Quantum Theory and Its Stochastic Limit

By: Luigi Accardi, Yun Gang Lu, Igor Volovich

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Hardcover | 7 August 2002

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Nowadays it is becoming clearer and clearer that, in the description of natural phenomena, the triadic scheme - microseopie, mesoscopic, macroscopic - is only a rough approximation and that there are many levels of description, probably an infinite hierarchy, in which the specific properties of a given level express some kind of cumulative or collective behaviour of properties or sys- tems corresponding to the lower levels. One of the most interesting challenges for contemporary natural sciences is the comprehension of the connections among these different levels of description of reality and the deduction of the laws of higher levels in this hierarchy from basic laws corresponding to lower levels. Since these cumulative or collective phenomena are, typically, nonlin- ear effects, the transition from this general program to concrete scientific achievements requires the developement of techniques which allow physical information to be extracted from nonlinear quantum systems. Explicitly in- tegrable examples of such systems are rare, and the most interesting physical phenomena are not captured by them. Even in the case of linear systems the fact that an explicit solution is formally available is often useless, since it is impossible to interpret interesting physical phenomena from it.

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"The authors have produced a very important and detailed book on the subject of the stochastic limit of quantum field theory. In particular, they formulate the stochastic limit in the framework of an algebraic central limit theory using the sort of scaling limits encountered in quantum transport phenomena. [...] The book is well written and covers many related problems [...] The audience will be mathematical and theoretical physicists. The authors are to be congratulated for showing the power and range of application of quantum probability to quantum physics." (Mathematical Reviews 2003h)

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Non-Fiction Mathematics Applied Mathematics Stochastics Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Probability & Statistics Computing & I.T.Computer Science

