libristo functional integrals 1 2260794
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Radon Integrals, 1 Springer, Berlin
Książki / Literatura obcojęzyczna
In topological measure theory, Radon measures are the most important objects. In the context of locally compact spaces, there are two equivalent canonical definitions. As a set function, a Radon measure is an inner compact regular Borel measure, finite on compact sets. As a functional, it is simply a positive linear form, defined on the vector lattice of continuous real-valued functions with compact support. During the last few decades, in particular because of the developments of modem probability theory and mathematical physics, attention has been focussed on measures on general topological spaces which are no longer locally compact, e.g. spaces of continuous functions or Schwartz distributions. For a Radon measure on an arbitrary Hausdorff space, essentially three equivalent definitions have been proposed: As a set function, it was defined by L. Schwartz as an inner compact regular Borel measure which is locally bounded. G. Choquet considered it as a strongly additive right continuous content on the lattice of compact subsets. Following P.A. Meyer, N. Bourbaki defined a Radon measure as a locally uniformly bounded family of compatible positive linear forms, each defined on the vector lattice of continuous functions on some compact subset.
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Techniques and Applications of Path Integration Dover Publications Inc.
Książki / Literatura obcojęzyczna
Part I Introduction 1 Introduction and Defining the Path Integral Appendix: The Trotter Product Formula 2 Probabilities and Probability Amplitudes for Paths 3 Correspondence Limit for the Path Integral (Heuristic) Appendix: Useful Integrals 4 Vector Potentials and Another Proof of the Path Integral Formula 5 The Ito Integral and Gauge Transformations 6 Doing the Integral: Free Particle and Quadratic Lagrangians Appendix: Exactness of the Sum over Classical Paths 7 Properties of Green's Functions; the Feynman-Kac Formula 8 Functional Derivatives and Commutation Relations 9 Brownian Motion and the Wiener Integral; Kac's Proof 10 Perturbation Theory and Feynman DiagramsPart II Selected Applications of the Path Integral 11 Asymptotic Analysis 12 The Calculus of Variations 13 The WKB Approximation and its Application to the Anharmonic Oscillator 14 Detailed Presentation of the WKB Approximation 15 WKB Near Caustics 16 Caustics and Uniform Asymptotic Approximations 17 The Phase of the Semiclassical Amplitude 18 The Semiclassical Propagator as a Function of Energy 19 Scattering Theory 20 Geometrical Optics 21 The Polaron 22 Spin and Related Matters 22.1 The Direct Method-Product Integrals or Time Ordered Products 22.2 Continuous Models for Spin 23 Path Integrals for Multiply Connected Spaces 23.1 Particle Constrained to a Circle 23.2 Rudiments of Homotopy Theory 23.3 Homotopy Applied to the Path Integral 23.4 Extensions of Symmetric Operators 24 Quantum Mechanics on Curved Spaces 25 Relativistic Propagators and Black Holes 26 Applications to Statistical Mechanics 27 Coherent State Representation 28 Systems with Random Impurities 29 Critical Droplets, Alias Instantons, and Metastability Appendix: Small Oscillations about the Instanton 30 Renormalization and Scaling for Critical Phenomena 31 Phase Space Path Integral 32 Omissions, Miscellany, and Prejudices 32.1 Field Theory 32.2 Uncompleting the Square 32.3 Rubber: Path Integral Formulation of a Polymer as a Random Walk 32.4 Hard Sphere Gas Second Virial Coefficient 32.5 Adding Paths by Computer 32.6 A Perturbation Expansion Using the Path Integral 32.7 Solvable Path Integral with the Potential ax2 + b/x2 32.8 Superfluidity 32.9 Fermions 32.10 Books and Review Papers on Path IntegralsAuthor IndexSubject IndexSupplements
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Quantum Field Theory Dover Publications Inc.
