Relativistic Many-Body Theory (eBook)

Preface 8
Contents 12
Chapter 1: Introduction 
20 
1.1 Standard Many-Body Perturbation Theory 20
1.2 Quantum-Electrodynamics 21
1.3 Bethe–Salpeter Equation 22
1.4 Helium Atom: Analytical Approach 23
1.5 Field-Theoretical Approach to Many-Body Perturbation Theory 24
References 26
Part I Basics: Standard Many-Body Perturbation Theory 29
Chapter 2: Time-Independent Formalism 
30 
2.1 First Quantization 30
2.1.1 De Broglie's Relations 30
2.1.2 The Schrödinger Equation 31
2.2 Second Quantization 33
2.2.1 Schrödinger Equation in Second Quantization 33
2.2.2 Particle–Hole Formalism: Normal Order andContraction 35
2.2.3 Wick's Theorem 36
2.3 Time-Independent Many-Body Perturbation Theory 37
2.3.1 Bloch Equation 37
2.3.2 Partitioning of the Hamiltonian 38
2.4 Graphical Representation 42
2.4.1 Goldstone Diagrams 42
2.4.2 Linked-Diagram Expansion 45
2.4.2.1 Complete Model Space 45
2.4.2.2 Incomplete Model Spaces 47
2.5 All-Order Methods: Coupled-Cluster Approach 47
2.5.1 Pair Correlation 47
2.5.2 Exponential Ansatz 50
2.5.3 Various Models for Coupled-Cluster Calculations: Intruder-State Problem 53
2.6 Relativistic MBPT: No-Virtual-Pair Approximation 54
2.6.1 QED Effects 56
2.7 Some Numerical Results of Standard MBPT and CC Calculations, Applied to Atoms 57
References 60
Chapter 3: Time-Dependent Formalism 
64 
3.1 Evolution Operator 64
3.2 Adiabatic Damping: Gell-Mann–Low Theorem 68
3.2.1 Gell-Mann–Low Theorem 69
3.3 Extended Model Space: The Generalized Gell-Mann–Low Relation 69
References 73
Part II Quantum-Electrodynamics: One-Photonand Two-Photon Exchange 74
Chapter 4: S-Matrix 
75 
4.1 Definition of the S-Matrix: Feynman Diagrams 76
4.2 Electron Propagator 77
4.3 Photon Propagator 81
4.3.1 Feynman Gauge 82
4.3.2 Coulomb Gauge 84
4.4 Single-Photon Exchange 85
4.4.1 Covariant Gauge 85
4.4.1.1 Feynman Gauge 88
4.4.2 Noncovariant Coulomb Gauge 88
4.4.3 Single-Particle Potential 90
4.5 Two-Photon Exchange 91
4.5.1 Two-Photon Ladder 91
4.5.2 Two-Photon Cross 94
4.6 QED Corrections 95
4.6.1 Bound-Electron Self-Energy 95
4.6.1.1 Covariant Gauge 96
4.6.1.2 Coulomb Gauge 97
4.6.2 Vertex Correction 98
4.6.2.1 Covariant Gauge 99
4.6.2.2 Coulomb Gauge 99
4.6.3 Vacuum Polarization 100
4.6.4 Photon Self-Energy 103
4.7 Feynman Diagrams for the S-Matrix: Feynman Amplitude 103
4.7.1 Feynman Diagrams 103
4.7.2 Feynman Amplitude 104
References 105
Chapter 5: Green's Functions 
106 
5.1 Classical Green's Function 106
5.2 Field-Theoretical Green's Function: Closed-Shell Case 107
5.2.1 Definition of the Field-Theoretical Green's Function 107
5.2.2 Single-Photon Exchange 110
5.2.3 Fourier Transform of the Green's Function 111
5.2.3.1 Single-Particle Green's Function 111
5.2.3.2 Electron Propagator 113
5.2.3.3 Two-Particle Green's Function in the Equal-Time Approximation 114
5.3 Graphical Representation of the Green's Function 115
5.3.1 Single-Particle Green's Function 115
5.3.1.1 One-Body Interaction 120
5.3.2 Many-Particle Green's Function 120
5.3.3 Self-Energy: Dyson Equation 123
5.3.4 Numerical Illustration 125
5.4 Field-Theoretical Green's Function: Open-Shell Case 125
5.4.1 Definition of the Open-Shell Green's Function 125
5.4.2 Two-Times Green's Function of Shabaev 126
5.4.3 Single-Photon Exchange 129
References 132
Chapter 6: Covariant Evolution Operator and Green's Operator 
133 
6.1 Definition of the Covariant Evolution Operator 133
6.2 Single-Photon Exchange in theCovariant-Evolution-Operator Formalism 136
6.2.1 Single-Photon Ladder 140
6.3 Multiphoton Exchange 141
6.3.1 General 141
6.3.2 Irreducible Two-Photon Exchange 142
6.3.2.1 Uncrossing Photons 142
6.3.2.2 Crossing Photons 144
6.3.3 Potential with Radiative Parts 144
6.4 Relativistic Form of the Gell-Mann–Low Theorem 145
6.5 Field-Theoretical Many-Body Hamiltonian 146
