Quantum Theory and Its Stochastic Limit

by Accardi, L.; Volovich, Igor; Lu, Yun Gang

• ISBN13: 9783540419280
• ISBN10: 3540419284
• Format: Hardcover
• Copyright: 2002-09-01
• Publisher: Springer Verlag

Summary

The subject of this book is a new mathematical technique, the stochastic limit, developed for solving nonlinear problems in quantum theory involving systems with infinitely many degrees of freedom (typically quantum fields or gases in the thermodynamic limit). This technique is condensed into some easily applied rules (called "stochastic golden rules"), which allow us to single out the dominating contributions to the dynamical evolution of systems in regimes involving long times and small effects. In the stochastic limit the original Hamiltonian theory is approximated using a new Hamiltonian theory which is singular. These singular Hamiltonians still define a unitary evolution, and the new equations give much more insight into the relevant physical phenomena than the original Hamiltonian equations. Especially, one can explicitly compute multi-time correlations (e.g. photon statistics) or coherent vectors, which are beyond the reach of typical asymptotic techniques.

Table of Contents

Part I. Statement of the Problem and Simplest Models
Notations and Statement of the Problem	3	(54)
The Schrodinger Equation	3	(2)
White Noise Approximation for a Free Particle: The Basic Formula	5	(4)
The Interaction Representation: Propagators	9	(1)
The Heisenberg Equation	10	(1)
Dynamical Systems and Their Perturbations	11	(1)
Asymptotic Behaviour of Dynamical Systems: The Stochastic Limit	12	(1)
Slow and Fast Degrees of Freedom	13	(2)
The Quantum Transport Coefficient: Why Just the t/λ2 Scaling?	15	(3)
Emergence of White Noise Hamiltonian Equations from the Stochastic Limit	18	(3)
From Interaction to Heisenberg Evolutions: Conditions on the State	21	(1)
From Interaction to Heisenberg Evolutions: Conditions on the Observable	22	(1)
Forward and Backward Langevin Equations	23	(1)
The Open System Approach to Dissipation and Irreversibility: Master Equation	24	(3)
Classical Processes Driving Quantum Phenomena	27	(1)
The Basic Steps of the Stochastic Limit	27	(3)
Connection with the Central Limit Theorems	30	(2)
Classical Systems	32	(1)
The Backward Transport Coefficient and the Arrow of Time	33	(1)
The Master and Fokker--Planck Equations: Projection Techniques	34	(2)
The Master Equation in Open Systems: an Heuristic Derivation	36	(2)
The Semiclassical Approximation for the Master Equation	38	(2)
Beyond the Master Equation	40	(2)
On the Meaning of the Decomposition H = H0 + H1: Discrete Spectrum	42	(2)
Other Rescalings when the Time Correlations Are not Integrable	44	(1)
Connections with the Classical Homogeneization Problem	45	(1)
Algebraic Formulation of the Stochastic Limit	46	(3)
Notes	49	(8)
Quantum Fields	57	(28)
Creation and Annihilation Operators	57	(2)
Gaussianity	59	(1)
Types of Gaussian States: Gauge-Invariant, Squeezed, Fock and Anti-Fock	60	(1)
Free Evolutions of Quantum Fields	61	(2)
States Invariant Under Free Evolutions	63	(2)
Existence of Squeezed Stationary Fields	65	(2)
Positivity of the Covariance	67	(1)
Dynamical Systems in Equilibrium: the KMS Condition	68	(1)
Equilibrium States: the KMS Condition	69	(1)
q-Gaussian Equilibrium States	70	(1)
Boson Gaussianity	71	(1)
Boson Fock Fields	72	(3)
