Note: Texts and Monographs in Physics -- Solvable Models in Quantum Mechanics -- Copyright -- Preface -- Contents -- Introduction -- PART I THE ONE-CENTER POINT INTERACTION -- CHAPTER I.1 The One-Center Point Interaction in Three Dimensions -- CHAPTER I.2 Coulomb Plus One-Center Point Interaction in Three Dimensions -- CHAPTER I.3 The One-Center δ-Interaction in One Dimension -- CHAPTER I.4 The One-Center δ'-Interaction in One Dimension -- CHAPTER I.5 The One-Center Point Interaction in Two Dimensions -- PART II POINT INTERACTIONS WITH A FINITE NUMBER OF CENTERS -- CHAPTER II.1 Finitely Many Point Interactions in Three Dimensions -- CHAPTER II.2 Finitely Many δ-lnteractions in One Dimension -- CHAPTER II.3 Finitely Many δ'-Interactions in One Dimension -- CHAPTER II.4 Finitely Many Point Interactions in Two Dimensions -- PART III POINT INTERACTIONS WITH INFINITELY MANY CENTERS -- CHAPTER III.1 Infinitely Many Point Interactions in Three Dimensions -- CHAPTER III.2 Infinitely Many δ-Interactions in One Dimension -- CHAPTER III.3 Infinitely Many δ'-Interactions in One Dimension -- CHAPTER III.4 Infinitely Many Point Interactions in Two Dimensions -- CHAPTER III.5 Random Hamiltonians with Point Interactions -- APPENDICES -- APPENDIX A Self-Adjoint Extensions of Symmetric Operators -- APPENDIX B Spectral Properties of Hamiltonians Defined as Quadratic Forms -- APPENDIX C Schrödinger Operators with Interactions Concentrated Around Infinitely Many Centers -- APPENDIX D Boundary Conditions for Schrodinger Operators on (0, ∞) -- APPENDIX E Time-Dependent Scattering Theory for Point Interactions -- APPENDIX F Dirichlet Forms for Point Interactions -- APPENDIX G Point Interactions and Scales of Hilbert Spaces -- APPENDIX H Nonstandard Analysis and Point Interactions -- APPENDIX I Elements of Probability Theory. , APPENDIX J Relativistic Point Interactions in One Dimension -- References -- Author Index.

Note: Intro -- Direction of Time -- Preface -- Introduction to Time, Its Arrow and the Present Book -- Time and Physics -- Time, Philosophy, and Psychology -- Time, Mathematics and Information Theory -- References -- Contents -- Contributors -- List of Participants -- Chapter 1: Is Time Real? -- 1 The Representations of Change -- 1.1 From Change to Time -- 1.1.1 Nothing Changes/Everything Changes -- 1.1.2 Objective Time, Subjective Time -- 1.1.3 Time's Arrow, Time's Cycle -- 1.1.4 Causality and the Two Forms of Time -- 1.2 Reversible and Irreversible Changes -- 1.2.1 Reversibility of Motion: Galileo -- 1.2.2 Reversibility and Irreversibility in the Earth's History: Burnet, Hutton, and Lyell -- 1.2.3 Irreversibility of Thermodynamical Transformations: Carnot -- 1.2.4 Irreversibility of Biological Evolution: Darwin -- 2 From Macro to Micro -- 2.1 From Reversibility to Irreversibility and Back -- 2.1.1 From Dichotomy to Statistics -- 2.1.2 Boltzmann, Loschmidt and Zermelo: Time's Arrow or Time's Cycle? -- 2.1.3 Order, Disorder, and Information -- 2.2 Reversibility/Irreversibility in the Quantum World -- 2.2.1 The Role of Chance in Quantum Mechanics -- 2.2.2 Irreversibility of Quantum Measurement -- 2.2.3 Ontic and Epistemic Uncertainties -- 2.2.4 A Uni ed Statistical Description of the Quantum World -- References -- Chapter 2: A Simple Model for Decoherence -- References -- Chapter 3: On the Different Aspects of Time in the Fundamental Theories of Physics -- 1 Classical Physics-Time Without Importance -- 2 Special Relativity-His Own Time for Everybody -- 3 General Relativity-The Flow of Time