6.6 Green's Operator 148
6.6.1 Definition 148
6.6.2 Relation Between the Green's Operatorand Many-Body Perturbation Procedures 150
6.7 Model-Space Contribution 152
6.7.1 Lowest Orders 153
6.7.2 All Orders 156
6.7.2.1 Linkedness of the Green's Operator 160
6.8 Bloch Equation for Green's Operator 160
6.9 Time Dependence of the Green's Operator. Connection to the Bethe–Salpeter Equation 165
6.9.1 Single-Reference Model Space 165
6.9.2 Multireference Model Space 168
References 170
Chapter 7: Numerical Illustrations to Part II 
171 
7.1 S-Matrix 171
7.1.1 Electron Self-Energy of Hydrogen-Like Ions 171
7.1.2 Lamb Shift of Hydrogen-Like Uranium 172
7.1.3 Lamb Shift of Lithium-Like Uranium 173
7.1.4 Two-Photon Nonradiative Exchangein Helium-Like Ions 174
7.1.5 Electron Correlation and QED Calculations on Ground States of Helium-Like Ions 177
7.1.6 g-Factor of Hydrogen-Like Ions: Mass of the Free Electron 178
7.2 Green's-Function and Covariant-Evolution-Operator Methods 179
7.2.1 Fine-Structure of Helium-Like Ions 179
7.2.2 Energy Calculations of 1s2s Levelsof Helium-Like Ions 181
References 182
Part III Quantum-Electrodynamics Beyond Two-Photon Exchange: Field-Theoretical Approach to Many-Body Perturbation Theory 185
Chapter 8: Covariant Evolution Combined with Electron Correlation 
186 
8.1 General Single-Photon Exchange 186
8.1.1 Transverse Part 187
8.1.2 Coulomb Interaction 194
8.2 General QED Potential 194
8.2.1 Single Photon with Crossed Coulomb Interaction 194
8.2.2 Electron Self-Energy and Vertex Correction 199
8.2.2.1 General Two-Electron Self-Energy 200
8.2.2.2 General Vertex Correction 203
8.2.3 Vertex Correction with Further Coulomb Iterations 205
8.2.4 General Two-Body Potential 206
8.3 Unification of the MBPT and QED Procedures: Connection to Bethe–Salpeter Equation 206
8.3.1 MBPT–QED Procedure 206
8.4 Coupled-Cluster-QED Expansion 209
References 211
Chapter 9: The Bethe–Salpeter Equation 
212 
9.1 The Original Derivations by the Bethe–Salpeter Equation 212
9.1.1 Derivation by Salpeter and Bethe 212
9.1.2 Derivation by Gell-Mann and Low 215
9.1.3 Analysis of the Derivations of the Bethe–Salpeter Equation 216
9.2 Quasi-Potential and Effective-Potential Approximations: Single-Reference Case 218
9.3 Bethe–Salpeter–Bloch Equation: Multireference Case 219
9.4 Problems with the Bethe–Salpeter Equation 221
References 222
Chapter 10: Implementation of the MBPT–QED Procedure with Numerical Results 
224 
10.1 The Fock-Space Bloch Equation 224
10.2 Single-Photon Potential in Coulomb Gauge: No Virtual Pairs 226
10.3 Single-Photon Exchange: Virtual Pairs 229
10.3.1 Illustration 229
10.3.2 Full Treatment 232
10.3.3 Higher Orders 234
10.4 Numerical Results 234
10.4.1 Two-Photon Exchange 234
10.4.2 Beyond Two Photons 234
10.4.3 Outlook 237
References 237
Chapter 11: Analytical Treatment of the Bethe–Salpeter Equation 
238 
11.1 Helium Fine Structure 238
11.2 The Approach of Sucher 239
11.3 Perturbation Expansion of the BS Equation 244
11.4 Diagrammatic Representation 246
11.5 Comparison with the Numerical Approach 248
References 248
Chapter 12: Regularization and Renormalization 
250 
12.1 The Free-Electron QED 250
12.1.1 The Free-Electron Propagator 250
12.1.2 The Free-Electron Self-Energy 252
12.1.3 The Free-Electron Vertex Correction 253
12.2 Renormalization Process 255
12.2.1 Mass Renormalization 255
12.2.2 Charge Renormalization 257
12.2.2.1 Electron Self-Energy 257
12.2.2.2 Vertex Correction 259
12.2.2.3 Photon Self-Energy 259
12.2.2.4 Higher-Order Renormalization 261
12.3 Bound-State Renormalization: Cutoff Procedures 261
12.3.1 Mass Renormalization 261
12.3.2 Evaluation of the Mass Term 262
12.3.3 Bethe's Nonrelativistic Treatment 264
12.3.4 Brown–Langer–Schaefer Regularization 265
12.3.5 Partial-Wave Regularization 268
12.3.5.1 Feynman Gauge 268