Statement of the Problem and Simplest Models
Notations and Statement of the Problem	p. 3
The Schrödinger Equation	p. 3
White Noise Approximation for a Free Particle: The Basic Formula	p. 5
The Interaction Representation: Propagators	p. 9
The Heisenberg Equation	p. 10
Dynamical Systems and Their Perturbations	p. 11
Asymptotic Behaviour of Dynamical Systems: The Stochastic Limit	p. 12
Slow and Fast Degrees of Freedom	p. 13
The Quantum Transport Coefficient: Why Just the t/² Scaling?	p. 15
Emergence of White Noise Hamiltonian Equations from the Stochastic Limit	p. 18
From Interaction to Heisenberg Evolutions: Conditions on the State	p. 21
From Interaction to Heisenberg Evolutions: Conditions on the Observable	p. 22
Forward and Backward Langevin Equations	p. 23
The Open System Approach to Dissipation and Irreversibility: Master Equation	p. 24
Classical Processes Driving Quantum Phenomena	p. 27
The Basic Steps ofthe Stochastic Limit	p. 27
Connection with the Central Limit Theorems	p. 30
Classical Systems	p. 32
The Backward Transport Coefficient and the Arrow of Time	p. 33
The Master and Fokker-Planck Equations: Projection Techniques	p. 34
The Master Equation in Open Systems: an Heuristic Derivation	p. 36
The Semiclassical Approximation for the Master Equation	p. 38
Beyond the Master Equation	p. 40
On the Meaning of the Decomposition H = H₀ + H_I: Discrete Spectrum	p. 42
Other Rescalings when the Time Correlations Are not Integrable	p. 44
Connections with the Classical Homogeneization Problem	p. 45
Algebraic Formulation of the Stochastic Limit	p. 46
Notes	p. 49
Quantum Fields	p. 57
Creation and Annihilation Operators	p. 57
Gaussianity	p. 59
Types of Gaussian States: Gauge-Invariant, Squeezed, Fock and Anti-Fock	p. 60
Free Evolutions of Quantum Fields	p. 61
States Invariant Under Free Evolutions	p. 63
Existence of Squeezed Stationary Fields	p. 65
Positivity of the Covariance	p. 67
Dynamical Systems in Equilibrium: the KMS Condition	p. 68
Equilibrium States: the KMS Condition	p. 69
q-Gaussian Equilibrium States	p. 70
Boson Gaussianity	p. 71
Boson Fock Fields	p. 72
Free Hamiltonians for Boson Fock Fields	p. 75
White Noises	p. 76
Boson Fock White Noises and Classical White Noises	p. 76
Boson Fock White Noises and Classical Wiener Processes	p. 77
Boson Thermal Statistics and Thermal White Noises	p. 77
Canonical Representation of the Boson Thermal States	p. 79
Spectral Representation of Quantum White Noise	p. 81
Locality of Quantum Fields and Ultralocality of Quantum White Noises	p. 83
Those Kinds of Fields We Call Noises	p. 85
Convergence of Fields in the Sense of Correlators	p. 85
Generalized White Noises as the Stochastic Limit ofGaussian Fields	p. 86
Existence of Fock, Temperature and Squeezed White Noises	p. 90
Convergence of the Field Operator to a Classical White Noise	p. 92
Beyond the Master Equation: The Master Field	p. 93
Discrete Spectrum Embedded in theContinuum	p. 94
The Stochastic Limit of a Classical Gaussian Random Field	p. 97
Semiclassical Versus Semiquantum Approximation	p. 98
An Historical Example: The Damped Harmonic Oscillator	p. 100
Emergence of the White Noise: A Traditional Derivation	p. 102
Heuristic Origins of Quantum White Noise	p. 103
Relativistic Quantum White Noises	p. 104
Space-Time Rescalings: Multidimensional White Noises	p. 106
The Chronological Stochastic Limit	p. 108
Notes	p. 111
Open Systems	p. 113
The Nonrelativistic QED Hamiltonian	p. 113
The Dipole Approximation	p. 116
The Rotating-Wave Approximation	p. 117
Composite Systems	p. 118
Assumptions on the Environment (Field, Gas, Reservoir, etc.)	p. 119
Assumptions on the System Hamiltonian	p. 120
The Free Hamiltonian	p. 121
Multiplicative (Dipole-Type) Interactions: Canonical Form	p. 121
Approximations of the Multiplicative Hamiltonian	p. 122
Rotating-Wave Approximation Hamiltonians	p. 123
No Rotating-Wave Approximation Hamiltonians with Cutoff	p. 123
Neither Dipole nor Rotating-Wave Approximation Hamiltonians Without Cutoff	p. 124
The Generalized Rotating-WaveApproximation	p. 125
