Statement of the Problem and Simplest Models / Part I: |
Notations and Statement of the Problem / 1: |
The Schrödinger Equation / 1.1: |
White Noise Approximation for a Free Particle: The Basic Formula / 1.2: |
The Interaction Representation: Propagators / 1.3: |
The Heisenberg Equation / 1.4: |
Dynamical Systems and Their Perturbations / 1.5: |
Asymptotic Behaviour of Dynamical Systems: The Stochastic Limit / 1.6: |
Slow and Fast Degrees of Freedom / 1.7: |
Emergence of White Noise Hamiltonian Equations from the Stochastic Limit / 1.8: |
From Interaction to Heisenberg Evolutions: Conditions on the State / 1.10: |
From Interaction to Heisenberg Evolutions: Conditions on the Observable / 1.11: |
Forward and Backward Langevin Equations / 1.12: |
The Open System Approach to Dissipation and Irreversibility: Master Equation / 1.13: |
Classical Processes Driving Quantum Phenomena / 1.14: |
The Basic Steps ofthe Stochastic Limit / 1.15: |
Connection with the Central Limit Theorems / 1.16: |
Classical Systems / 1.17: |
The Backward Transport Coefficient and the Arrow of Time / 1.18: |
The Master and Fokker-Planck Equations: Projection Techniques / 1.19: |
The Master Equation in Open Systems: an Heuristic Derivation / 1.20: |
The Semiclassical Approximation for the Master Equation / 1.21: |
Beyond the Master Equation / 1.22: |
Other Rescalings when the Time Correlations Are not Integrable / 1.23: |
Connections with the Classical Homogeneization Problem / 1.25: |
Algebraic Formulation of the Stochastic Limit / 1.26: |
Notes / 1.27: |
Quantum Fields / 2: |
Creation and Annihilation Operators / 2.1: |
Gaussianity / 2.2: |
Types of Gaussian States: Gauge-Invariant, Squeezed, Fock and Anti-Fock / 2.3: |
Free Evolutions of Quantum Fields / 2.4: |
States Invariant Under Free Evolutions / 2.5: |
Existence of Squeezed Stationary Fields / 2.6: |
Positivity of the Covariance / 2.7: |
Dynamical Systems in Equilibrium: the KMS Condition / 2.8: |
Equilibrium States: the KMS Condition / 2.9: |
q-Gaussian Equilibrium States / 2.10: |
Boson Gaussianity / 2.11: |
Boson Fock Fields / 2.12: |
Free Hamiltonians for Boson Fock Fields / 2.13: |
White Noises / 2.14: |
Boson Fock White Noises and Classical White Noises / 2.15: |
Boson Fock White Noises and Classical Wiener Processes / 2.16: |
Boson Thermal Statistics and Thermal White Noises / 2.17: |
Canonical Representation of the Boson Thermal States / 2.18: |
Spectral Representation of Quantum White Noise / 2.19: |
Locality of Quantum Fields and Ultralocality of Quantum White Noises / 2.20: |
Those Kinds of Fields We Call Noises / 3: |
Convergence of Fields in the Sense of Correlators / 3.1: |
Generalized White Noises as the Stochastic Limit ofGaussian Fields / 3.2: |
Existence of Fock, Temperature and Squeezed White Noises / 3.3: |
Convergence of the Field Operator to a Classical White Noise / 3.4: |
Beyond the Master Equation: The Master Field / 3.5: |
Discrete Spectrum Embedded in theContinuum / 3.6: |
The Stochastic Limit of a Classical Gaussian Random Field / 3.7: |
Semiclassical Versus Semiquantum Approximation / 3.8: |
An Historical Example: The Damped Harmonic Oscillator / 3.9: |
Emergence of the White Noise: A Traditional Derivation / 3.10: |
Heuristic Origins of Quantum White Noise / 3.11: |
Relativistic Quantum White Noises / 3.12: |
Space-Time Rescalings: Multidimensional White Noises / 3.13: |
The Chronological Stochastic Limit / 3.14: |
Open Systems / 3.15: |
The Nonrelativistic QED Hamiltonian / 4.1: |
The Dipole Approximation / 4.2: |
The Rotating-Wave Approximation / 4.3: |
Composite Systems / 4.4: |
Assumptions on the Environment (Field, Gas, Reservoir, etc.) / 4.5: |
Assumptions on the System Hamiltonian / 4.6: |
The Free Hamiltonian / 4.7: |
Multiplicative (Dipole-Type) Interactions: Canonical Form / 4.8: |
Approximations of the Multiplicative Hamiltonian / 4.9: |
Rotating-Wave Approximation Hamiltonians / 4.9.1: |
No Rotating-Wave Approximation Hamiltonians with Cutoff / 4.9.2: |
Neither Dipole nor Rotating-Wave Approximation Hamiltonians Without Cutoff / 4.9.3: |