12.3.5.2 Coulomb Gauge 269
12.4 Dimensional Regularization in Feynman Gauge 270
12.4.1 Evaluation of the Renormalized Free-Electron Self-Energy in Feynman Gauge 271
12.4.2 Free-Electron Vertex Correction in Feynman Gauge 274
12.5 Dimensional Regularization in Coulomb Gauge 276
12.5.1 Free-Electron Self-Energy in the Coulomb Gauge 276
12.6 Direct Numerical Regularization of the Bound-State Self-Energy 280
12.6.1 Feynman Gauge 281
12.6.2 Coulomb Gauge 281
References 282
Chapter 13: Summary and Conclusions 
283 
Appendix A: Notations and Definitions 
285 
A.1 Four-Component Vector Notations 285
A.2 Vector Spaces 287
A.2.1 Notations 287
A.2.2 Basic Definitions 287
A.2.3 Special Spaces 289
A.2.3.1 Banach Space 289
A.2.3.2 Hilbert Space 289
A.2.3.3 Fock Space 289
A.3 Special Functions 289
A.3.1 Dirac Delta Function 289
A.3.2 Integrals over Functions 291
A.3.3 The Heaviside Step Function 294
References 294
Appendix B: Second Quantization 
295 
B.1 Definitions 295
B.2 Heisenberg and Interaction Pictures 298
References 299
Appendix C: Representations of States and Operators 
300 
C.1 Vector Representation of States 300
C.2 Matrix Representation of Operators 302
C.3 Coordinate Representations 303
C.3.1 Representation of Vectors 303
C.3.2 Closure Property 303
C.3.3 Representation of Operators 304
Appendix D: Dirac Equation and the Momentum Representation 
305 
D.1 Dirac Equation 305
D.1.1 Free Particles 305
D.1.2 Dirac Equation in an Electromagnetic Field 310
D.2 Momentum Representation 310
D.2.1 Representation of States 310
D.2.2 Representation of Operators 311
D.2.3 Closure Property for Momentum Functions 312
D.3 Relations for the Alpha and Gamma Matrices 312
Reference 313
Appendix E: Lagrangian Field Theory 
314 
E.1 Classical Mechanics 314
E.1.1 Electron in External Field 316
E.2 Classical Field Theory 317
E.3 Dirac Equation in Lagrangian Formalism 318
References 319
Appendix F: Semiclassical Theory of Radiation 
320 
F.1 Classical Electrodynamics 320
F.1.1 Maxwell's Equations in Covariant Form 320
F.1.1.1 Electromagnetic-Field Lagrangian 321
F.1.1.2 Lorenz Condition 323
F.1.1.3 Continuity Equation 323
F.1.1.4 Gauge Invariance 323
F.1.2 Coulomb Gauge 324
F.1.2.1 Transverse and Longitudinal Field Components 324
F.2 Quantized Radiation Field 326
F.2.1 Transverse Radiation Field 326
F.2.2 Breit Interaction 327
F.2.3 Transverse Photon Propagator 330
F.2.4 Comparison with the Covariant Treatment 331
References 333
Appendix G: Covariant Theory of Quantum ElectroDynamics 
334 
G.1 Covariant Quantization: Gupta–Bleuler Formalism 334
G.2 Gauge Transformation 336
G.2.1 General 336
G.2.2 Covariant Gauges 337
G.2.2.1 Feynman Gauge 337
G.2.2.2 Landau Gauge 337
G.2.2.3 Fried–Yennie Gauge 338
G.2.3 Noncovariant Gauge 338
G.2.3.1 Coulomb Gauge 338
G.2.3.2 Covariant Gauge 339
G.2.3.3 Noncovariant Gauge 340
G.3 Gamma Function 341
G.3.1 z=-1- 342
G.3.2 z=-2- 342
References 343
Appendix H: Feynman Diagrams and Feynman Amplitude 
344 
H.1 Feynman Diagrams 344
H.1.1 S-Matrix 344
H.1.2 Green's Function 345
H.1.3 Covariant Evolution Operator 345
H.2 Feynman Amplitude 346
Appendix I: Evaluation Rules for Time-Ordered Diagrams 
349 
I.1 Single-Photon Exchange 350
I.2 Two-Photon Exchange 351
I.2.1 No Virtual Pair 352
I.2.2 Single Hole 353
I.2.3 Double Holes 354
I.3 General Evaluation Rules 355
References 356
Appendix J: Some Integrals 
357 
J.1 Feynman Integrals 357
J.2 Evaluation of the Integral d3k(2)3ei kr12q2-k2+i 359
J.3 Evaluation of the Integral d3k(2)3(1)(2)eikr12q2-k2+i 360
References 362
Appendix K: Unit Systems and Dimensional Analysis 
363 
K.1 Unit Systems 363
K.1.1 SI System 363
K.1.2 Relativistic or ``Natural'' Unit System 363
K.1.3 Hartree Atomic Unit System 364
K.1.4 cgs Unit Systems 365
K.2 Dimensional Analysis 365
Abbreviations 369
Index 370