Lindgren Relativistic Many-Body Theory

A New Field-Theoretical Approach
1. Auflage 2011
ISBN: 978-1-4419-8309-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 63, 366 Seiten

Reihe: Springer Series on Atomic, Optical, and Plasma Physics

This book gives a comprehensive account of relativistic many-body perturbation theory, based upon field theory. After some introductory chapters about time-independent and time dependent many-body perturbation theory (MBPT), the standard techniques of S-matrix and Green's functions are reviewed. Next, the newly introduced covariant-evolution-operator method is described, which can be used, like the S-matrix method, for calculations in quantum electrodynamics (QED). Unlike the S-matrix method, this has a structure that is similar to that of MBPT and therefore can serve as basis for a unified theory. Such an approach is developed in the final chapters, and its equivalence to the Bethe-Salpeter equation is demonstrated. Possible applications are discussed and numerical illustrations given.

Autoren/Hrsg.

Lindgren, Ingvar

Inhaltsverzeichnis

1;Preface;8
2;Contents;12
3;Chapter 1: Introduction
;20
3.1;1.1 Standard Many-Body Perturbation Theory;20
3.2;1.2 Quantum-Electrodynamics;21
3.3;1.3 Bethe–Salpeter Equation;22
3.4;1.4 Helium Atom: Analytical Approach;23
3.5;1.5 Field-Theoretical Approach to Many-Body Perturbation Theory;24
3.6;References;26
4;Part I Basics: Standard Many-Body Perturbation Theory;29
4.1;Chapter 2: Time-Independent Formalism
;30
4.1.1;2.1 First Quantization;30
4.1.1.1;2.1.1 De Broglie's Relations;30
4.1.1.2;2.1.2 The Schrödinger Equation;31
4.1.2;2.2 Second Quantization;33
4.1.2.1;2.2.1 Schrödinger Equation in Second Quantization;33
4.1.2.2;2.2.2 Particle–Hole Formalism: Normal Order andContraction;35
4.1.2.3;2.2.3 Wick's Theorem;36
4.1.3;2.3 Time-Independent Many-Body Perturbation Theory;37
4.1.3.1;2.3.1 Bloch Equation;37
4.1.3.2;2.3.2 Partitioning of the Hamiltonian;38
4.1.4;2.4 Graphical Representation;42
4.1.4.1;2.4.1 Goldstone Diagrams;42
4.1.4.2;2.4.2 Linked-Diagram Expansion;45
4.1.4.2.1;2.4.2.1 Complete Model Space;45
4.1.4.2.2;2.4.2.2 Incomplete Model Spaces;47
4.1.5;2.5 All-Order Methods: Coupled-Cluster Approach;47
4.1.5.1;2.5.1 Pair Correlation;47
4.1.5.2;2.5.2 Exponential Ansatz;50
4.1.5.3;2.5.3 Various Models for Coupled-Cluster Calculations: Intruder-State Problem;53
4.1.6;2.6 Relativistic MBPT: No-Virtual-Pair Approximation;54
4.1.6.1;2.6.1 QED Effects;56
4.1.7;2.7 Some Numerical Results of Standard MBPT and CC Calculations, Applied to Atoms;57
4.1.8;References;60
4.2;Chapter 3: Time-Dependent Formalism
;64
4.2.1;3.1 Evolution Operator;64
4.2.2;3.2 Adiabatic Damping: Gell-Mann–Low Theorem;68
4.2.2.1;3.2.1 Gell-Mann–Low Theorem;69
4.2.3;3.3 Extended Model Space: The Generalized Gell-Mann–Low Relation;69
4.2.4;References;73
5;Part II Quantum-Electrodynamics: One-Photonand Two-Photon Exchange;74