Książki / Literatura obcojęzyczna
PrefaceGeneral References1. Classical Theory 1.1 Principle of Least Action 1.1.1 Classical Motion 1.1.2 Electromagnetic Field as an Infinite Dynamical System 1.1.3 Electromagnetic Interaction of a Point Particle 1.2 Symmetries and Conservation Laws 1.2.1 Fundamental Invariants 1.2.2 Energy Momentum Tensor 1.2.3 Internal Symmetries 1.3 Propagation and Radiation 1.3.1 Green Functions 1.3.2 Radiation2. The Dirac Equation 2.1 Toward a Relativistic Wave Equation 2.1.1 Quantum Mechanics and Relativity 2.1.2 The Dirac Equation 2.1.3 Relativistic Covariance 2.2 Physical Content 2.2.1 Plane Wave Solutions and Projectors 2.2.2 Wave Packets 2.2.3 Electromagnetic Coupling 2.2.4 Foldy-Wouthuysen Transformation 2.3 Hydrogen-like Atoms 2.3.1 Nonrelativistic versus Relativistic Spectrum 2.3.2 Dirac Theory 2.4 Hole Theory and Charge Conjugation 2.4.1 Reinterpretation of Negative Energy Solutions 2.4.2 Charge Conjugation 2.4.3 Zero-Mass Particles 2.5 Dirac Propagator 2.5.1 Free Propagator 2.5.2 Propagation in an Arbitrary External Electromagnetic Field 2.5.3 Application to the Coulomb Scattering 2.5.4 Fock-Schwinger Proper Time Method3. Quantization--Free Fields 3.1 Canonical Quantization 3.1.1 General Formulation 3.1.2 Scalar Field 3.1.3 Charged Scalar Field 3.1.4 Time-Ordered Product 3.1.5 Thermodynamic Equilibrium 3.2 Quantized Radiation Field 3.2.1 Indefinite Metric 3.2.2 Propagator 3.2.3 Massive Vector Field 3.2.4 Vacuum Fluctuations 3.3 Dirac Field and Exclusion Principle 3.3.1 Anticommutators 3.3.2 Fock Space for Fermions 3.3.3 Relation between Spin and Statistics--Propagator 3.4 Discrete Symmetries 3.4.1 Parity 3.4.2 Charge Conjugation 3.4.3 Time Reversal 3.4.4 Summary4. Interaction with an External Field 4.1 Quantized Electromagnetic Field Interacting with a Classical Source 4.1.1 Emission Probabilities 4.1.2 Emitted Energy and the Infrared Catastrophe 4.1.3 Induced Absorption and Emission 4.1.4 S Matrix and Evolution Operator 4.2 Wick's Theorem 4.2.1 Bose Fields 4.2.2 Fermi Fields 4.2.3 General Case 4.3 Quantized Dirac Field Interacting with a Classical Potential 4.3.1 General Formalism 4.3.2 Emission Rate to Lowest Order 4.3.3 Pair Creation in a Constant Uniform Electric Field 4.3.4 The Euler-Heisenberg Effective Lagrangian5. Elementary Processes 5.1 S Matrix and Asymptotic Theory 5.1.1 Cross Sections 5.1.2 Asymptotic Theory 5.1.3 Reduction Formulas 5.1.4 Generating Functional 5.1.5 Connected Parts 5.1.6 Fermions 5.1.7 Photons 5.2 Applications 5.2.1 Compton Effect 5.2.2 Pair Annihilation 5.2.3 Positronium Lifetime 5.2.4 Bremsstrahlung 5.3 Unitarity and Causality 5.3.1 Unitarity and Partial Wave Decomposition 5.3.2 Causality and Analyticity 5.3.3 The Jost-Lehmann-Dyson Representation 5.3.4 Forward Dispersion Relations 5.3.5 Momentum Transfer Analyticity6. Perturbation Theory 6.1 Interaction Representation and Feynman Rules 6.1.1 Self-Interacting Scalar Field 6.1.2 Feynman Rules for Spinor Electrodynamics 6.1.3 Electron-Electron and Electron-Positron Scattering 6.1.4 Scalar Electrodynamics 6.2 Diagrammatics 6.2.1 Loopwise Expansion 6.2.2 Truncated and Proper Diagrams 6.2.3 Parametric Representation 6.2.4 Euclidean Green Functions 6.3 Analyticity Properties 6.3.1 Landau Equations 6.3.2 Real Singularities 6.3.3 Real Singularities of Simple Diagrams 6.3.4 Physical-Region Singularities. Cutkosky Rules7. Radiative Corrections 7.1 One-Loop Renormalization 7.1.1 Vacuum Polarization 7.1.2 Electron Propagator 7.1.3 Vertex Function 7.1.4 Summary 7.2 Radiative Corrections to the Interaction with an