Holdings: Quantum theory and its stochastic limit

Quantum theory and its stochastic limit:

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Bibliographic Details
Main Authors:	Accardi, Luigi 1947- (Author), Lu, Yun Gang 1955- (Author), Volovič, Igorʹ V. 1946- (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2002
Series:	Physics and astronomy online library
Subjects:	Kwantummechanica Niet-lineaire problemen Stochastische analyse Quantentheorie Quantum theory Stochastic processes Quantenmechanik Stochastischer Prozess Stochastik
Online Access:	Inhaltsverzeichnis
Physical Description:	XX, 473 S.
ISBN:	3540419284

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650		7	\|a Kwantummechanica \|2 gtt
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650		4	\|a Quantentheorie
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650		4	\|a Stochastic processes
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Record in the Search Index

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adam_text	CONTENTS PART I. STATEMENT OF THE PROBLEM AND SIMPLEST MODELS 1. NOTATIONS AND STATEMENT OF THE PROBLEM ................. 3 1.1 THE SCHR¨ ODINGER EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 WHITE NOISE APPROXIMATION FOR A FREE PARTICLE: THE BASIC FORMULA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 THE INTERACTION REPRESENTATION: PROPAGATORS . . . . . . . . . . . . . 9 1.4 THE HEISENBERG EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 DYNAMICAL S YSTEMS AND THEIR PERTURBATIONS . . . . . . . . . . . . . 11 1.6 ASYMPTOTIC BEHAVIOUR OF DYNAMICAL SYSTEMS: THE S TOCHASTIC LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 S LOW AND FAST DEGREES OF FREEDOM . . . . . . . . . . . . . . . . . . . . . . 13 1.8 THE QUANTUM TRANSPORT COEFFICIENT: WHY JUST THE T/* 2 S CALING? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.9 EMERGENCE OF WHITE NOISE HAMILTONIAN EQUATIONS FROM THE S TOCHASTIC LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.10 FROM INTERACTION TO HEISENBERG EVOLUTIONS: CONDITIONS ON THE S TATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.11 FROM INTERACTION TO HEISENBERG EVOLUTIONS: CONDITIONS ON THE OBSERVABLE. . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.12 FORWARD AND BACKWARD LANGEVIN EQUATIONS . . . . . . . . . . . . . . 23 1.13 THE OPEN SYSTEM APPROACH TO DISSIPATION AND IRREVERSIBILITY: MASTER EQUATION . . . . . . . . . . . . . . . . . . . . . 24 1.14 CLASSICAL PROCESSES DRIVING QUANTUM PHENOMENA . . . . . . . . . 27 1.15 THE BASIC S TEPS OF THE S TOCHASTIC LIMIT . . . . . . . . . . . . . . . . . . 27 1.16 CONNECTION WITH THE CENTRAL LIMIT THEOREMS . . . . . . . . . . . . . 30 1.17 CLASSICAL S YSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.18 THE BACKWARD TRANSPORT COEFFICIENT AND THE ARROW OF TIME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.19 THE MASTER AND FOKKERPLANCK EQUATIONS: PROJECTION TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.20 THE MASTER EQUATION IN OPEN SYSTEMS: AN HEURISTIC DERIVATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 XIV CONTENTS 1.21 THE SEMICLASSICAL APPROXIMATION FOR THE MASTER EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.22 BEYOND THE MASTER EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.23 ON THE MEANING OF THE DECOMPOSITION H = H 0 + H I : DISCRETE S PECTRUM . . . . . . . . . . . . . . . . . . . . . . . 42 1.24 OTHER RESCALINGS WHEN THE TIME CORRELATIONS ARE NOT INTEGRABLE . . . . . . . . . . . 