5.1;Chapter 4: S-Matrix
;75
5.1.1;4.1 Definition of the S-Matrix: Feynman Diagrams;76
5.1.2;4.2 Electron Propagator;77
5.1.3;4.3 Photon Propagator;81
5.1.3.1;4.3.1 Feynman Gauge;82
5.1.3.2;4.3.2 Coulomb Gauge;84
5.1.4;4.4 Single-Photon Exchange;85
5.1.4.1;4.4.1 Covariant Gauge;85
5.1.4.1.1;4.4.1.1 Feynman Gauge;88
5.1.4.2;4.4.2 Noncovariant Coulomb Gauge;88
5.1.4.3;4.4.3 Single-Particle Potential;90
5.1.5;4.5 Two-Photon Exchange;91
5.1.5.1;4.5.1 Two-Photon Ladder;91
5.1.5.2;4.5.2 Two-Photon Cross;94
5.1.6;4.6 QED Corrections;95
5.1.6.1;4.6.1 Bound-Electron Self-Energy;95
5.1.6.1.1;4.6.1.1 Covariant Gauge;96
5.1.6.1.2;4.6.1.2 Coulomb Gauge;97
5.1.6.2;4.6.2 Vertex Correction;98
5.1.6.2.1;4.6.2.1 Covariant Gauge;99
5.1.6.2.2;4.6.2.2 Coulomb Gauge;99
5.1.6.3;4.6.3 Vacuum Polarization;100
5.1.6.4;4.6.4 Photon Self-Energy;103
5.1.7;4.7 Feynman Diagrams for the S-Matrix: Feynman Amplitude;103
5.1.7.1;4.7.1 Feynman Diagrams;103
5.1.7.2;4.7.2 Feynman Amplitude;104
5.1.8;References;105
5.2;Chapter 5: Green's Functions
;106
5.2.1;5.1 Classical Green's Function;106
5.2.2;5.2 Field-Theoretical Green's Function: Closed-Shell Case;107
5.2.2.1;5.2.1 Definition of the Field-Theoretical Green's Function;107
5.2.2.2;5.2.2 Single-Photon Exchange;110
5.2.2.3;5.2.3 Fourier Transform of the Green's Function;111
5.2.2.3.1;5.2.3.1 Single-Particle Green's Function;111
5.2.2.3.2;5.2.3.2 Electron Propagator;113
5.2.2.3.3;5.2.3.3 Two-Particle Green's Function in the Equal-Time Approximation;114
5.2.3;5.3 Graphical Representation of the Green's Function;115
5.2.3.1;5.3.1 Single-Particle Green's Function;115
5.2.3.1.1;5.3.1.1 One-Body Interaction;120
5.2.3.2;5.3.2 Many-Particle Green's Function;120
5.2.3.3;5.3.3 Self-Energy: Dyson Equation;123
5.2.3.4;5.3.4 Numerical Illustration;125
5.2.4;5.4 Field-Theoretical Green's Function: Open-Shell Case;125
5.2.4.1;5.4.1 Definition of the Open-Shell Green's Function;125
5.2.4.2;5.4.2 Two-Times Green's Function of Shabaev;126
5.2.4.3;5.4.3 Single-Photon Exchange;129
5.2.5;References;132
5.3;Chapter 6: Covariant Evolution Operator and Green's Operator
;133
5.3.1;6.1 Definition of the Covariant Evolution Operator;133
5.3.2;6.2 Single-Photon Exchange in theCovariant-Evolution-Operator Formalism;136
5.3.2.1;6.2.1 Single-Photon Ladder;140
5.3.3;6.3 Multiphoton Exchange;141
5.3.3.1;6.3.1 General;141
5.3.3.2;6.3.2 Irreducible Two-Photon Exchange;142
5.3.3.2.1;6.3.2.1 Uncrossing Photons;142
5.3.3.2.2;6.3.2.2 Crossing Photons;144
5.3.3.3;6.3.3 Potential with Radiative Parts;144
5.3.4;6.4 Relativistic Form of the Gell-Mann–Low Theorem;145
5.3.5;6.5 Field-Theoretical Many-Body Hamiltonian;146
5.3.6;6.6 Green's Operator;148
5.3.6.1;6.6.1 Definition;148
5.3.6.2;6.6.2 Relation Between the Green's Operatorand Many-Body Perturbation Procedures;150
5.3.7;6.7 Model-Space Contribution;152
5.3.7.1;6.7.1 Lowest Orders;153
