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Tables of Contents for Renormalization
Chapter/Section Title
Page #
Page Count
1. Field Theory
1
26
1.1 A Motivation for Path Integrals
1
5
1.2 Gaussian Integrals and Random Variables
6
5
1.2.1 Preliminaries
6
1
1.2.2 Gaussian Integrals in Finitely Many Variables
7
3
1.2.3 The Covariance Splitting Formula
10
1
1.3 Field Theory on a Lattice
11
8
1.3.1 Discretization
12
3
1.3.2 The Partition Function and Correlations
15
2
1.3.3 The Ising Model
17
2
1.4 Free Fields
19
2
1.5 Properties of the Free Covariance
21
2
1.6 Problems With and Without Cutoffs
23
4
2. Techniques
27
36
2.1 Integration by Parts
27
1
2.2 Wick Ordering
28
5
2.2.1 Definition and Main Properties
28
2
2.2.2 Further Results
30
3
2.3 Evaluation of Gaussian Integrals
33
8
2.3.1 Labelled Feynman Graphs
33
5
2.3.2 Symmetry Factors and Topological Feynman Graphs
38
1
2.3.3 Motivations for Taking the Logarithm
39
2
2.4 Polymer Systems
41
6
2.4.1 Preparation: Graphs and Partitions
42
2
2.4.2 The Logarithm of the Polymer Partition Function
44
3
2.5 The Effective Action and Connected Graphs
47
10
2.5.1 Definition and Semigroup Property
47
2
2.5.2 Derivation of the Graphical Representation
49
6
2.5.3 Result and Discussion
55
2
2.6 Graphical Representations: Conclusions
57
6
3. The Renormalization Group
63
50
3.1 A Cutoff in Momentum Space
64
1
3.2 The Semigroup Structure of Renormalization
65
3
3.3 The Renormalization Group Equation
68
4
3.3.1 The Functional Form
68
1
3.3.2 The Component Form
69
3
3.4 The Structure of the RG equation
72
3
3.4.1 The Graphical Representation
72
2
3.4.2 The Relation to the Feynman Graph Expansion
74
1
3.4.3 The Continuum Limit at Fixed XXX(0)
74
1
3.5 Differential Inequalities
75
3
3.6 Two Dimensions
78
7
3.6.1 Boundedness
78
2
3.6.2 Phi^(4)(2)
80
2
3.6.3 Convergence
82
3
3.7 Three Dimensions
85
14
3.7.1 Power Counting for the Truncated Equation
86
2
3.7.2 Renormalization: A Change of Boundary Conditions
88
3
3.7.3 Renormalized Phi^(4)(3)
91
8
3.8 Four Dimensions
99
10
3.8.1 Counterterms in Second Order
99
3
3.8.2 Power Counting (Skeleton Flow)
102
1
3.8.3 The Boundary Conditions for Renormalization
103
2
3.8.4 Renormalized Phi^(4) Theory
105
4
3.9 The RG Flow in the Ladder Approximation
109
4
4. The Fermi Surface Problem
113
68
4.1 Physical Motivation
113
1
4.2 Many-Fermion Systems on a Lattice
114
16
4.2.1 The Hamiltonian
115
3
4.2.2 The Grand Canonical Ensemble
118
1
4.2.3 The Fermi Gas
119
4
4.2.4 The Functional Integral Representation
123
2
4.2.5 RG Flow: Energy Scales
125
1
4.2.6 Model Assumptions
126
2
4.2.7 The Physical Significance of the Assumptions
128
1
4.2.8 The Role of the Initial Energy Scale
129
1
4.3 The Renormalization Group Differential Equation
130
7
4.3.1 The Effective Action
130
1
4.3.2 The RG Equation
131
5
4.3.3 The Component RGE in Fourier Space
136
1
4.4 Power Counting for Skeletons
137
8
4.4.1 Bounds for the Infinite-Volume Propagator
137
2
4.4.2 Sup Norm Estimates
139
3
4.4.3 Estimates in L^(1) Norm
142
3
4.5 The Four-Point Function
145
12
4.5.1 Motivation
146
1
4.5.2 The Parquet Four-Point Function
147
2
4.5.3 The One-Loop Volume Bound
149
2
4.5.4 The Particle-Particle Flow
151
4
4.5.5 The Particle-Hole Flow
155
2
4.5.6 The Combined Flow
157
1
4.6 Improved Power Counting
157
10
4.6.1 Overlapping Loops
157
3
4.6.2 Volume Improvement from Overlapping Loops
160
1
4.6.3 Volume Improvement in the RGE
161
1
4.6.4 Bounds on the Non-Ladder Skeletons
162
3
4.6.5 The Derivatives of the Skeleton Selfenergy
165
2
4.7 Renormalization Subtractions
167
8
4.7.1 Motivation; the Counterterm
167
1
4.7.2 Full Amputation
168
2
4.7.3 Bounds for a Truncation
170
3
4.7.4 The Meaning of K
173
2
4.8 Conclusion
175
6
4.8.1 Summary
175
1
4.8.2 A Fermi Liquid Criterion
176
2
4.8.3 How the Curvature Sets a Scale
178
3
A. Appendix to Chapters 1-3
181
10
A.1 A Topology on the Ring of Formal Power Series
181
1
A.2 Fourier Transformation
181
3
A.3 Properties of the Boson Propagator
184
1
A.4 Wick Reordering for Bosons
185
4
A.5 The Lower Bound for the Sunset Graph
189
2
B. Appendix to Chapter 4
191
36
B.1 Fermionic Fock Space
191
1
B.2 Calculus on Grassmann Algebras
192
4
B.3 Grassmann Gaussian Integrals
196
2
B.4 Gram's Inequality; Bounds for Gaussian Integrals
198
3
B.5 Grassmann Integrals for Fock Space Traces
201
13
B.5.1 Delta Functions and Integral Kernels
202
2
B.5.2 The Formula for the Trace
204
1
B.5.3 The Time Continuum Limit
205
7
B.5.4 Nambu Formalism
212
1
B.5.5 Matsubara Frequencies
213
1
B.6 Feynman Graph Expansions
214
3
B.7 The Thermodynamic Limit in Perturbation Theory
217
3
B.8 Volume Improvement Bounds
220
7
B.8.1 The One-Loop Volume Bound
220
2
B.8.2 The Two-Loop Volume Bound
222
5
References
227
2
Index
229