MODERN FUNCTIONAL QUANTUM FIELD THEORY: Summing Feynman Graphs

by Herbert Martin Fried

Overview

These pages offer a simple, analytic, functional approach to non-perturbative QFT, using a frequently overlooked functional representation of Fradkin to explicitly calculate relevant portions of the Schwinger Generating Functional (GF). In QED, this corresponds to summing all Feynman graphs representing virtual photon exchange between charged particles. It is then possible to see, analytically, the cancellation of an infinite number of perturbative, UV logarithmic divergences, leading to an approximate but most reasonable statement of finite charge renormalization.A similar treatment of QCD, with the addition of a long-overlooked but simple rearrangement of the Schwinger GF which displays Manifest Gauge Invariance, is then able to produce a simple, analytic derivation of quark-binding potentials without any approximation of infinite quark masses. A crucial improvement of previous QCD theory takes into account the experimental fact that asymptotic quarks are always found in bound states; and therefore that their transverse coordinates can never be measured, nor specified, exactly. And this change of formalism permits a clear and simple realization of true quark binding, into mesons and nucleons. An extension into the QCD binding of two nucleons into an effective deuteron presents a simple, analytic derivation of nuclear forces.Finally, a new QED-based solution of Vacuum Energy is displayed as a possible candidate for Dark Energy. An obvious generalization to include Inflation, which automatically suggests a model for Dark Matter, is immediately possible; and one more obvious generalization produces an understanding of the origin of the Big Bang, and of the Birth (and Death) of a Universe. If nothing else, this illustrates the Power and the Reach of Quantum Field Theory.

Product Details

ISBN-13:	9789814415903
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/10/2014
Sold by:	Barnes & Noble
Format:	eBook
Pages:	280
File size:	16 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface vii

Basic Preliminaries 1

1 Quantum Field Theory - Why and How 3

1.1 Schwinger's Action Principle 5

1.2 Free-Field Kinematics 10

2 Functional Preliminaries 15

2.1 Functional Differentiation 16

2.2 Linear Translation 17

2.3 Quadratic (Gaussian) Translation 18

2.4 Functional Integration 23

2.5 Cluster Decomposition 27

2.6 Two Useful Relations 30

3 Functional Field Theory 33

3.1 The Generating Functional 33

3.2 A Gauge Digression: SU(N)-QCD 38

3.3 Coupled Fermion and Boson Fields 41

3.4 Fields at the Same Point 44

4 The Generating Functional and the 5-Matrix 51

4.1 The Generating Functional Operator 51

4.2 Asymptotic Conditions 54

4.3 The S-Matrix 55

4.4 A Bremsstrahlung Example 59

5 Schwinger/Fradkin Representations 61

5.1 Formalism 61

5.2 Gauge Structure 66

5.3 The Bloch-Nordsieck - IR - Eikonal Approximation 69

5.4 A Convenient Reformulation 73

Quantum Electrodynamics 77

Quantum Electrodynamics 79

6 Radiative Corrections of the Photon Propagator 83

6.1 Functional Approach to the Photon Propagator 83

6.2 Higher-order Radiative Correction 87

6.3 The DP Model for a Single Closed Fermion Loop 90

6.4 The Extended DP Model, and the Finiteness of Z₃ 96

6.5 Summary 103

7 Radiative Corrections to the Electron Propagator 107

7.1 Introduction 107

7.2 Formulation 108

7.3 Computation 111

7.4 Summary 115

8 A QED Symmetry-Breaking Model of Vacuum Energy 117

8.1 Introduction 117

8.2 Formulation 120

8.3 Approximation 123

8.4 Computation 124

8.5 Application to Dark Energy 127

8.6 Partial Summary 129

Quantum Chromodynamics 131

Quantum Chromodynamics 133

9 Explicit, Non-Perturbative Gauge Invariance 137

9.1 From QED to QCD 137

9.2 Gluon Summations and Explicit Gauge Invariance 142

9.3 Effective Locality 147

10 QCD Transverse Fluctuations 151

10.1 Introduction 151

10.2 A Phenomenological Expression of Transverse Imprecision 155

10.3 Bundle Diagrams 157

11 Quark Binding Potential 161

11.1 Use of the Eikonal Approximation 161

11.2 Estimation of the "Model Pion" Mass 167

12 Nucleon Scattering and Binding 171

12.1 Introduction 171

12.2 Formulation 172

12.3 A Qualitative Binding Potential 184

12.4 Binding Estimations 189

12.5 Summary and Speculation 190

Astrophysical Speculations 193

Astrophysical Speculations 195

13 Inflation as the Precursor of Dark Energy 197

13.1 Introduction 197

13.2 Computation 200

13.3 A Cosmological Speculation 202

13.4 Summary 203

14 Quantum Tachyon Dynamics 205

14.1 Introduction 206

14.2 QTD as a QFT 210

14.3 Functional QTD 215

14.4 Photon Emission and Reabsorption 218

14.5 Kinematic of Tachyon-Particle Reactions 220

14.5.1 Blocking the reaction γ → T + T 221

14.5.2 Scattering: T + P = T' + P' 222

14.6 From Ehrenfest's Theorem to Loop Annihilation 223

14.7 NASA's Fermi Bubbles 227

14.8 Summary 229

Acknowledgment 233

Appendix A Equivalence Example of the DP Model 235

Appendix B Intuitive Justification of the DP Model 237

Appendix C Connected Cluster-Expansion Functionals 239

Appendix D Fradkin's Representations for non-Abelian G_c [A] and L[A] 241

Appendix E Effective Locality and Transverse Imprecision 245

Appendix F Tachyonic Photon Emission 249

Appendix G Relativistic Tachyon Notation 251

G.1 Free Tachyon Equation 251

G.2 Plane Wave Solutions 252

G.2.1 Positive energy solutions 252

G.2.2 Negative energy solutions 253

G.3 Some Useful Formulae, Valid for P ≥ M 254

G.4 "Spin" of the Tachyon 254

G.5 Quantization 255

G.6 Tachyonic States 257

First Epilogue 259

Second Epilogue 261

Bibliography 263