5.3.7.2;6.7.2 All Orders;156
5.3.7.2.1;6.7.2.1 Linkedness of the Green's Operator;160
5.3.8;6.8 Bloch Equation for Green's Operator;160
5.3.9;6.9 Time Dependence of the Green's Operator. Connection to the Bethe–Salpeter Equation;165
5.3.9.1;6.9.1 Single-Reference Model Space;165
5.3.9.2;6.9.2 Multireference Model Space;168
5.3.10;References;170
5.4;Chapter 7: Numerical Illustrations to Part II
;171
5.4.1;7.1 S-Matrix;171
5.4.1.1;7.1.1 Electron Self-Energy of Hydrogen-Like Ions;171
5.4.1.2;7.1.2 Lamb Shift of Hydrogen-Like Uranium;172
5.4.1.3;7.1.3 Lamb Shift of Lithium-Like Uranium;173
5.4.1.4;7.1.4 Two-Photon Nonradiative Exchangein Helium-Like Ions;174
5.4.1.5;7.1.5 Electron Correlation and QED Calculations on Ground States of Helium-Like Ions;177
5.4.1.6;7.1.6 g-Factor of Hydrogen-Like Ions: Mass of the Free Electron;178
5.4.2;7.2 Green's-Function and Covariant-Evolution-Operator Methods;179
5.4.2.1;7.2.1 Fine-Structure of Helium-Like Ions;179
5.4.2.2;7.2.2 Energy Calculations of 1s2s Levelsof Helium-Like Ions;181
5.4.3;References;182
6;Part III Quantum-Electrodynamics Beyond Two-Photon Exchange: Field-Theoretical Approach to Many-Body Perturbation Theory;185
6.1;Chapter 8: Covariant Evolution Combined with Electron Correlation
;186
6.1.1;8.1 General Single-Photon Exchange;186
6.1.1.1;8.1.1 Transverse Part;187
6.1.1.2;8.1.2 Coulomb Interaction;194
6.1.2;8.2 General QED Potential;194
6.1.2.1;8.2.1 Single Photon with Crossed Coulomb Interaction;194
6.1.2.2;8.2.2 Electron Self-Energy and Vertex Correction;199
6.1.2.2.1;8.2.2.1 General Two-Electron Self-Energy;200
6.1.2.2.2;8.2.2.2 General Vertex Correction;203
6.1.2.3;8.2.3 Vertex Correction with Further Coulomb Iterations;205
6.1.2.4;8.2.4 General Two-Body Potential;206
6.1.3;8.3 Unification of the MBPT and QED Procedures: Connection to Bethe–Salpeter Equation;206
6.1.3.1;8.3.1 MBPT–QED Procedure;206
6.1.4;8.4 Coupled-Cluster-QED Expansion;209
6.1.5;References;211
6.2;Chapter 9: The Bethe–Salpeter Equation
;212
6.2.1;9.1 The Original Derivations by the Bethe–Salpeter Equation;212
6.2.1.1;9.1.1 Derivation by Salpeter and Bethe;212
6.2.1.2;9.1.2 Derivation by Gell-Mann and Low;215
6.2.1.3;9.1.3 Analysis of the Derivations of the Bethe–Salpeter Equation;216
6.2.2;9.2 Quasi-Potential and Effective-Potential Approximations: Single-Reference Case;218
6.2.3;9.3 Bethe–Salpeter–Bloch Equation: Multireference Case;219
6.2.4;9.4 Problems with the Bethe–Salpeter Equation;221
6.2.5;References;222
6.3;Chapter 10: Implementation of the MBPT–QED Procedure with Numerical Results
;224
6.3.1;10.1 The Fock-Space Bloch Equation;224
6.3.2;10.2 Single-Photon Potential in Coulomb Gauge: No Virtual Pairs;226
6.3.3;10.3 Single-Photon Exchange: Virtual Pairs;229
6.3.3.1;10.3.1 Illustration;229
6.3.3.2;10.3.2 Full Treatment;232
6.3.3.3;10.3.3 Higher Orders;234
6.3.4;10.4 Numerical Results;234
