sections are included:

• Pure States
  • Gisin Theorem
  • Popescu/Rohrlich Theorem

• Mixed States
  • Werner States
  • Andås Theorem
  • Succession of Measurements of Werner States
  • The Bell Operator

• Purification Procedures

The following sections are included:

• Pykacz/Santos Theorems

• A Probabilistic Treatment of GHSZ States

• Beltrametti/Maczyński Theorems

• Stochastic Quantum Mechanics and Bell Inequalities
  • Informational Completeness and Bell inequalities
  • Other Developments

• Garg/Mermin Model
  • General Statement of the Problem
  • Spin 1/2
  • Spin 1
  • Spin 3/2
  • Conclusion

• Concluding Remark

The following sections are included:

• Space-Time Entanglement
  • The Theory
  • The Experiments

• Time Entanglement for Fields

• Polarisation and Space-Time Entanglements

• Phase-Momentum Entanglement

• Photon-Number Correlation

• Violation of Bell Inequalities in Macroscopic Domain

• Violation in Vacuum State

The following sections are included:

• Non-Locality by Single Particles and by Independent Sources
  • Non-Locality of Single Photons
  • Entanglement by Independent Sources

• Aharonov/Bohm Effect
  • Theory
  • Interpretation of Aharonov/Bohm-Effect
  • Experiments

• Tunnelling Time

The tenth part is devoted to the most recent QM domain — and surely one of the most promising ones: Quantum information.

• Chapter 42 is centered on the relation between information and entropy. As we shall see, many themes and areas which have already been discussed (particularly problems connected with Measurement) can be treated in a new form in this context.

• Chapter 43 is devoted to quantum cryptography and teleportation, another very interesting and recent area.

• Chapter 44 deals with an entirely new method of computation: quantum computation, based on the quantum-mechanical principles and aiming to overcome the limitations of classical computers.

The following sections are included:

• General Features of Quantum Entropy
  • Von Neumann Entropy
  • Properties of the von Neumann Entropy

• Uncertainty and Entropy
  • Deutsch Entropic Uncertainty
  • Kraus Entropic Uncertainty
  • Maassen/Uffink Entropic Uncertainty
  • POVM Entropic Uncertainty
  • Uncertainty and Measurement

• Quantum Entropy for Open Systems
  • Relative Entropy
  • Increasing of Entropy
  • Increasing of Information
  • Is the Information Conserved?
  • Expectation value of Final Entropy

• Bounds for Information
  • Upper Bound
  • Lower Bound
  • Systematic Comparison Between the Different Bounds

• No-Cloning, Broadcasting and Copying
  • The Problem
  • Violation of Unitarity
  • Broadcasting
  • Universal Copying Machine

• Which Path and Information
  • Zeilinger's Analysis
  • Relationship Between Zeilinger's Measure and that of Greenberger/Yasin
  • Another Model

• Entropy and Bell Inequalities
  • Information-Theoretic Bell Inequality
  • Quadrilateral Information-Distance Bell Inequality
  • Covariance-Distance Bell Inequalities
  • Generalization of Information-Distance Bell Inequalities

• Measure of Entanglement
  • Barnett and Phoenix's Proposal
  • Bennett and Co-workers' Proposal
  • Vedral and Co-workers' Measure
  • Schlienz and Mahler's Measure
  • Popescu and Rohrlich's Measure
  • Decompression

The following sections are included:

• Quantum Cryptography
  • Some Basic Concepts and a Brief History
  • Bennett/Brassard Model
  • Ekert's Model

• Teleportation
  • General Formalism
  • Teleportation is the Reverse of an Operation
  • Experimental Proposals and Realizations
  • Superdense Coding

• Fidelity, Disturbance and Information
  • Fidelity of Teleportation
  • Relationship Between Information and Disturbance
  • Eavesdropping

The following sections are included:

• Classical and Quantum Computation

• Quantum Information
  • Classical Operations
  • Qubits
  • First Example of Quantum Information-Processing

• Quantum Transmission
  • Fidelity of a Quantum Transmission
  • Two Lemmas*
  • Transmission, Entropy and Information
  • Discussion
  • An Example of Quantum Data Compression

• More on Quantum Computation
  • The Factorization of Large Numbers

• Decoherence and Quantum Information
  • General Problems
  • Quantum Error-Correcting Codes
  • Von Neumann Capacity

• A Generalization

• Concluding Remarks

As we have said [see p. 2], the two points which we need as preliminary ones for the accomplishment of the task of a global understanding of the theory are a better development of QM's foundations and a first global interpretation. We have already dealt with the historical developments concerning the interpretation [mainly in chapters 7 and 8] and the foundations [mainly in chapter 9]. After a long examination covering different areas and problems, we can now summarize our conclusions by trying to give a general orientation about both issues.

• In chapter 45 we discuss the problem of foundations. We try to present a foundational synthesis of a theory whose constituent parts are a logico-probabilistic structure and a theory of open systems.

• In chapter 46 we discuss the problem of the interpretation of QM. In the introduction [p. 3] we distinguished between the physical interpretation of the theory and the philosophical interpretation or philosophical aspects of the interpretational problem. Here we try to summarize a consistent physical interpretation and to see some general philosophical consequences. On the other hand, we are not essentially concerned here with specific metatheoretical problems — principles or other assumptions which have been made during the examination.

The following sections are included:

• Introduction and Contents

• Foundations of Quantum Mechanics
  • Logico-Algebraic Structure
  • POVM Theory

• Measurement Theory
  • General Features of the Measurement Process
  • Theory of Measurement
  • Operational Definitions of Observable and State

• Complementarity
  • CP4 Again
  • POVM Again

The following sections are included:

• Locality and Totality

• Dynamics

• Three ‘Levels’

• Reversibility and Irreversibility

• Micro and Macro

• Final Philosophical Considerations
  • Epistemology
  • Philosophy and Quantum Mechanics

The following sections are included:

• BIBLIOGRAPHY
  • General Information Concerning the Use of the Bibliography
  • Bibliography

• LIST OF SYMBOLS
  • Letters
    • Latin Letters
    • Greek Letters
  • Other Symbols
    • Logic
    • Mathematics
    • Set theory
    • Physics

• LIST OF ABBREVIATIONS

• LIST OF COROLLARIES, DEFINITIONS, LEMMAS, POSTULATES, PRINCIPLES, PROPOSITIONS, AND THEOREMS
  • Corollaries
  • Definitions
  • Lemmas
  • Postulates
  • Principles
  • Propositions
  • Theorems

• LIST OF EXPERIMENTS
  • Atomic-Level Experiments
  • Coincidence Cascade Experiments
  • Interferometry
  • Microcavity Experiments
  • One-Hole-Experiments
  • Parametric Down Conversion
  • SGM-Experiments
  • Tunnelling
  • Two-Slit-Experiments
  • Other Experiments

• Index

• Author Index

• List of Figures

• List of Tables