Depends on the Situation -- 4 Quantum Theory-The Disappearance of Time -- 5 The Real Structure of Time-The Facts Are Created on the Border Between Classical Physics and Quantum -- 6 Cosmology-Time Becomes Universal and Receives a Beginning -- 7 Possible Conclusions. , References -- Chapter 4: Quantum Events and Irreversibility -- Chapter 5: Physics and Our Intuitive Outlook on Time -- 1 Our Intuitive Outlook on Time -- 1.1 Physics -- 1.2 The Eternal Universe -- 2 The Eternal Universe and Our Intuitive Notion of Time -- 2.1 The Present Moment -- 2.2 Asymmetry of Time and Free Will -- 3 Conclusion -- References -- Chapter 6: The Direction of Time in Quantum Mechanics -- 1 Decoherence and Irreversibility -- 1.1 The Theory of Irreversible Processes -- 1.2 The Case of Decoherence -- 1.3 The General Behavior of Decoherence -- 1.4 The Direction of Time -- 1.5 The Logical Direction of Time -- References -- Chapter 7: Two Arrows of Time in Nonlocal Particle Dynamics -- References -- Chapter 8: Boundary Conditions, Time Reversal and Measurements -- 1 The (Deterministic) Time We Know -- 2 The (Random) Time We Would Like to Know -- References -- Chapter 9: On Abuse of Time-Metaphors -- 1 Introduction -- 2 Time in Different Cultures -- 3 Time Metaphors -- References -- Chapter 10: The Direction of Time Ensured by Cosmology -- 1 The Cosmic Arrow of Time -- 2 The Effects of the Cosmic Time Arrow -- 3 Conclusion -- References -- Chapter 11: Asymmetries, Irreversibility, and Dynamics of Time -- 1 From CPT Invariance Violation and Cosmic Asymmetry to the Fundamental Concept of Entropy -- 1.1 Arrows of Time and Their Relations -- 1.2 The Fundamental Principle of Entropy in Thermodynamics Theory -- 1.3 Irreversibility of Complex Systems -- 2 Symmetry and Symmetry Breaking in Nature -- 2.1 The Meanings of Symmetry -- 2.2 Examples of Symmetry Breaking -- 3 Spontaneous Symmetry Breakdown, Gauge Fields and Particle Physics -- 4 Some Mathematical Aspects of Bifurcations, Singularities and Universality -- 5 Brief Remarks on Conservative and Dissipative Systems -- 6 Some Qualitative and Geometrical Properties of Psychological Time. , References -- Cited Literature -- Further Reading -- Chapter 12: Is Time Directed? -- 1 Ontology of Time -- 2 The Difference Between Past and Future -- 3 Statistical Thermodynamics, Part 1 -- 4 Questions -- 5 Probability -- 6 Classical De nition -- 7 Predicted Relative Frequency -- 8 Axiomatic -- 9 Property of What? -- 10 Statistical Thermodynamics, Part Two -- 11 Thermodynamic Probability -- 12 Predictions -- 13 Direction of Time? -- 14 Quantum Mechanics -- 15 Necessitiy of Classical Concepts -- 16 Process of Measurement -- 17 Objectivity -- References -- Chapter 13: Time in Modern Philosophy of Physics-A Survey -- 1 Philosophical Preliminaries -- 1.1 Time and Temporality-Being and Becoming -- 1.2 The Metaphysics of Time -- 1.3 Zeno's Paradoxes -- 2 Physical Preliminaries -- 2.1 Newtonian Space-Time and Time Reversal (Reversal of Motion) -- 2.2 Arrows of Time -- 3 Relativity Theory -- 3.1 Special Relativity -- 3.2 General Relativity -- 4 Thermodynamics -- 4.1 The Second Theorem-A Law? -- 4.2 Maxwell's Demon, Entropy and Information -- 5 Quantum Mechanics -- 5.1 The Measurement Problem -- 5.2 Interpretations of QM -- 6 Conclusion -- References -- Chapter 14: The Direction of Time in Dynamical Systems -- 1 Time in Classical and Relativistic Dynamics -- 2 Time in Quantum Dynamics -- 3 Time in Thermodynamics -- 4 Time in Evolutionary Dynamics -- 5 Time in Computation and Information