6.3.4.1;10.4.1 Two-Photon Exchange;234
6.3.4.2;10.4.2 Beyond Two Photons;234
6.3.4.3;10.4.3 Outlook;237
6.3.5;References;237
6.4;Chapter 11: Analytical Treatment of the Bethe–Salpeter Equation
;238
6.4.1;11.1 Helium Fine Structure;238
6.4.2;11.2 The Approach of Sucher;239
6.4.3;11.3 Perturbation Expansion of the BS Equation;244
6.4.4;11.4 Diagrammatic Representation;246
6.4.5;11.5 Comparison with the Numerical Approach;248
6.4.6;References;248
6.5;Chapter 12: Regularization and Renormalization
;250
6.5.1;12.1 The Free-Electron QED;250
6.5.1.1;12.1.1 The Free-Electron Propagator;250
6.5.1.2;12.1.2 The Free-Electron Self-Energy;252
6.5.1.3;12.1.3 The Free-Electron Vertex Correction;253
6.5.2;12.2 Renormalization Process;255
6.5.2.1;12.2.1 Mass Renormalization;255
6.5.2.2;12.2.2 Charge Renormalization;257
6.5.2.2.1;12.2.2.1 Electron Self-Energy;257
6.5.2.2.2;12.2.2.2 Vertex Correction;259
6.5.2.2.3;12.2.2.3 Photon Self-Energy;259
6.5.2.2.4;12.2.2.4 Higher-Order Renormalization;261
6.5.3;12.3 Bound-State Renormalization: Cutoff Procedures;261
6.5.3.1;12.3.1 Mass Renormalization;261
6.5.3.2;12.3.2 Evaluation of the Mass Term;262
6.5.3.3;12.3.3 Bethe's Nonrelativistic Treatment;264
6.5.3.4;12.3.4 Brown–Langer–Schaefer Regularization;265
6.5.3.5;12.3.5 Partial-Wave Regularization;268
6.5.3.5.1;12.3.5.1 Feynman Gauge;268
6.5.3.5.2;12.3.5.2 Coulomb Gauge;269
6.5.4;12.4 Dimensional Regularization in Feynman Gauge;270
6.5.4.1;12.4.1 Evaluation of the Renormalized Free-Electron Self-Energy in Feynman Gauge;271
6.5.4.2;12.4.2 Free-Electron Vertex Correction in Feynman Gauge;274
6.5.5;12.5 Dimensional Regularization in Coulomb Gauge;276
6.5.5.1;12.5.1 Free-Electron Self-Energy in the Coulomb Gauge;276
6.5.6;12.6 Direct Numerical Regularization of the Bound-State Self-Energy;280
6.5.6.1;12.6.1 Feynman Gauge;281
6.5.6.2;12.6.2 Coulomb Gauge;281
6.5.7;References;282
6.6;Chapter 13: Summary and Conclusions
;283
7;Appendix A: Notations and Definitions
;285
7.1;A.1 Four-Component Vector Notations;285
7.2;A.2 Vector Spaces;287
7.2.1;A.2.1 Notations;287
7.2.2;A.2.2 Basic Definitions;287
7.2.3;A.2.3 Special Spaces;289
7.2.3.1;A.2.3.1 Banach Space;289
7.2.3.2;A.2.3.2 Hilbert Space;289
7.2.3.3;A.2.3.3 Fock Space;289
7.3;A.3 Special Functions;289
7.3.1;A.3.1 Dirac Delta Function;289
7.3.2;A.3.2 Integrals over Functions;291
7.3.3;A.3.3 The Heaviside Step Function;294
7.4;References;294
8;Appendix B: Second Quantization
;295
8.1;B.1 Definitions;295
8.2;B.2 Heisenberg and Interaction Pictures;298
8.3;References;299
9;Appendix C: Representations of States and Operators
;300
9.1;C.1 Vector Representation of States;300
9.2;C.2 Matrix Representation of Operators;302
9.3;C.3 Coordinate Representations;303
9.3.1;C.3.1 Representation of Vectors;303
9.3.2;C.3.2 Closure Property;303
9.3.3;C.3.3 Representation of Operators;304
10;Appendix D: Dirac Equation and the Momentum Representation