44 1.25 CONNECTIONS WITH THE CLASSICAL HOMOGENEIZATION PROBLEM . . 45 1.26 ALGEBRAIC FORMULATION OF THE S TOCHASTIC LIMIT . . . . . . . . . . . . 46 1.27 NOTES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2. QUANTUM FIELDS .......................................... 57 2.1 CREATION AND ANNIHILATION OPERATORS . . . . . . . . . . . . . . . . . . . . 57 2.2 GAUSSIANITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 TYPES OF GAUSSIAN STATES: GAUGE-INVARIANT, SQUEEZED, FOCK AND ANTI-FOCK . . . . . . . . . . . 60 2.4 FREE EVOLUTIONS OF QUANTUM FIELDS . . . . . . . . . . . . . . . . . . . . . . 61 2.5 S TATES INVARIANT UNDER FREE EVOLUTIONS . . . . . . . . . . . . . . . . . . 63 2.6 EXISTENCE OF SQUEEZED STATIONARY FIELDS . . . . . . . . . . . . . . . . . . 65 2.7 POSITIVITY OF THE COVARIANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.8 DYNAMICAL SYSTEMS IN EQUILIBRIUM: THE KMS CONDITION . . . 68 2.9 EQUILIBRIUM STATES: THE KMS CONDITION . . . . . . . . . . . . . . . . . . 69 2.10 Q -GAUSSIAN EQUILIBRIUM STATES . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.11 BOSON GAUSSIANITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.12 BOSON FOCK FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.13 FREE HAMILTONIANS FOR BOSON FOCK FIELDS . . . . . . . . . . . . . . . . . 75 2.14 WHITE NOISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.15 BOSON FOCK WHITE NOISES AND CLASSICAL WHITE NOISES . . . . . . 76 2.16 BOSON FOCK WHITE NOISES AND CLASSICAL WIENER PROCESSES . . 77 2.17 BOSON THERMAL S TATISTICS AND THERMAL WHITE NOISES . . . . . . 77 2.18 CANONICAL REPRESENTATION OF THE BOSON THERMAL S TATES . . . . 79 2.19 S PECTRAL REPRESENTATION OF QUANTUM WHITE NOISE . . . . . . . . . 81 2.20 LOCALITY OF QUANTUM FIELDS AND ULTRALOCALITY OF QUANTUM WHITE NOISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3. THOSE KINDS OF FIELDS WE CALL NOISES .................... 85 3.1 CONVERGENCE OF FIELDS IN THE S ENSE OF CORRELATORS . . . . . . . . . 85 3.2 GENERALIZED WHITE NOISES AS THE STOCHASTIC LIMIT OF GAUSSIAN FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3 EXISTENCE OF FOCK, TEMPERATURE AND SQUEEZED WHITE NOISES 90 3.4 CONVERGENCE OF THE FIELD OPERATOR TO A CLASSICAL WHITE NOISE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.5 BEYOND THE MASTER EQUATION: THE MASTER FIELD . . . . . . . . . . . 93 3.6 DISCRETE S PECTRUM EMBEDDED IN THE CONTINUUM . . . . . . . . . . 94 3.7 THE STOCHASTIC LIMIT OF A CLASSICAL GAUSSIAN RANDOM FIELD 97 CONTENTS XV 3.8 S EMICLASSICAL VERSUS S EMIQUANTUM APPROXIMATION . . . . . . . . 98 3.9 AN HISTORICAL EXAMPLE: THE DAMPED HARMONIC OSCILLATOR. . 100 3.10 EMERGENCE OF THE WHITE NOISE: A TRADITIONAL DERIVATION . . . 102 3.11 HEURISTIC ORIGINS OF QUANTUM WHITE NOISE . . . . . . . . . . . . . . . 103 3.12 RELATIVISTIC QUANTUM WHITE NOISES . . . . . . . . . . . . . . . . . . . . . 104 3.13 S PACETIME RESCALINGS: MULTIDIMENSIONAL WHITE NOISES . . . . 106 3.14 THE CHRONOLOGICAL S TOCHASTIC LIMIT . . . . . . . . . . . . . . . . . . . . . 108 3.15 NOTES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4 . OPEN SYSTEMS ............................................ 113 4.1 THE NONRELATIVISTIC QED HAMILTONIAN . . . . . . . . . . . . . . . . . . . 113 4.2 THE DIPOLE APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3 THE ROTATING-WAVE APPROXIMATION . . . . . . . . . . . . . . . . . . . . . 