Dynamics -- References -- Chapter 15: The Philosophical Signi cance of the Relativistic Conception of Time -- Chapter 16: Geometry of Psychological Time -- 1 Introduction -- 2 Examples of Psychopathology of Time -- 2.1 Near-Death Experiences -- 2.2 Drug-Induced States -- 2.3 Mental Psychoses -- 2.4 Mystical States -- 3 An Outline of the Algebraic Geometrical Model of Time Dimension -- 4 Conclusion -- References. , Chapter 17: Where to Put It? The Direction of Time Between Axioms and Supplementary Conditions -- 1 How to Understand Boltzmann's Legacy -- 2 Hilbert Versus Gödel on General Relativity and Cosmology -- References -- Chapter 18: Models of Time -- 1 How Old is an Electron? -- 2 Time as a One-Dimensional Connected Continuum -- 3 A Non-archimedean Model of Time -- 4 Time in Classical Physics -- 5 Reversibility and Homogeneity -- 6 Arrows of Time -- 7 Time Reversal: Axiomatic Approach -- 8 Examples -- 9 Time Re ections in von Neumann Algebras -- 10 Time Reversibility and Time Reversal Invariance -- 11 Evolutions -- 12 Time Reversal in the Schrödinger Representation -- 13 Time Re ection and Positivity of the Spectrum -- 14 Antilinear Anti-automorphisms -- 15 Anti-states -- 16 Automorphisms with Anti-invariant States -- 17 Motivations for the De nition of Time Reversal -- 18 Time Re ections in Local Algebras -- 19 The Adjoint and the Time Reversed of a Markov Semigroup -- 20 Field Algebras -- 21 Existence of a Time Re ection on the Field Algebra -- 22 rho0-Anti-invariant Gaussian States -- References -- Chapter 19: The Arrow of Time and Information Theory -- 1 Introduction -- 1.1 The Arrows of Time -- 1.2 The Aim of This Paper -- 2 Review of the Main Notions -- 2.1 Clausius and Boltzmann Entropy -- 2.2 Shannon Information and Entropy -- 2.2.1 Shannon Versus Boltzmann Entropy -- 2.2.2 A Mathematical De nition of Shannon Entropy -- 2.3 Information Content -- 2.4 Algorithmic Information Content -- 3 Computable Information Content -- 3.1 The Idea of Computable Information Content -- 3.2 The De nition of CIC -- 4 Information and Dynamics -- 4.1 Physical Systems -- 4.2 Chaotic Reversible Systems -- 4.3 Gradient-Like Systems -- 4.4 Self-organizing Systems -- 5 An Exorcism of the Maxwell Demon -- 5.1 Szilard -- 5.2 Brillouin and Gabor -- 5.3 Landauer -- 5.4 Bennet. , 5.5 Our Point of View -- 6 Conclusions -- References -- Chapter 20: Two-Way Thermodynamics: Could It Really Happen? -- 1 Context -- 2 Opposite Arrows -- 3 Dialogue on Opposite Arrows -- 3.1 Solving the Two-Time Boundary Value Problem -- 3.2 Isolation -- 3.3 Closed Timelike Curves -- 3.4 Entropy Calculation -- 4 Causality -- References -- Chapter 21: Remarks on the Compatibility of Opposite Arrows of Time -- 1 Introduction -- 2 Retarded and Advanced Fields -- 3 Cat Maps -- 4 (Anti-)Causality -- 5 Cosmology and Gravitation -- 6 Quantum Aspects -- References -- Chapter 22: A Computer's Arrow of Time -- 1 Introduction -- 2 The Thermodynamic Arrow: Causality and Entropy Increase -- 2.1 Symmetric Behavior of Entropy in the Two-Time Boundary Condition Context -- 2.2 Symmetric Behavior of Causality in the Two-Time Boundary Condition Context -- 2.3 Analysis of a Macroscopic System in This Context -- 2.4 Prediction and Retrodiction -- 3 A Computer -- 4 Arrow-Like Features in Computer States -- 5 Conclusions -- References -- Chapter 23: Remarks on the Compatibility of Opposite Arrows of Time II -- 1 Introduction -- 2 Wet Carpets and Detective Stories -- 3 Phase Space Volume -- 4 Quantum Measurement -- 5 Gravity -- References -- Index.