External Field 7.2.1 Effective Interaction and Anomalous Magnetic Moment 7.2.2 Radiative Corrections to Coulomb Scattering 7.2.3 Soft Bremsstrahlung 7.2.4 Finite Inclusive Cross Section 7.3 New Effects 7.3.1 Photon-Photon Scattering 7.3.2 Lamb Shift 7.3.3 Van der Waals Forces at Large Distances8. Renormalization 8.1 Regularization and Power Counting 8.1.1 Introduction 8.1.2 Regularization 8.1.3 Power Counting 8.1.4 Convergence Theorem 8.2 Renormalization 8.2.1 Normalization Conditions and Structure of the Counterterms 8.2.2 Bogoliubov's Recursion Formula 8.2.3 Zimmermann's Explicit Solution 8.2.4 Renormalization in Parametric Space 8.2.5 Finite Renormalizations 8.2.6 Composite Operators 8.3 Zero-Mass Limit, Asymptotic Behavior, and Weinberg's Theorem 8.3.1 Massless Theories 8.3.2 Ultraviolet Behavior and Weinberg's Theorem 8.4 The Case of Quantum Electrodynamics 8.4.1 Formal Derivation of the Ward-Takahashi Identities 8.4.2 Pauli-Villars Regularization to All Orders 8.4.3 Renormalization 8.4.4 Two-Loop Vacuum Polarization9. Functional Methods 9.1 Path Integrals 9.1.1 The Role of the Classical Action in Quantum Mechanics 9.1.2 Trajectories in the Bargmann-Fock Space 9.1.3 Fermion Systems 9.2 Relativistic Formulation 9.2.1 S Matrix and Green Functions in Terms of Path Integrals 9.2.2 Effective Action and Steepest-Descent Method 9.3 Constrained Systems 9.3.1 General Discussion 9.3.2 The Electromagnetic Field as an Example 9.4 Large Orders in Perturbation Theory 9.4.1 Introduction 9.4.2 Anharmonic Oscillator10. Integral Equations and Bound-State Problems 10.1 The Dyson-Schwinger Equations 10.1.1 Field Equations 10.1.2 Renormalization 10.2 Relativistic Bound States 10.2.1 Homogeneous Bethe-Salpeter Equation 10.2.2 The Wick Rotation 10.2.3 Scalar Massless Exchange in the Ladder Approximation&n 12.3 The Effective Action at the One-Loop Order 12.3.1 General Form 12.3.2 Two-Point Function 12.3.3 Other Functions 12.3.4 One-Loop Renormalization 12.4 Renormalization 12.4.1 Slavnov-Taylor Identities 12.4.2 Identities for Proper Functions 12.4.3 Recursive Construction of the Counterterms 12.4.4 Gauge Dependence of Green Functions 12.4.5 Anomalies 12.5 Massive Gauge Fields 12.5.1 Historical Background 12.5.2 Massive Gauge Theory 12.5.3 Spontaneous Symmetry Breaking 12.5.4 Renormalization of Spontaneously Broken Gauge 12.5.5 Gauge Independence and Unitarity of the S Matrix 12.6 The Weinberg-Salam Model 12.6.1 The Model for Leptons 12.6.2 Electron-Neutrino Cross Sections 12.6.3 Higher-Order Corrections 12.6.4 Incorporation of Hadrons13. Asymptotic Behavior 13.1 Effective Charge in Electrodynamics 13.1.1 The Gell-Mann and Low Function 13.1.2 The Callan-Symanzik Equation 13.2 Broken Scale Invariance 13.2.1 Scale and Conformal Invariance 13.2.2 Modified Ward Identities 13.2.3 Callan-Symanzik Coefficients to Lowest Order 13.3 Scale Invariance Recovered 13.3.1 Coupling Constant Flow 13.3.2 Asymptotic Freedom 13.3.3 Mass Corrections 13.4 Deep Inelastic Lepton-Hadron Scattering and Electron-Positron Annihilation into Hadrons 13.4.1 Electroproduction 13.4.2 Light-Cone Dynamics 13.4.3 Electron-Positron Annihilation 13.5 Operator Product Expansions 13.5.1 Short-Distance Expansion 13.5.2 Dominant and Subdominant Operators, Operator Mixing, and Conservation Laws 13.5.3 Light-Cone ExpansionAppendixA-1 MetricA-2 Dirac Matrices and SpinorsA-3 Normalization of States, S Matrix, Unitarity, and Cross SectionsA-4 Feynman RulesIndex
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Catastrophe Theory and Its Applications Dover Publications Inc.