Free Hamiltonians for Boson Fock Fields	75	(1)
White Noises	76	(1)
Boson Fock White Noises and Classical White Noises	76	(1)
Boson Fock White Noises and Classical Wiener Processes	77	(1)
Boson Thermal Statistics and Thermal White Noises	77	(2)
Canonical Representation of the Boson Thermal States	79	(2)
Spectral Representation of Quantum White Noise	81	(2)
Locality of Quantum Fields and Ultralocality of Quantum White Noises	83	(2)
Those Kinds of Fields We Call Noises	85	(28)
Convergence of Fields in the Sense of Correlators	85	(1)
Generalized White Noises as the Stochastic Limit of Gaussian Fields	86	(4)
Existence of Fock, Temperature and Squeezed White Noises	90	(2)
Convergence of the Field Operator to a Classical White Noise	92	(1)
Beyond the Master Equation: The Master Field	93	(1)
Discrete Spectrum Embedded in the Continuum	94	(3)
The Stochastic Limit of a Classical Gaussian Random Field	97	(1)
Semiclassical Versus Semiquantum Approximation	98	(2)
An Historical Example: The Damped Harmonic Oscillator	100	(2)
Emergence of the White Noise: A Traditional Derivation	102	(1)
Heuristic Origins of Quantum White Noise	103	(1)
Relativistic Quantum White Noises	104	(2)
Space--Time Rescalings: Multidimensional White Noises	106	(2)
The Chronological Stochastic Limit	108	(3)
Notes	111	(2)
Open Systems	113	(40)
The Nonrelativistic QED Hamiltonian	113	(3)
The Dipole Approximation	116	(1)
The Rotating-Wave Approximation	117	(1)
Composite Systems	118	(1)
Assumptions on the Environment (Field, Gas, Reservoir, etc.)	119	(1)
Assumptions on the System Hamiltonian	120	(1)
The Free Hamiltonian	121	(1)
Multiplicative (Dipole-Type) Interactions: Canonical Form	121	(1)
Approximations of the Multiplicative Hamiltonian	122	(3)
Rotating-Wave Approximation Hamiltonians	123	(1)
No Rotating-Wave Approximation Hamiltonians with Cutoff	123	(1)
Neither Dipole nor Rotating-Wave Approximation Hamiltonians Without Cutoff	124	(1)
The Generalized Rotating-Wave Approximation	125	(1)
The Stochastic Limit of the Multiplicative Interaction	126	(1)
The Normal Form of the White Noise Hamiltonian Equation	127	(1)
Invariance of the Ito Correction Term Under Free System Evolution	128	(1)
The Stochastic Golden Rule: Langevin and Master Equations	129	(3)
Classical Stochastic Processes Underlying Quantum Processes	132	(1)
The Fluctuation-Dissipation Relation	133	(1)
Vacuum Transition Amplitudes	134	(2)
Mass Gap of D+ D and Speed of Decay	136	(1)
How to Avoid Decoherence	137	(1)
The Energy Shell Scalar Product: Linewidths	138	(2)
Dispersion Relations and the Ito Correction Term	140	(1)
The Case: H1 = |k2|	141	(1)
Multiplicative Coupling with the Rotating Wave Approximation: Arbitrary Gaussian State	142	(1)
Multiplicative Coupling with RWA: Gauge Invariant State	143	(1)
Red Shifts and Blue Shifts	144	(1)
The Free Evolution of the Master Field	145	(2)
Algebras Invariant Under the Flow	147	(1)
Notes	148	(5)
Spin-Boson Systems	153	(52)
Dropping the Rotating-Wave Approximation	154	(1)
The Master Field	154	(3)
The White Noise Hamiltonian Equation	157	(1)
The Operator Transport Coefficient: no Rotating-Wave Approximation, Arbitrary Gaussian Reference State	158	(1)
Different Roles of the Positive and Negative Bohr Frequencies	159	(2)
No Rotating-Wave Approximation with Cutoff: Gauge Invariant States	161	(1)
No Rotating-Wave Approximation with Cutoff: Squeezing States	161	(1)