Note: Cover -- Half Title -- Title Page -- Copyright Page -- Contents -- Preface -- Introduction -- Chapter 1 Basic Functional Groups and Lie Algebras -- 1.1 NotationsandPreliminaries -- 1.2 TheLieAlgebrasC (X, G) and D(X, G) -- 1.3 ThcHilbert-SoboIev Lie Algebras W (X, G) -- 1.4 Technical Results About Some Continuous Sums of Operators -- 1.5 The Hausdorff Series On D (X, G) -- 1.6 TheNuclearGroup D (X, G) -- 1.7 The Lie Algebra ? (X, G) -- The Groups C (X, G) -- 1.8 The Hilbert-Sobolev Lie Groups W (X, G) -- 1.9 The Gauge Groups and Their Various Sobolev CompletionsofOrderl -- 1.10 On Some Subgroups of D (X, G) and Related Topics -- References -- Chapter 2 Multiplicative G-Distributions on a Riemannian Manifold -- 2.1 Introduction -- 2.2 Basic Materials for a Theory of G-Distributions on a Riemannian Manifold -- 2.3 The Action of Diff(X) on D (X, G) -- 2.4 A Basic Tool: Representations ofType (S) -- 2.5 Nonlocated Multiplicative G-Integrals -- The Continuous Tensor Products of Representations of G -- 2.6 The Case Where G Is a Nilpotent Lie Group -- References -- Chapter 3 The Energy Representations of Gauge Groups -- 3.1 Introduction -- 3.2 The Basic Energy Representation Associated with a Riemannian Flag -- 3.3 The Representation U and Its Spectral Measure -- 3.4 T-Property -- T-Property -- 3.5 On the Irreducibility ofthe Basic Energy Representation UF -- 3.6 Rings of Generalized Energy Representations -- References -- Chapter 4 Energy Representations ofPath Groups -- 4.1 Introduction -- 4.2 Standard Brownian Measure on Path Spaces -- 4.3 Reduction ofthe Energy Representation -- 4.4 Irreducibility of Almost All Ua's and Properties of the Right and Left Versions Ur, UE of the Energy Representation -- 4.5 Comments on Further Developments -- References -- Chapter 5 The Algebraic Level: Representations of Current Algebras -- 5.1 Introduction. , 5.2 Affine KAC-Moody Algebras -- 5.3 Some Connected Problems and Applications -- 5.4 Beyond Affine Lie Algebras -- References -- Index.

Note: Cover -- Title -- Copyright -- Contents -- Preface -- Dedication -- I.0 Introduction -- Acknowledgements -- I.1 The two-dimensional Plateau problem -- I.2 Topological and metric structures on the space of mappings and metrics -- Appendix to I.2: ILH-structures -- I.3 Harmonic maps and global structures -- I.4 Cauchy-Riemann operators -- I.5 Zeta-function and heat-kernel determinants of an operator -- I.6 The Faddeev-Popov procedure -- I.6.1. The Faddeev-Popov map -- I.6.2. The Faddeev-Popov determinant: the case G=H -- I.6.3. The Faddeev-Popov determinant: the general case -- I.7 Determinant bundles -- I.8 Chern classes of determinant bundles -- I.9 Gaussian measures and random fields -- I.10 Functional quantization of the Hoegh-Krohn and Liouville models on a compact surface -- I.11 Small time asymptotics for heat-kernel regularized determinants -- II.1 Quantization by functional integrals -- II. 2 The Polyakov measure -- Step 1: The integration on the space of embeddings -- Step 2: The Faddeev-Popov procedure -- Step 3: The Polyakov measure in noncritical dimensions and the Liouville measure -- Step 4: The Polyakov measure in critical dimension -- Step 5: Correlation functions -- II.3 Formal Lebesgue measures on Hilbert spaces -- II.4 The Gaussian integration on the space of embeddings -- II.5 The Faddeev-Popov procedure for bosonic strings -- II.6 The Polyakov measure in noncritical dimension and the Liouville measure -- II.7 The Polyakov measure in the critical dimension d = 26 -- II.8 Correlation functions -- References -- Index.