117 4.4 COMPOSITE S YSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.5 ASSUMPTIONS ON THE ENVIRONMENT (FIELD, GAS, RESERVOIR, ETC.) 119 4.6 ASSUMPTIONS ON THE S YSTEM HAMILTONIAN . . . . . . . . . . . . . . . . . 120 4.7 THE FREE HAMILTONIAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.8 MULTIPLICATIVE (DIPOLE-TYPE) INTERACTIONS:CANONICAL FORM . . 121 4.9 APPROXIMATIONS OF THE MULTIPLICATIVE HAMILTONIAN . . . . . . . . 122 4.9.1 ROTATING-WAVE APPROXIMATION HAMILTONIANS . . . . . . 123 4.9.2 NO ROTATING-WAVE APPROXIMATION HAMILTONIANS WITH CUTOFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.9.3 NEITHER DIPOLE NOR ROTATING-WAVE APPROXIMATION HAMILTONIANS WITHOUT CUTOFF . . . . . . . . . . . . . . . . . . . . 124 4.10 THE GENERALIZED ROTATING-WAVE APPROXIMATION . . . . . . . . . . . 125 4.11 THE S TOCHASTIC LIMIT OF THE MULTIPLICATIVE INTERACTION . . . . . 126 4.12 THE NORMAL FORM OF THE WHITE NOISE HAMILTONIAN EQUATION 127 4.13 INVARIANCE OF THE ITO CORRECTION TERM UNDER FREE S YSTEM EVOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.14 THE STOCHASTIC GOLDEN RULE: LANGEVIN AND MASTER EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . 129 4.15 CLASSICAL STOCHASTIC PROCESSES UNDERLYING QUANTUM PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.16 THE FLUCTUATIONDISSIPATION RELATION . . . . . . . . . . . . . . . . . . . 133 4.17 VACUUM TRANSITION AMPLITUDES . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.18 MASS GAP OF D + D AND S PEED OF DECAY . . . . . . . . . . . . . . . . . . 136 4.19 HOW TO AVOID DECOHERENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.20 THE ENERGY S HELL S CALAR PRODUCT: LINEWIDTHS . . . . . . . . . . . . . 138 4.21 DISPERSION RELATIONS AND THE ITO CORRECTION TERM . . . . . . . . . 140 4.22 THE CASE: H 1 = \| K 2 \| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.23 MULTIPLICATIVE COUPLING WITH THE ROTATING WAVE APPROXIMATION: ARBITRARY GAUSSIAN S TATE . . . . . . . . . . . . . . . . 142 4.24 MULTIPLICATIVE COUPLING WITH RWA: GAUGE INVARIANT STATE . 143 4.25 RED S HIFTS AND BLUE S HIFTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.26 THE FREE EVOLUTION OF THE MASTER FIELD . . . . . . . . . . . . . . . . . . 145 XVI CONTENTS 4.27 ALGEBRAS INVARIANT UNDER THE FLOW . . . . . . . . . . . . . . . . . . . . . . 147 4.28 NOTES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5. SPINBOSON SYSTEMS ...................................... 153 5.1 DROPPING THE ROTATING-WAVE APPROXIMATION . . . . . . . . . . . . . 154 5.2 THE MASTER FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.3 THE WHITE NOISE HAMILTONIAN EQUATION . . . . . . . . . . . . . . . . . 157 5.4 THE OPERATOR TRANSPORT COEFFICIENT: NO ROTATING-WAVE APPROXIMATION, ARBITRARY GAUSSIAN REFERENCE S TATE . . . . . . . 158 5.5 DIFFERENT ROLES OF THE POSITIVE AND NEGATIVE BOHR FREQUENCIES . . . . . . . . . . . . 159 5.6 NO ROTATING-WAVE APPROXIMATION WITH CUTOFF: GAUGE INVARIANT S TATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.7 NO ROTATING-WAVE APPROXIMATION WITH CUTOFF: SQUEEZING STATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.8 THE STOCHASTIC GOLDEN RULE FOR DIPOLE TYPE INTERACTIONS AND GAUGE-INVARIANT S TATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.9 THE S TOCHASTIC GOLDEN RULE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.10 THE LANGEVIN EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.11 S UBALGEBRAS INVARIANT UNDER THE GENERATOR. . . . . . . . . . . . . . . 172 5.12 THE LANGEVIN EQUATION: GENERIC S YSTEMS . . . . . . . . . . . . . . . . 