The Stochastic Golden Rule for Dipole Type Interactions and Gauge-Invariant States	162	(2)
The Stochastic Golden Rule	164	(6)
The Langevin Equation	170	(2)
Subalgebras Invariant Under the Generator	172	(1)
The Langevin Equation: Generic Systems	173	(3)
The Stochastic Golden Rule Versus Standard Perturbation Theory	176	(2)
Spin-Boson Hamiltonian	178	(3)
The Damping and Oscillating Regimes: Fock Case	181	(2)
The Damping and the Oscillating Regimes: Nonzero Temperature	183	(1)
No Rotating-Wave Approximation Without Cutoff	184	(1)
The Drift Term for Gauge-Invariant States	185	(2)
The Free Evolution of the Master Field	187	(1)
The Stochastic Limit of the Generalized Spin-Boson Hamiltonian	187	(3)
The Langevin Equation	190	(1)
Convergence to Equilibrium: Connections with Quantum Measurement	191	(3)
Control of Coherence	194	(2)
Dynamics of Spin Systems	196	(5)
Nonstationary White Noises	201	(1)
Notes	202	(3)
Measurements and Filtering Theory	205	(14)
Input--Output Channels	205	(1)
The Filtering Problem in Classical Probability	206	(1)
Field Measurements	207	(3)
Properties of the Input and Output Processes	210	(3)
The Filtering Problem in Quantum Theory	213	(1)
Filtering of a Quantum System Over a Classical Process	214	(1)
Nondemolition Processes	215	(2)
Standard Scheme to Construct Examples of Nondemolition Measurements	217	(1)
Discrete Time Nondemolition Processes	217	(1)
Notes	218	(1)
Idea of the Proof and Causal Normal Order	219	(18)
Term-by-Term Convergence of the Series	219	(1)
Vacuum Transition Amplitude: The Fourth-Order Term	220	(3)
Vacuum Transition Amplitude: Non-Time-Consecutive Diagrams	223	(2)
The Causal δ-Function and the Time-Consecutive Principle	225	(2)
Theory of Distributions on the Standard Simplex	227	(3)
The Second-Order Term of the Limit Vacuum Amplitude	230	(1)
The Fourth-Order Term of the Limit Vacuum Amplitude	230	(1)
Higher-Order Terms of the Vacuum-Vacuum Amplitude	231	(1)
Proof of the Normal Form of the White Noise Hamiltonian Equation	231	(2)
The Unitarity Condition for the Limit Equation	233	(1)
Normal Form of the Thermal White Noise Equation: Boson Case	234	(1)
From White Noise Calculus to Stochastic Calculus	235	(2)
Chronological Product Approach to the Stochastic Limit	237	(10)
Chronological Products	237	(1)
Chronological Product Approach to the Stochastic Limit	238	(4)
The Limit of the nth Term, Time-Ordered Product Approach: Vacuum Expectation	242	(2)
The Stochastic Limit, Time-Ordered Product Approach: General Case	244	(3)
Functional Integral Approach to the Stochastic Limit	247	(8)
Statement of the Problem	247	(1)
The Stochastic Limit of the Free Massive Scalar Field	248	(2)
The Stochastic Limit of the Free Massless Scalar Field	250	(1)
Polynomial Interactions	251	(2)
The Stochastic Limit of the Electromagnetic Field	253	(2)
Low-Density Limit: The Basic Idea	255	(6)
The Low-Density Limit: Fock Case, No System	255	(2)
The Low-Density Limit: Fock Case, Arbitrary System Operator	257	(1)
Comparison of the Distribution and the Stochastic Approach	258	(1)
LDL General, Fock Case, No System Operator, ω = 0	259	(2)
Six Basic Principles of the Stochastic Limit	261	(24)
Polynomial Interactions with Cutoff	262	(1)
Assumptions on the Dynamics: Standard Models	263	(3)
Polynomial Interactions: Canonical Forms, Fock Case	266	(3)