Note: Intro -- Preface to the Volume Advanced School on Complexity and Emergence: Ideas, Methods, with a Special Attention to Mathematics and Its Application to Economics and Finance -- Acknowledgments -- Contents -- A Topological and Dynamical Approach to the Study of Complex Living Systems -- 1 Introduction: Complex Structures in Physics -- 2 Topological Stability in Condensed Matter and Phase Transition -- 3 Epistemological Remarks on the Limits of the Hierarchical Paradigm of Science -- 4 The Spin-Like Representation of the KT Topological Phase Transition -- 5 The Effectiveness of the Concept of Winding -- 6 Topological Changes and Emergence of Physical Patterns -- 7 Complexity of Time and of the Universe -- 8 The Complexity and the Breaking-Symmetry Phenomenon -- 9 The Specificity of Biological Complexity, Emergence and Causation -- 10 Systems Biology and Complexity -- 11 Self-Organization and the Causal Role Played by Systemic Properties in Biology -- 12 Systems Biology, Reductionism and Emergence -- 13 Chaos, Fractality and Complexity -- 14 The Difference Between Self-Assembly and Self-Organization -- 15 Complexity of Biological Systems: The Genome as a Complex System -- 16 The Need for a Systems Biology Approach -- 17 The Complex Topology of Biological Structures: From Topoisomerases to Supercoiling -- 18 The Functional Role of Topological Plasticity -- 19 The Topological State of Supercoiling and Its Biological Functions -- 20 Conclusive Remarks -- References -- The Complexity Theory and Financial Systems Regulation -- 1 The Characteristics of Complex Systems -- 2 Complexity Theory and the Law: Complex Versus Complicated -- 3 Complexity Theory and Legal Thinking -- 4 Complexity Theory and the Financial System -- 5 Complexity Jurisprudence and Financial Systems Regulation -- 6 The Financial System Regulation and the Systemic Risk. , 7 Complex Systems Models and the Role of Fraud -- References -- The Emergence of the Order Parameter in the Interpolating Replica Trick for Disordered Statistical Mechanics Systems -- 1 Introduction -- 2 Replicas and the Auxiliary Function -- 3 Interpolating on the Number of Replicas -- 4 The Replica Trick in the Random Energy Model and the Emergence of the Universal Order Parameter -- 5 The Emergence of the Order Parameter and the Trial Function in the Sherrington-Kirkpatrick Model -- 6 Conclusion and Outlook -- References -- Information and Complexity, Or: Where Is the Information? -- 1 Introduction -- 2 Background: Principles of Information Theory -- 2.1 Shannon Information -- 2.2 Mutual and Conditional Information -- 2.3 Maximum Entropy -- 2.4 Kullback-Leibler Divergence -- 3 Complexity -- 3.1 Algorithmic Complexity -- 3.2 External and Internal Complexity -- 3.3 Optimization Principles -- 3.4 Correlations in Time Series -- 3.5 Complementarity -- 3.6 Hierarchical Models and Complexity Measures -- 3.7 Interactions Between Levels -- 4 Information Decomposition -- References -- Complex Systems: From the Presocratics to Pension Funds -- 1 Introduction -- 2 Presocratic Philosophy Revisited -- 2.1 Heraclitus and the Philosophy of Nature -- 2.2 Parmenides and the Modern Idea of Science -- 2.3 Democritus and Atomism -- 2.4 Markov Chains -- 3 Models and Causes -- 3.1 The Example of Feynman Diagrams -- 3.2 Aristotle on Causality -- 3.3 Aristotle's Causes in Modern Science -- 4 Mathematical Atomism -- 4.1 Logical and Mathematical Atoms -- 4.2 The Principle of Reason -- 4.3 Mathematics as a Dynamical System -- 5 Pension Schemes as Complex Systems -- 5.1 Defined Benefit Pension Funds -- 5.2 Wrong Way Risk -- 5.3 A Merton Model for Defaults -- 5.4 Quantitative Results -- 5.5 Financial Conclusions -- 5.6 Epistemological Lessons -- References. , From Complex Dynamics to the Architecture of the City -- 1 Introduction -- 2 Basic Classic Theories on Urban Systems -- 3 Cities: Design and Self-organization -- 4 The City Designed -- 5 The Spontaneous Self-organizing City -- 6 The Artificial Self-organized City -- 7 The Scientific and Architectonic Way of Knowledge -- 8 From Object to Patterns -- 9 Imitation and Time -- 10 Communication Is the Main Problem that a City Faces and Solves -- 11 From Communication to the Basic Center-Area Pattern -- 12 The Channeling of Movement and the Establishment of a Center -- 13 The Two Basic Patterns -- 14 The Self-similarity of the System of Patterns -- 15 The Spatial Organization of the System of Patterns -- 16 The Tracing of the Roads Network -- 17 The Topological Aspects of the Roads Network, and the Production of Urban Space -- 18 Final Remarks -- 19 Conclusions -- References -- Randomness, Emergence and Causation: A Historical Perspective of Simulation in the Social Sciences -- 1 Introduction -- 2 Experiments with Early Computers -- 2.1 ENIAC -- 2.2 Intermezzo -- 2.3 FERMIAC -- 2.4 MONIAC -- 3 System Dynamics -- 3.1 Intermezzo -- 4 Discrete-Event Simulation -- 5 Microsimulation -- 5.1 Microsimulation in Economics -- 5.2 Intermezzo -- 5.3 Early Simulation of Voting Behavior -- 6 Cellular Automata -- 6.1 Intermezzo -- 7 Agent-Based Models -- 7.1 A Menagerie of Names -- 8 Conclusions -- References.

Note: Includes bibliographies and index