173 5.13 THE STOCHASTIC GOLDEN RULE VERSUS S TANDARD PERTURBATION THEORY . . . . . . . . . . . . . . . . . . . 176 5.14 S PINBOSON HAMILTONIAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.15 THE DAMPING AND OSCILLATING REGIMES: FOCK CASE . . . . . . . . 181 5.16 THE DAMPING AND THE OSCILLATING REGIMES: NONZERO TEMPERATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.17 NO ROTATING-WAVE APPROXIMATION WITHOUT CUTOFF . . . . . . . . 184 5.18 THE DRIFT TERM FOR GAUGE-INVARIANT S TATES . . . . . . . . . . . . . . . 185 5.19 THE FREE EVOLUTION OF THE MASTER FIELD . . . . . . . . . . . . . . . . . . 187 5.20 THE STOCHASTIC LIMIT OF THE GENERALIZED S PINBOSON HAMILTONIAN . . . . . . . . . . . . . . 187 5.21 THE LANGEVIN EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.22 CONVERGENCE TO EQUILIBRIUM: CONNECTIONS WITH QUANTUM MEASUREMENT . . . . . . . . . . . . . . . . 191 5.23 CONTROL OF COHERENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.24 DYNAMICS OF S PIN S YSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.25 NONSTATIONARY WHITE NOISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.26 NOTES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6. MEASUREMENTS AND FILTERING THEORY ...................... 205 6.1 INPUTOUTPUT CHANNELS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.2 THE FILTERING PROBLEM IN CLASSICAL PROBABILITY . . . . . . . . . . . . 206 6.3 FIELD MEASUREMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.4 PROPERTIES OF THE INPUT AND OUTPUT PROCESSES . . . . . . . . . . . . 210 CONTENTS XVII 6.5 THE FILTERING PROBLEM IN QUANTUM THEORY . . . . . . . . . . . . . . . 213 6.6 FILTERING OF A QUANTUM SYSTEM OVER A CLASSICAL PROCESS . . . 214 6.7 NONDEMOLITION PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.8 STANDARD SCHEME TO CONSTRUCT EXAMPLES OF NONDEMOLITION MEASUREMENTS . . . . . . . . . . . . . . . . . . . . . . . . 217 6.9 DISCRETE TIME NONDEMOLITION PROCESSES . . . . . . . . . . . . . . . . . . 217 6.10 NOTES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7. IDEA OF THE PROOF AND CAUSAL NORMAL ORDER .............. 219 7.1 TERM-BY-TERM CONVERGENCE OF THE S ERIES . . . . . . . . . . . . . . . . . 219 7.2 VACUUM TRANSITION AMPLITUDE: THE FOURTH-ORDER TERM . . . 220 7.3 VACUUM TRANSITION AMPLITUDE: NON-TIME-CONSECUTIVE DIAGRAMS . . . . . . . . . . . . . . . . . . . . . . . . 223 7.4 THE CAUSAL -FUNCTION AND THE TIME-CONSECUTIVE PRINCIPLE 225 7.5 THEORY OF DISTRIBUTIONS ON THE S TANDARD S IMPLEX . . . . . . . . . 227 7.6 THE SECOND-ORDER TERM OF THE LIMIT VACUUM AMPLITUDE . . 230 7.7 THE FOURTH-ORDER TERM OF THE LIMIT VACUUM AMPLITUDE . . 230 7.8 HIGHER-ORDER TERMS OF THE VACUUMVACUUM AMPLITUDE . . . 231 7.9 PROOF OF THE NORMAL FORM OF THE WHITE NOISE HAMILTONIAN EQUATION . . . . . . . . . . . . . . . . 231 7.10 THE UNITARITY CONDITION FOR THE LIMIT EQUATION . . . . . . . . . . 233 7.11 NORMAL FORM OF THE THERMAL WHITE NOISE EQUATION: BOSON CASE . . . . . . . . 234 7.12 FROM WHITE NOISE CALCULUS TO S TOCHASTIC CALCULUS . . . . . . . . 235 8. CHRONOLOGICAL PRODUCT APPROACH TO THE STOCHASTIC LIMIT . 237 8.1 CHRONOLOGICAL PRODUCTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.2 CHRONOLOGICAL PRODUCT APPROACH TO THE STOCHASTIC LIMIT . . . 238 8.3 THE LIMIT OF THE N TH TERM, TIME-ORDERED PRODUCT APPROACH: VACUUM EXPECTATION . . . . 242 8.4 THE STOCHASTIC LIMIT, TIME-ORDERED PRODUCT APPROACH: GENERAL CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9. FUNCTIONAL INTEGRAL APPROACH TO THE STOCHASTIC LIMIT ..... 