Polynomial Interactions: Canonical Forms, Gauge-Invariant Case	269	(6)
The Stochastic Universality Class Principle	275	(2)
The Case of Many Independent Fields	277	(1)
The Block Principle: Fock Case	278	(2)
The Stochastic Resonance Principle	280	(1)
The Orthogonalization Principle	280	(1)
The Stochastic Bosonization Principle	280	(2)
The Time-Consecutive Principle	282	(3)
Part II. Strongly Nonlinear Regimes
Particles Interacting with a Boson Field	285	(66)
A Single Particle Interacting with a Boson Field	287	(2)
Dynamical q-Deformation: Emergence of the Entangled Commutation Relations	289	(2)
The Two-Point and Four-Point Correlators	291	(2)
The Stochastic Limit of the N-Point Correlator	293	(3)
The q-Deformed Module Wick Theorem	296	(5)
The Wick Theorem for the QED Module	301	(1)
The Limit White Noise Hamiltonian Equation	302	(2)
Free Independence of the Increments of the Master Field	304	(2)
Boltzmannian White Noise Hamiltonian Equations: Normal Form	306	(3)
Unitarity Conditions	309	(1)
Matrix Elements of the Solution	310	(2)
Normal Form of the QED Module Hamiltonian Equation	312	(1)
Unitarity of the Solution: Direct Proof	313	(1)
Matrix Elements of the Limit Evolution Operator	314	(3)
Nonexponential Decays	317	(4)
Equilibrium States	321	(1)
The Master Field	321	(1)
Proof of the Result for the Two- and Four-Point Correlators	322	(2)
The Vanishing of the Crossing Diagrams	324	(9)
The Hot Free Algebra	333	(2)
Interaction of the QEM Field with a Nonfree Particle	335	(3)
The Limit Two-Point Function	338	(4)
The Limit Four-Point Function	342	(2)
The Limit Hilbert Module	344	(3)
The Limit Stochastic Process	347	(1)
The Stochastic Differential Equation	348	(1)
Notes	349	(2)
The Anderson Model	351	(18)
Nonrelativistic Fermions in External Potential: The Anderson Model	352	(3)
The Limit of the Connected Correlators	355	(1)
The Four-Point Function	356	(3)
The Limit of the Connected Transition Amplitude	359	(3)
Proof of (13.4.3)	362	(3)
Solution of the Nonlinear Equation (13.4.2)	365	(2)
Notes	367	(2)
Field--Field Interactions	369	(24)
Interacting Commutation Relations	369	(4)
The Tri-linear Hamiltonian with Momentum Conservation	373	(2)
Proof of Theorem 14.2.1	375	(4)
Example: Four Internal Lines	379	(1)
The Stochastic Limit for Green Functions	380	(1)
Second Quantized Representation of the Nonrelativistic QED Hamiltonian	381	(2)
Interacting Commutation Relations and QED Module Algebra	383	(1)
Decay and the Universality Class of the QED Hamiltonian	384	(2)
Photon Splitting Cascades and New Statistics	386	(7)
Part III. Estimates and Proofs
Analytical Theory of Feynman Diagrams	393	(60)
The Connected Component Theorem	395	(7)
The Factorization Theorem	402	(9)
The Case of Many Independent Fields	411	(1)
The Fermi Block Theorem	411	(5)
Non-Time-Consecutive Terms: The First Vanishing Theorem	416	(3)
Non-Time Consecutive Terms: The Second Vanishing Theorem	419	(4)
The Type-I Term Theorem	423	(8)
The Double Integral Lemma	431	(4)
The Multiple-Simplex Theorem	435	(3)
The Multiple Integral Lemma	438	(1)
The Second Multiple-Simplex Theorem	439	(7)
Some Combinatorial Facts and the Block Normal Ordering Theorem	446	(7)
Term-by-Term Convergence	453	(8)
The Universality Class Principle and Effective Interaction Hamiltonians	455	(3)
Block and Orthogonalization Principles	458	(1)
The Stochastic Resonance Principle	459	(2)
References	461	(8)
Index	469