The Stochastic Limit of the Multiplicative Interaction	p. 126
The Normal Form of the White Noise Hamiltonian Equation	p. 127
Invariance of the Ito Correction Term Under Free System Evolution	p. 128
The Stochastic Golden Rule: Langevin and Master Equations	p. 129
Classical Stochastic Processes Underlying Quantum Processes	p. 132
The Fluctuation-Dissipation Relation	p. 133
Vacuum Transition Amplitudes	p. 134
Mass Gap of D⁺ D and Speed of Decay	p. 136
How to Avoid Decoherence	p. 137
The Energy Shell Scalar Product: Linewidths	p. 138
Dispersion Relations and the Ito Correction Term	p. 140
The Case: H₁ = k²	p. 141
Multiplicative Coupling with the Rotating Wave Approximation: ArbitraryGaussianState	p. 142
Multiplicative Coupling with RWA: Gauge Invariant State1	p. 43
RedShifts and Blue Shifts	p. 144
The Free Evolution of the Master Field	p. 145
Algebras Invariant Under the Flow	p. 147
Notes	p. 148
Spin-Boson Systems	p. 153
Dropping the Rotating-Wave Approximation	p. 154
The Master Field	p. 154
The White Noise Hamiltonian Equation	p. 157
The Operator Transport Coefficient: no Rotating-Wave Approximation, Arbitrary Gaussian Reference State	p. 158
Different Roles of the Positive and Negative Bohr Frequencies	p. 159
No Rotating-Wave Approximation with Cutoff: Gauge Invariant States	p. 161
No Rotating-Wave Approximation with Cutoff: Squeezing States	p. 161
The Stochastic Golden Rule for Dipole Type Interactions and Gauge-Invariant States	p. 162
The Stochastic GoldenRule	p. 164
The Langevin Equation	p. 170
Subalgebras Invariant Under the Generator	p. 172
The Langevin Equation: Generic Systems	p. 173
The Stochastic Golden Rule Versus Standard Perturbation Theory	p. 176
Spin-Boson Hamiltonian	p. 178
The Damping and Oscillating Regimes: Fock Case	p. 181
The Damping and the Oscillating Regimes: Nonzero Temperature	p. 183
No Rotating-Wave Approximation Without Cutoff	p. 184
The Drift Term for Gauge-Invariant States	p. 185
The Free Evolution of the Master Field	p. 187
The Stochastic Limit of the Generalized Spin-Boson Hamiltonian	p. 187
The Langevin Equation	p. 190
Convergence to Equilibrium: Connections with Quantum Measurement	p. 191
Control of Coherence	p. 194
Dynamics of Spin Systems	p. 196
Nonstationary White Noises	p. 201
Notes	p. 202
Measurements and Filtering Theory	p. 205
Input-Output Channels	p. 205
The Filtering Problem in Classical Probability	p. 206
Field Measurements	p. 207
Properties of the Input and Output Processes	p. 210
The Filtering Problem in Quantum Theory	p. 213
Filtering of a Quantum System Over a Classical Process	p. 214
Nondemolition Processes	p. 215
Standard Scheme to Construct Examples of Nondemolition Measurements	p. 217
Discrete Time Nondemolition Processes	p. 217
Notes	p. 218
Idea of the Proof and Causal Normal Order	p. 219
Term-by-Term Convergence of the Series	p. 219
Vacuum Transition Amplitude: The Fourth-Order Term	p. 220
Vacuum Transition Amplitude: Non-Time-ConsecutiveDiagrams	p. 223
The Causal -Function and the Time-Consecutive Principle	p. 225
Theory of Distributions on the Standard Simplex	p. 227
The Second-Order Term of the Limit Vacuum Amplitude	p. 230
The Fourth-Order Term of the Limit Vacuum Amplitude	p. 230
Higher-Order Terms of the Vacuum-Vacuum Amplitude	p. 231
Proof of the Normal Form of the White Noise Hamiltonian Equation	p. 231
The Unitarity Condition for the Limit Equation	p. 233
Normal Form ofthe Thermal White Noise Equation: Boson Case	p. 234
From White Noise Calculus to Stochastic Calculus	p. 235
Chronological Product Approach to the Stochastic Limit	p. 237
Chronological Products	p. 237
Chronological Product Approach to the Stochastic Limit	p. 238
The Limit of the nth Term, Time-Ordered Product Approach: Vacuum Expectation	p. 242
The Stochastic Limit, Time-Ordered Product Approach: General Case	p. 244
Functional Integral Approach to the Stochastic Limit	p. 247
Statement of the Problem	p. 247
The StochasticLimit of the Free Massive Scalar Field	p. 248
The StochasticLimit of the Free Massless Scalar Field	p. 250
Polynomial Interactions	p. 251