The Generalized Rotating-WaveApproximation / 4.10: |
The Stochastic Limit of the Multiplicative Interaction / 4.11: |
The Normal Form of the White Noise Hamiltonian Equation / 4.12: |
Invariance of the Ito Correction Term Under Free System Evolution / 4.13: |
The Stochastic Golden Rule: Langevin and Master Equations / 4.14: |
Classical Stochastic Processes Underlying Quantum Processes / 4.15: |
The Fluctuation-Dissipation Relation / 4.16: |
Vacuum Transition Amplitudes / 4.17: |
How to Avoid Decoherence / 4.18: |
The Energy Shell Scalar Product: Linewidths / 4.20: |
Dispersion Relations and the Ito Correction Term / 4.21: |
Multiplicative Coupling with the Rotating Wave Approximation: ArbitraryGaussianState / 4.22: |
Multiplicative Coupling with RWA: Gauge Invariant State1 / 4.24: |
RedShifts and Blue Shifts / 4.25: |
The Free Evolution of the Master Field / 4.26: |
Algebras Invariant Under the Flow / 4.27: |
Spin-Boson Systems / 4.28: |
Dropping the Rotating-Wave Approximation / 5.1: |
The Master Field / 5.2: |
The White Noise Hamiltonian Equation / 5.3: |
The Operator Transport Coefficient: no Rotating-Wave Approximation, Arbitrary Gaussian Reference State / 5.4: |
Different Roles of the Positive and Negative Bohr Frequencies / 5.5: |
No Rotating-Wave Approximation with Cutoff: Gauge Invariant States / 5.6: |
No Rotating-Wave Approximation with Cutoff: Squeezing States / 5.7: |
The Stochastic Golden Rule for Dipole Type Interactions and Gauge-Invariant States / 5.8: |
The Stochastic GoldenRule / 5.9: |
The Langevin Equation / 5.10: |
Subalgebras Invariant Under the Generator / 5.11: |
The Langevin Equation: Generic Systems / 5.12: |
The Stochastic Golden Rule Versus Standard Perturbation Theory / 5.13: |
Spin-Boson Hamiltonian / 5.14: |
The Damping and Oscillating Regimes: Fock Case / 5.15: |
The Damping and the Oscillating Regimes: Nonzero Temperature / 5.16: |
No Rotating-Wave Approximation Without Cutoff / 5.17: |
The Drift Term for Gauge-Invariant States / 5.18: |
The Stochastic Limit of the Generalized Spin-Boson Hamiltonian / 5.19: |
Convergence to Equilibrium: Connections with Quantum Measurement / 5.21: |
Control of Coherence / 5.23: |
Dynamics of Spin Systems / 5.24: |
Nonstationary White Noises / 5.25: |
Measurements and Filtering Theory / 5.26: |
Input-Output Channels / 6.1: |
The Filtering Problem in Classical Probability / 6.2: |
Field Measurements / 6.3: |
Properties of the Input and Output Processes / 6.4: |
The Filtering Problem in Quantum Theory / 6.5: |
Filtering of a Quantum System Over a Classical Process / 6.6: |
Nondemolition Processes / 6.7: |
Standard Scheme to Construct Examples of Nondemolition Measurements / 6.8: |
Discrete Time Nondemolition Processes / 6.9: |
Idea of the Proof and Causal Normal Order / 6.10: |
Term-by-Term Convergence of the Series / 7.1: |
Vacuum Transition Amplitude: The Fourth-Order Term / 7.2: |
Vacuum Transition Amplitude: Non-Time-ConsecutiveDiagrams / 7.3: |
The Causal ?-Function and the Time-Consecutive Principle / 7.4: |
Theory of Distributions on the Standard Simplex / 7.5: |
The Second-Order Term of the Limit Vacuum Amplitude / 7.6: |
The Fourth-Order Term of the Limit Vacuum Amplitude / 7.7: |
Higher-Order Terms of the Vacuum-Vacuum Amplitude / 7.8: |
Proof of the Normal Form of the White Noise Hamiltonian Equation / 7.9: |
The Unitarity Condition for the Limit Equation / 7.10: |
Normal Form ofthe Thermal White Noise Equation: Boson Case / 7.11: |
From White Noise Calculus to Stochastic Calculus / 7.12: |
Chronological Product Approach to the Stochastic Limit / 8: |
Chronological Products / 8.1: |
The Limit of the nth Term, Time-Ordered Product Approach: Vacuum Expectation / 8.2: |
The Stochastic Limit, Time-Ordered Product Approach: General Case / 8.4: |
Functional Integral Approach to the Stochastic Limit / 9: |
Statement of the Problem / 9.1: |
The StochasticLimit of the Free Massive Scalar Field / 9.2: |
The StochasticLimit of the Free Massless Scalar Field / 9.3: |
Polynomial Interactions / 9.4: |