247 9.1 S TATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 9.2 THE S TOCHASTIC LIMIT OF THE FREE MASSIVE S CALAR FIELD . . . . . 248 9.3 THE S TOCHASTIC LIMIT OF THE FREE MASSLESS S CALAR FIELD . . . . . 250 9.4 POLYNOMIAL INTERACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.5 THE S TOCHASTIC LIMIT OF THE ELECTROMAGNETIC FIELD . . . . . . . . 253 10. LOW-DENSITY LIMIT: THE BASIC IDEA ....................... 255 10.1 THE LOW-DENSITY LIMIT: FOCK CASE, NO S YSTEM . . . . . . . . . . . 255 10.2 THE LOW-DENSITY LIMIT: FOCK CASE, ARBITRARY S YSTEM OPERATOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 XVIII CONTENTS 10.3 COMPARISON OF THE DISTRIBUTION AND THE S TOCHASTIC APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 10.4 LDL GENERAL, FOCK CASE, NO SYSTEM OPERATOR, = 0 . . . . . 259 11. SIX BASIC PRINCIPLES OF THE STOCHASTIC LIMIT ............... 261 11.1 POLYNOMIAL INTERACTIONS WITH CUTOFF . . . . . . . . . . . . . . . . . . . . . 262 11.2 ASSUMPTIONS ON THE DYNAMICS: S TANDARD MODELS . . . . . . . . . . 263 11.3 POLYNOMIAL INTERACTIONS: CANONICAL FORMS, FOCK CASE . . . . . . 266 11.4 POLYNOMIAL INTERACTIONS: CANONICAL FORMS, GAUGE-INVARIANT CASE . . . . . . . . . . . . . . . . . . 269 11.5 THE S TOCHASTIC UNIVERSALITY CLASS PRINCIPLE . . . . . . . . . . . . . . . 275 11.6 THE CASE OF MANY INDEPENDENT FIELDS . . . . . . . . . . . . . . . . . . . 277 11.7 THE BLOCK PRINCIPLE: FOCK CASE . . . . . . . . . . . . . . . . . . . . . . . . . 278 11.8 THE S TOCHASTIC RESONANCE PRINCIPLE . . . . . . . . . . . . . . . . . . . . . 280 11.9 THE ORTHOGONALIZATION PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . 280 11.10 THE S TOCHASTIC BOSONIZATION PRINCIPLE . . . . . . . . . . . . . . . . . . . 280 11.11 THE TIME-CONSECUTIVE PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . 282 PART II. STRONGLY NONLINEAR REGIMES 12. PARTICLES INTERACTING WITH A BOSON FIELD .................. 285 12.1 A S INGLE PARTICLE INTERACTING WITH A BOSON FIELD . . . . . . . . . . 287 12.2 DYNAMICAL Q -DEFORMATION: EMERGENCE OF THE ENTANGLED COMMUTATION RELATIONS . . . . . . . . . . . . . . . . 289 12.3 THE TWO-POINT AND FOUR-POINT CORRELATORS . . . . . . . . . . . . . . . 291 12.4 THE STOCHASTIC LIMIT OF THE N -POINT CORRELATOR . . . . . . . . . . . 293 12.5 THE Q -DEFORMED MODULE WICK THEOREM . . . . . . . . . . . . . . . . . 296 12.6 THE WICK THEOREM FOR THE QED MODULE . . . . . . . . . . . . . . . . . 301 12.7 THE LIMIT WHITE NOISE HAMILTONIAN EQUATION . . . . . . . . . . . . 302 12.8 FREE INDEPENDENCE OF THE INCREMENTS OF THE MASTER FIELD . . . 304 12.9 BOLTZMANNIAN WHITE NOISE HAMILTONIAN EQUATIONS: NORMAL FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 12.10 UNITARITY CONDITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 12.11 MATRIX ELEMENTS OF THE S OLUTION . . . . . . . . . . . . . . . . . . . . . . . . 310 12.12 NORMAL FORM OF THE QED MODULE HAMILTONIAN EQUATION. . . 312 12.13 UNITARITY OF THE S OLUTION: DIRECT PROOF . . . . . . . . . . . . . . . . . . . 313 12.14 MATRIX ELEMENTS OF THE LIMIT EVOLUTION OPERATOR . . . . . . . . . 314 12.15 NONEXPONENTIAL DECAYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 12.16 EQUILIBRIUM STATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 12.17 THE MASTER FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 12.18 PROOF OF THE RESULT FOR THE TWO- AND FOUR-POINT CORRELATORS . . . . . . . . . . . . . . . . . 