What Is a Quantum Field Theory?: A First Introduction for Mathematicians

Description

Quantum field theory (QFT) is one of the great achievements of physics, of profound interest to mathematicians. Most pedagogical texts on QFT are geared toward budding professional physicists, however, whereas mathematical accounts are abstract and difficult to relate to the physics. This book bridges the gap. While the treatment is rigorous whenever possible, the accent is not on formality but on explaining what the physicists do and why, using precise mathematical language. In particular, it covers in detail the mysterious procedure of renormalization. Written for readers with a mathematical background but no previous knowledge of physics and largely self-contained, it presents both basic physical ideas from special relativity and quantum mechanics and advanced mathematical concepts in complete detail. It will be of interest to mathematicians wanting to learn about QFT and, with nearly 300 exercises, also to physics students seeking greater rigor than they typically find in their courses. Frontmatter Dedication Contents Introduction Part I Basics 1 Preliminaries 2 Basicsof Non-relativisticQuantumMechanics 3 Non-relativisticQuantumFields 4 The Lorentz Group and thePoincaréGroup 5 The MassiveScalarFree Field 6 Quantization 7 The Casimir Effect Part II Spin 8 Representationsof theOrthogonaland theLorentzGroup 9 Representationsof thePoincaréGroup 10 BasicFree Fields Part III Interactions 11 PerturbationTheory 12 Scattering,theScatteringMatrix and Cross-Sections 13 The ScatteringMatrix in PerturbationTheory 14 InteractingQuantumFields Part IV Renormalization 15 Prologue:PowerCounting 16 The Bogoliubov–Parasiuk–Hepp–ZimmermannScheme 17 Counter-terms 18 ControllingSingularities 19 Proof of Convergenceof theBPHZ Scheme Part V Complements AppendixA Complementson Representations AppendixB End ofProof ofStone’s Theorem AppendixC CanonicalCommutationRelations AppendixD ACrash Course on LieAlgebras AppendixE SpecialRelativity AppendixF Does a PositionOperatorExist? AppendixG More on theRepresentationsof thePoincaréGroup AppendixH HamiltonianFormalismfor ClassicalFields AppendixI Quantizationofthe ElectromagneticFieldthroughthe Gupta–BleulerApproach AppendixJ Lippmann–SchwingerEquationsand ScatteringStates AppendixK Functionson Surfaces andDistributions AppendixL What Is a Tempered DistributionReally? AppendixM Wightman Axiomsand Haag’s Theorem AppendixN FeynmanPropagatorand Klein-Gordon Equation AppendixO YukawaPotential AppendixP PrincipalValuesandDeltaFunctions Solutions to Selected Exercises Reading Suggestions References Index