;305
10.1;D.1 Dirac Equation;305
10.1.1;D.1.1 Free Particles;305
10.1.2;D.1.2 Dirac Equation in an Electromagnetic Field;310
10.2;D.2 Momentum Representation;310
10.2.1;D.2.1 Representation of States;310
10.2.2;D.2.2 Representation of Operators;311
10.2.3;D.2.3 Closure Property for Momentum Functions;312
10.3;D.3 Relations for the Alpha and Gamma Matrices;312
10.4;Reference;313
11;Appendix E: Lagrangian Field Theory
;314
11.1;E.1 Classical Mechanics;314
11.1.1;E.1.1 Electron in External Field;316
11.2;E.2 Classical Field Theory;317
11.3;E.3 Dirac Equation in Lagrangian Formalism;318
11.4;References;319
12;Appendix F: Semiclassical Theory of Radiation
;320
12.1;F.1 Classical Electrodynamics;320
12.1.1;F.1.1 Maxwell's Equations in Covariant Form;320
12.1.1.1;F.1.1.1 Electromagnetic-Field Lagrangian;321
12.1.1.2;F.1.1.2 Lorenz Condition;323
12.1.1.3;F.1.1.3 Continuity Equation;323
12.1.1.4;F.1.1.4 Gauge Invariance;323
12.1.2;F.1.2 Coulomb Gauge;324
12.1.2.1;F.1.2.1 Transverse and Longitudinal Field Components;324
12.2;F.2 Quantized Radiation Field;326
12.2.1;F.2.1 Transverse Radiation Field;326
12.2.2;F.2.2 Breit Interaction;327
12.2.3;F.2.3 Transverse Photon Propagator;330
12.2.4;F.2.4 Comparison with the Covariant Treatment;331
12.3;References;333
13;Appendix G: Covariant Theory of Quantum ElectroDynamics
;334
13.1;G.1 Covariant Quantization: Gupta–Bleuler Formalism;334
13.2;G.2 Gauge Transformation;336
13.2.1;G.2.1 General;336
13.2.2;G.2.2 Covariant Gauges;337
13.2.2.1;G.2.2.1 Feynman Gauge;337
13.2.2.2;G.2.2.2 Landau Gauge;337
13.2.2.3;G.2.2.3 Fried–Yennie Gauge;338
13.2.3;G.2.3 Noncovariant Gauge;338
13.2.3.1;G.2.3.1 Coulomb Gauge;338
13.2.3.2;G.2.3.2 Covariant Gauge;339
13.2.3.3;G.2.3.3 Noncovariant Gauge;340
13.3;G.3 Gamma Function;341
13.3.1;G.3.1 z=-1-;342
13.3.2;G.3.2 z=-2-;342
13.4;References;343
14;Appendix H: Feynman Diagrams and Feynman Amplitude
;344
14.1;H.1 Feynman Diagrams;344
14.1.1;H.1.1 S-Matrix;344
14.1.2;H.1.2 Green's Function;345
14.1.3;H.1.3 Covariant Evolution Operator;345
14.2;H.2 Feynman Amplitude;346
15;Appendix I: Evaluation Rules for Time-Ordered Diagrams
;349
15.1;I.1 Single-Photon Exchange;350
15.2;I.2 Two-Photon Exchange;351
15.2.1;I.2.1 No Virtual Pair;352
15.2.2;I.2.2 Single Hole;353
15.2.3;I.2.3 Double Holes;354
15.3;I.3 General Evaluation Rules;355
15.4;References;356
16;Appendix J: Some Integrals
;357
16.1;J.1 Feynman Integrals;357
16.2;J.2 Evaluation of the Integral d3k(2)3ei kr12q2-k2+i;359
16.3;J.3 Evaluation of the Integral d3k(2)3(1)(2)eikr12q2-k2+i;360
16.4;References;362
17;Appendix K: Unit Systems and Dimensional Analysis
;363
17.1;K.1 Unit Systems;363
17.1.1;K.1.1 SI System;363
17.1.2;K.1.2 Relativistic or ``Natural'' Unit System;363
17.1.3;K.1.3 Hartree Atomic Unit System;364
17.1.4;K.1.4 cgs Unit Systems;365
17.2;K.2 Dimensional Analysis;365
18;Abbreviations;369
19;Index;370