322 12.19 THE VANISHING OF THE CROSSING DIAGRAMS . . . . . . . . . . . . . . . . . 324 12.20 THE HOT FREE ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 CONTENTS XIX 12.21 INTERACTION OF THE QEM FIELD WITH A NONFREE PARTICLE . . . . . . 335 12.22 THE LIMIT TWO-POINT FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . 338 12.23 THE LIMIT FOUR-POINT FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . 342 12.24 THE LIMIT HILBERT MODULE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 12.25 THE LIMIT S TOCHASTIC PROCESS . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 12.26 THE S TOCHASTIC DIFFERENTIAL EQUATION . . . . . . . . . . . . . . . . . . . . 348 12.27 NOTES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 13. THE ANDERSON MODEL ..................................... 351 13.1 NONRELATIVISTIC FERMIONS IN EXTERNAL POTENTIAL: THE ANDERSON MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 13.2 THE LIMIT OF THE CONNECTED CORRELATORS . . . . . . . . . . . . . . . . . 355 13.3 THE FOUR-POINT FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 13.4 THE LIMIT OF THE CONNECTED TRANSITION AMPLITUDE . . . . . . . . 359 13.5 PROOF OF (13.4.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 13.6 S OLUTION OF THE NONLINEAR EQUATION (13.4.2) . . . . . . . . . . . . . . 365 13.7 NOTES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 14. FIELD*FIELD INTERACTIONS .................................. 369 14.1 INTERACTING COMMUTATION RELATIONS . . . . . . . . . . . . . . . . . . . . . 369 14.2 THE TRI-LINEAR HAMILTONIAN WITH MOMENTUM CONSERVATION . 373 14.3 PROOF OF THEOREM 14.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 14.4 EXAMPLE: FOUR INTERNAL LINES . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 14.5 THE S TOCHASTIC LIMIT FOR GREEN FUNCTIONS . . . . . . . . . . . . . . . . 380 14.6 SECOND QUANTIZED REPRESENTATION OF THE NONRELATIVISTIC QED HAMILTONIAN . . . . . . . . . . . . . . . . . 381 14.7 INTERACTING COMMUTATION RELATIONS AND QED MODULE ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 14.8 DECAY AND THE UNIVERSALITY CLASS OF THE QED HAMILTONIAN . 384 14.9 PHOTON S PLITTING CASCADES AND NEW S TATISTICS . . . . . . . . . . . . 386 PART III. ESTIMATES AND PROOFS 15. ANALYTICAL THEORY OF FEYNMAN DIAGRAMS ................. 393 15.1 THE CONNECTED COMPONENT THEOREM . . . . . . . . . . . . . . . . . . . . 395 15.2 THE FACTORIZATION THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 15.3 THE CASE OF MANY INDEPENDENT FIELDS . . . . . . . . . . . . . . . . . . . 411 15.4 THE FERMI BLOCK THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 15.5 NON-TIME-CONSECUTIVE TERMS: THE FIRST VANISHING THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . 416 15.6 NON-TIME CONSECUTIVE TERMS: THE S ECOND VANISHING THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . 419 15.7 THE TYPE-I TERM THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 15.8 THE DOUBLE INTEGRAL LEMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 XX CONTENTS 15.9 THE MULTIPLE-S IMPLEX THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . 435 15.10 THE MULTIPLE INTEGRAL LEMMA . . . . . . . . . . . . . . . . . . . . . . . . . . 438 15.11 THE S ECOND MULTIPLE-S IMPLEX THEOREM . . . . . . . . . . . . . . . . . . 439 15.12 SOME COMBINATORIAL FACTS AND THE BLOCK NORMAL ORDERING THEOREM . . . . . . . . . . . . . . . . 446 16. TERM-BY-TERM CONVERGENCE .............................. 453 16.1 THE UNIVERSALITY CLASS PRINCIPLE AND EFFECTIVE INTERACTION HAMILTONIANS . . . . . . . . . . . . . . . . . . . 455 16.2 BLOCK AND ORTHOGONALIZATION PRINCIPLES . . . . . . . . . . . . . . . . . . 458 16.3 THE S TOCHASTIC RESONANCE PRINCIPLE . . . . . . . . . . . . . . . . . . . . . 459 REFERENCES .................................................... 461 INDEX ......................................................... 469