The StochasticLimit of the Electromagnetic Field	p. 253
Low-Density Limit: The Basic Idea	p. 255
The Low-Density Limit: Fock Case, NoSystem	p. 255
The Low-Density Limit: Fock Case, Arbitrary System Operator	p. 257
Comparison of the Distribution and the Stochastic Approach	p. 258
LDL General, Fock Case, No System Operator, = 0	p. 259
Six Basic Principles of the Stochastic Limit	p. 261
Polynomial Interactions with Cutoff	p. 262
Assumptions on the Dynamics: Standard Models	p. 263
Polynomial Interactions: Canonical Forms, FockCase	p. 266
Polynomial Interactions: Canonical Forms, Gauge-Invariant Case	p. 269
The Stochastic Universality Class Principle	p. 275
The Case of Many Independent Fields	p. 277
The Block Principle: Fock Case	p. 278
The Stochastic Resonance Principle	p. 280
The Orthogonalization Principle	p. 280
The Stochastic Bosonization Principle	p. 280
The Time-Consecutive Principle	p. 282
Strongly Nonlinear Regimes
Particles Interacting with a Boson Field	p. 285
A Single Particle Interacting with a Boson Field	p. 287
Dynamical q-Deformation: Emergence of the Entangled Commutation Relations	p. 289
The Two-Point and Four-Point Correlators	p. 291
The Stochastic Limit of the N-Point Correlator	p. 293
The q-Deformed Module Wick Theorem	p. 296
The Wick Theorem for the QED Module	p. 301
The Limit White Noise Hamiltonian Equation	p. 302
Free Independence of the Increments of the Master Field	p. 304
Boltzmannian White Noise Hamiltonian Equations: Normal Form	p. 306
Unitarity Conditions	p. 309
Matrix Elements of the Solution	p. 310
Normal Form of the QED Module Hamiltonian Equation	p. 312
Unitarity ofthe Solution: Direct Proof	p. 313
Matrix Elements of the Limit Evolution Operator	p. 314
Nonexponential Decays	p. 317
Equilibrium States	p. 321
The Master Field321
Proof of the Result for the Two-and Four-Point Correlators	p. 322
The Vanishing ofthe Crossing Diagrams	p. 324
The Hot Free Algebra	p. 333
Interaction of the QEM Field with a Nonfree Particle	p. 335
The Limit Two-Point Function	p. 338
The Limit Four-Point Function	p. 342
The Limit Hilbert Module	p. 344
The Limit Stochastic Process	p. 347
The Stochastic Differential Equation	p. 348
Notes	p. 349
The Anderson Model	p. 351
Non relativistic Fermions in External Potential: The Anderson Model	p. 352
The Limit of the Connected Correlators	p. 355
The Four-Point Function	p. 356
The Limit of the Connected Transition Amplitude	p. 359
Proof of(13.4.3)	p. 362
Solution of the Nonlinear Equation(13.4.2)	p. 365
Notes	p. 367
Field-Field Interactions	p. 369
Interacting Commutation Relations	p. 369
The Tri-linear Hamiltonian with Momentum Conservation	p. 373
Proof of Theorem 14.2.1	p. 375
Example: Four Internal Lines	p. 379
The Stochastic Limit for Green Functions	p. 380
Second Quantized Representation of the Nonrelativistic QED Hamiltonian	p. 381
Interacting Commutation Relations and QED Module Algebra	p. 383
Decay and the Universality Class of the QED Hamiltonian	p. 384
Photon Splitting Cascades and New Statistics	p. 386
Estimates and Proofs
Analytical Theory of Feynman Diagrams	p. 393
The Connected Component Theorem	p. 395
The Factorization Theorem	p. 402
The Caseof Many Independent Fields	p. 411
The Fermi Block Theorem	p. 411
Non-Time-Consecutive Terms: The First Vanishing Theorem	p. 416
Non-Time Consecutive Terms: The Second Vanishing Theorem	p. 419
The Type-I Term Theorem	p. 423
The Double Integral Lemma	p. 431
The Multiple-Simplex Theorem	p. 435
The Multiple Integral Lemma	p. 438
The Second Multiple-Simplex Theorem	p. 439
Some Combinatorial Facts and the Block Normal Ordering Theorem	p. 446
Term-by-Term Convergence	p. 453
The Universality Class Principle and Effective Interaction Hamiltonians	p. 455
Block and Orthogonalization Principles	p. 458
The Stochastic Resonance Principle	p. 459
References	p. 461
Index	p. 469
Table of Contents provided by Publisher. All Rights Reserved.

Quantum Theory and Its Stochastic Limit

At a Glance

Hardcover

Industry Reviews

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