The StochasticLimit of the Electromagnetic Field / 9.5: |
Low-Density Limit: The Basic Idea / 10: |
The Low-Density Limit: Fock Case, NoSystem / 10.1: |
The Low-Density Limit: Fock Case, Arbitrary System Operator / 10.2: |
Comparison of the Distribution and the Stochastic Approach / 10.3: |
LDL General, Fock Case, No System Operator, ? = 0 / 10.4: |
Six Basic Principles of the Stochastic Limit / 11: |
Polynomial Interactions with Cutoff / 11.1: |
Assumptions on the Dynamics: Standard Models / 11.2: |
Polynomial Interactions: Canonical Forms, FockCase / 11.3: |
Polynomial Interactions: Canonical Forms, Gauge-Invariant Case / 11.4: |
The Stochastic Universality Class Principle / 11.5: |
The Case of Many Independent Fields / 11.6: |
The Block Principle: Fock Case / 11.7: |
The Stochastic Resonance Principle / 11.8: |
The Orthogonalization Principle / 11.9: |
The Stochastic Bosonization Principle / 11.10: |
The Time-Consecutive Principle / 11.11: |
Strongly Nonlinear Regimes / Part II: |
Particles Interacting with a Boson Field / 12: |
A Single Particle Interacting with a Boson Field / 12.1: |
Dynamical q-Deformation: Emergence of the Entangled Commutation Relations / 12.2: |
The Two-Point and Four-Point Correlators / 12.3: |
The Stochastic Limit of the N-Point Correlator / 12.4: |
The q-Deformed Module Wick Theorem / 12.5: |
The Wick Theorem for the QED Module / 12.6: |
The Limit White Noise Hamiltonian Equation / 12.7: |
Free Independence of the Increments of the Master Field / 12.8: |
Boltzmannian White Noise Hamiltonian Equations: Normal Form / 12.9: |
Unitarity Conditions / 12.10: |
Matrix Elements of the Solution / 12.11: |
Normal Form of the QED Module Hamiltonian Equation / 12.12: |
Unitarity ofthe Solution: Direct Proof / 12.13: |
Matrix Elements of the Limit Evolution Operator / 12.14: |
Nonexponential Decays / 12.15: |
Equilibrium States / 12.16: |
The Master Field321 / 12.17: |
Proof of the Result for the Two-and Four-Point Correlators / 12.18: |
The Vanishing ofthe Crossing Diagrams / 12.19: |
The Hot Free Algebra / 12.20: |
Interaction of the QEM Field with a Nonfree Particle / 12.21: |
The Limit Two-Point Function / 12.22: |
The Limit Four-Point Function / 12.23: |
The Limit Hilbert Module / 12.24: |
The Limit Stochastic Process / 12.25: |
The Stochastic Differential Equation / 12.26: |
The Anderson Model / 12.27: |
Non relativistic Fermions in External Potential: The Anderson Model / 13.1: |
The Limit of the Connected Correlators / 13.2: |
The Four-Point Function / 13.3: |
The Limit of the Connected Transition Amplitude / 13.4: |
Proof of(13.4.3) / 13.5: |
Solution of the Nonlinear Equation(13.4.2) / 13.6: |
Field-Field Interactions / 13.7: |
Interacting Commutation Relations / 14.1: |
The Tri-linear Hamiltonian with Momentum Conservation / 14.2: |
Proof of Theorem 14.2.1 / 14.3: |
Example: Four Internal Lines / 14.4: |
The Stochastic Limit for Green Functions / 14.5: |
Second Quantized Representation of the Nonrelativistic QED Hamiltonian / 14.6: |
Interacting Commutation Relations and QED Module Algebra / 14.7: |
Decay and the Universality Class of the QED Hamiltonian / 14.8: |
Photon Splitting Cascades and New Statistics / 14.9: |
Estimates and Proofs / Part III: |
Analytical Theory of Feynman Diagrams / 15: |
The Connected Component Theorem / 15.1: |
The Factorization Theorem / 15.2: |
The Caseof Many Independent Fields / 15.3: |
The Fermi Block Theorem / 15.4: |
Non-Time-Consecutive Terms: The First Vanishing Theorem / 15.5: |
Non-Time Consecutive Terms: The Second Vanishing Theorem / 15.6: |
The Type-I Term Theorem / 15.7: |
The Double Integral Lemma / 15.8: |
The Multiple-Simplex Theorem / 15.9: |
The Multiple Integral Lemma / 15.10: |
The Second Multiple-Simplex Theorem / 15.11: |
Some Combinatorial Facts and the Block Normal Ordering Theorem / 15.12: |
Term-by-Term Convergence / 16: |
The Universality Class Principle and Effective Interaction Hamiltonians / 16.1: |
Block and Orthogonalization Principles / 16.2: |
References / 16.3: |
Index |