any_adam_object	1
author	Accardi, Luigi 1947- Lu, Yun Gang 1955- Volovič, Igorʹ V. 1946-
author_GND	(DE-588)172488729 (DE-588)124092772 (DE-588)123091411
author_facet	Accardi, Luigi 1947- Lu, Yun Gang 1955- Volovič, Igorʹ V. 1946-
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spelling	Accardi, Luigi 1947- Verfasser (DE-588)172488729 aut Quantum theory and its stochastic limit Luigi Accardi ; Yun Gang Lu ; Igor Volovich Berlin [u.a.] Springer 2002 XX, 473 S. txt rdacontent n rdamedia nc rdacarrier Physics and astronomy online library Kwantummechanica gtt Niet-lineaire problemen gtt Stochastische analyse gtt Quantentheorie Quantum theory Stochastic processes Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Quantentheorie (DE-588)4047992-4 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastik (DE-588)4121729-9 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 s Stochastik (DE-588)4121729-9 s DE-604 Quantentheorie (DE-588)4047992-4 s Stochastischer Prozess (DE-588)4057630-9 s Lu, Yun Gang 1955- Verfasser (DE-588)124092772 aut Volovič, Igorʹ V. 1946- Verfasser (DE-588)123091411 aut SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009530233&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Accardi, Luigi 1947- Lu, Yun Gang 1955- Volovič, Igorʹ V. 1946- Quantum theory and its stochastic limit Kwantummechanica gtt Niet-lineaire problemen gtt Stochastische analyse gtt Quantentheorie Quantum theory Stochastic processes Quantenmechanik (DE-588)4047989-4 gnd Quantentheorie (DE-588)4047992-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Stochastik (DE-588)4121729-9 gnd
subject_GND	(DE-588)4047989-4 (DE-588)4047992-4 (DE-588)4057630-9 (DE-588)4121729-9
title	Quantum theory and its stochastic limit
title_auth	Quantum theory and its stochastic limit
title_exact_search	Quantum theory and its stochastic limit
title_full	Quantum theory and its stochastic limit Luigi Accardi ; Yun Gang Lu ; Igor Volovich
title_fullStr	Quantum theory and its stochastic limit Luigi Accardi ; Yun Gang Lu ; Igor Volovich
title_full_unstemmed	Quantum theory and its stochastic limit Luigi Accardi ; Yun Gang Lu ; Igor Volovich
title_short	Quantum theory and its stochastic limit
title_sort	quantum theory and its stochastic limit
topic	Kwantummechanica gtt Niet-lineaire problemen gtt Stochastische analyse gtt Quantentheorie Quantum theory Stochastic processes Quantenmechanik (DE-588)4047989-4 gnd Quantentheorie (DE-588)4047992-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Stochastik (DE-588)4121729-9 gnd
topic_facet	Kwantummechanica Niet-lineaire problemen Stochastische analyse Quantentheorie Quantum theory Stochastic processes Quantenmechanik Stochastischer Prozess Stochastik
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009530233&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT accardiluigi quantumtheoryanditsstochasticlimit AT luyungang quantumtheoryanditsstochasticlimit AT volovicigorʹv quantumtheoryanditsstochasticlimit

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