Książki / Literatura obcojęzyczna
Preface1 Smooth and sudden changes 1. Catastrophes 2. The Zeeman catastrophe machine 3. Gravitational catastrophe machines 4. Catastrophe theory2 Multidimensional geometry 1. Set-theoretic notation 2. Euclidean space 3. Linear transformations 4. Matrices 5. Quadratic forms 6. Two-variable cubic forms 7. Polynomial geometry3 Multidimensional calculus 1. Distance in Euclidean space 2. The derivative as tangent 3. Contours 4. Partial derivatives 5. Higher derivatives 6. Taylor series 7. Truncated algebra 8. The Inverse Function Theorem 9. The Implicit Function Theorem4 Critical points and transversality 1. Critical points 2. The Morse Lemma 3. Functions of a single variable 4. Functions of several variables 5. The Splitting Lemma 6. Structural stability 7. Manifolds 8. Transversality 9. Transversality and stability 10. Transversality for mappings 11. Codimension5 Machines revisited 1. The Zeeman machine 2. The canonical cusp catastrophe 3. Dynamics of the Zeeman machine 4. The gravitational machines 5. Formulation of a general problem6 Structural stability 1. Equivalence of families 2. Structural stabillty of families 3. Physical interpretations of structural stability 4. The Morse and Splitting Lemmas for families 5. Catastrophe geometry7 Thom's classification theorem 1. Functions and families of functions 2. One-parameter families 3. Non-transversaliity and symmetry 4. Two-parameter families 5. "Three-, four- and five-parameter families" 6. Higher catastrophes 7. Thom's theorem8 Determinacy and unfoldings 1. Determine and strong determinacy 2. One-variable jet spaces 3. Infinitesimal changes of variable 4. Weaker determinacy conditions 5. Transformations that move the origin 6. Tangency and transversality 7. Codimension and unfoldings 8. Transversality and universality 9. Strong equivalence of unfoldings 10. Numbers associated with singularities 11. Inequalities 12. Summary of results and calculation methods 13. Examples and calculations 14. Compulsory remarks on terminology9 The first seven catastrophe geometries 1. The objects of study 2. The fold catastrophe 3. The cusp catastrophe 4. The swallowtail catastrophe 5. The butterfly catastrophe 6. The elliptic umbilic 7. The hyperbolic umbilic 8. The parabolic umbilic 9. Ruled surfaces10 Stability of ships Static equilibrium 1. Buoyancy 2. Equilibrium 3. Stability 4. The vertical-sided ship 5. Geometry of the buoyancy locus 6. Metacentres Ship shapes 7. The elliptical ship 8. The rectangular ship 9. Three dimensions 10. Oil-rigs 11. Comparison with current methods11. The geometry of fluids Background on fluid mechanics 1. What we are describing 2. Stream functions 3. Examples of flows 4. Rotation 5. Complex variable methods Stability and experiment 6. Changes of variable 7. Heuristic programme 8. Experimental realization Combining polymer molecules 9. Non-Newtonian behaviour 10. Extensional flows Degenerate flows 11. The six-roll mill 12. The non-local bifurcation set of the elliptic umbilic 13. The six-roll mill with polymer solution 14. The 2n-roll mill12 Optics and scattering theory Ray optics 1. Caustics 2. The rainbow 3. Variational principles 4. Scattering Wave optics 5. Asymptotic solutions of wave equations 6. Oscillatory integrals 7. Universal unfoldings 8. Orders of caustics Applications 9. Scattering from a crystal lattice 10. Other caustics 11. Mirages 12. Sonic booms 13. Giant ocean waves13 Elastic structures General theory 1. Objects under stress 2. Elastic equilibria 3. Infinite-dimensional peculiarities Euler struts 4. Finite element vision 5. Classical (1744) variational version 6. Perturbation analysis 7. Modern functional analysis 8. The buckling of a spring 9. The pinned strut The geometry of collapse 10. Imperfection sensitivity 11. "(r, s)-Stability" 12. Optimization 13. Symmetry: rods and shells Buckling plates 14. The von Kármán equations 15. Unfolding a double eigenvalue Dynamics 16. Soft modes 17. Stiffness14 Thermodynamics and phase transitions Equations of state 1. van der Waals' equation 2. Ferromagnetism Thermodynamic potentials 3. Entropy 4. Transforming the maximum entropy principle 5. Legendre transformations 6. Explicit potentials 7. The Landau theory Fluctuations and critical exponents 8. Classical exponents 9. Topological tinkering 10. The rôle of fluctuations 11. Spatial variation 12. Partition functions 13. Renormalization group 14. Structural stability of renormalization The rôle of symmetry 15. Even functions 16. The shapes of rotating stars 17. Symmetry breaking 18. Tricritical points 19. Crystal symmetries 20. Spectrum singularities15 Laser physics Preliminaries 1. Atoms 2. Field 3. Interaction 4. Measurement The laser catastrophe 5. Unfolded Hamiltonian 6. Equations of motion 7. Mean field approximation 8. Boundary conditions 9. Non-equilibrium stationary manifold Experiments 10. Laser transition 11. Optical bistability 12. Photocount distributions Analytic correspondence 13. Equilibrium boundary conditions 14. Equilibrium manifold 15. Thermodynamic phase transition 16. Critical behaviour 17. Analytic correspondence of experiments &
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t1=0.013, t2=0, t3=0, t4=0, t=0.014