Table of Contents
1 Introduction 1
2 Second Quantization 7
2.1 Coordinate and Momentum Space 7
2.2 The Many-Electron System 9
2.3 Annihilation and Creation Operators 13
2.3.1 Coordinate Space 13
2.3.2 Momentum Space 16
3 Perturbation Theory 19
3.1 The Perturbation Series for eHo+λV 19
3.2 The Perturbation Series for the Partition Function 22
3.3 The Perturbation Series for the Correlation Functions 30
4 Gaussian Integration and Grassmann Integrals 33
4.1 Why Grassmann Integration? A Motivating Example 34
4.2 Grassmann Integral Representations 40
4.3 Ordinary Gaussian Integrals 44
4.4 Theory of Grassmann Integration 47
5 Bosonic Functional Integral Representation 59
5.1 The Hubbard-Stratonovich Transformation 60
5.2 The Effective Potential 65
6 BCS Theory and Spontaneous Symmetry Breaking 79
6.1 The Quadratic Mean Field Model 81
6.2 The Quartic BCS Model 88
6.3 BCS with Higher l-Wave Interaction 92
7 The Many-Electron System in a Magnetic Field 105
7.1 Solution of the Single Body Problem 105
7.1.1 Disk Geometry 106
7.1.2 Rectangular Geometry 111
7.2 Diagonalization of the Fractional Quantum Hall Hamiltonian in a Long Range Limit 113
8 Feynman Diagrams 129
8.1 The Typical Behavior of Field Theoretical Perturbation Series 129
8.2 Connected Diagrams and the Linked Cluster Theorem 131
8.3 Estimates on Feynman Diagrams 136
8.3.1 Elementary Bounds 136
8.3.2 Single Scale Bounds 147
8.3.3 Multiscale Bounds 151
8.4 Ladder Diagrams 162
9 Renormalization Group Methods 169
9.1 Integrating Out Scales 171
9.2 A Single Scale Bound on the Sum of All Diagrams 174
9.3 A Multiscale Bound on the Sum of Convergent Diagrams 186
9.4 Elimination of Divergent Diagrams 193
9.5 The Feldman-Knörrer-Trubowitz Fermi Liquid Construction 195
10 Resummation of Perturbation Series 197
10.1 Starting Point and Typical Examples 197
10.2 Computing Inverse Matrix Elements 200
10.2.1 An Inversion Formula 200
10.2.2 Field Theoretical Motivation of the Inversion Formula 204
10.3 The Averaged Green Function of the Anderson Model 211
10.3.1 Two-Loop Approximation 213
10.3.2 Higher Orders 215
10.4 The Many-Electron System with Attractive Delta Interaction 221
10.4.1 The Integral Equations in Two-Loop Approximation 221
10.4.2 Discussion 227
10.5 Application to Bosonic Models 230
10.5.1 The φ4-Model 230
10.5.2 Higher Orders 233
10.5.3 The φ2ψ2-Model 235
10.6 General Structure of the Integral Equations 239
The 'Many-Electron Millennium Problems' 243
References 245
Index 251
