UID:
almafu_9960143980202883
Umfang:
1 online resource (xv, 741 pages) :
,
digital, PDF file(s).
Ausgabe:
1st ed.
ISBN:
9781108225144
,
1-108-24711-3
,
1-108-22514-4
Inhalt:
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.
Anmerkung:
Title from publisher's bibliographic system (viewed on 23 Feb 2022).
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Cover -- Half-title -- Endorsements -- Title page -- Copyright information -- Epigraph -- Contents -- Introduction -- Part I Basics -- 1 Preliminaries -- 1.1 Dimension -- 1.2 Notation -- 1.3 Distributions -- 1.4 The Delta Function -- 1.5 The Fourier Transform -- 2 Basics of Non-relativistic Quantum Mechanics -- 2.1 Basic Setting -- 2.2 Measuring Two Different Observables on the Same System -- 2.3 Uncertainty -- 2.4 Finite versus Continuous Models -- 2.5 Position State Space for a Particle -- 2.6 Unitary Operators -- 2.7 Momentum State Space for a Particle -- 2.8 Dirac's Formalism -- 2.9 Why Are Unitary Transformations Ubiquitous? -- 2.10 Unitary Representations of Groups -- 2.11 Projective versus True Unitary Representations -- 2.12 Mathematicians Look at Projective Representations -- 2.13 Projective Representations of R -- 2.14 One-parameter Unitary Groups and Stone's Theorem -- 2.15 Time-evolution -- 2.16 Schrödinger and Heisenberg Pictures -- 2.17 A First Contact with Creation and Annihilation Operators -- 2.18 The Harmonic Oscillator -- 3 Non-relativistic Quantum Fields -- 3.1 Tensor Products -- 3.2 Symmetric Tensors -- 3.3 Creation and Annihilation Operators -- 3.4 Boson Fock Space -- 3.5 Unitary Evolution in the Boson Fock Space -- 3.6 Boson Fock Space and Collections of Harmonic Oscillators -- 3.7 Explicit Formulas: Position Space -- 3.8 Explicit Formulas: Momentum Space -- 3.9 Universe in a Box -- 3.10 Quantum Fields: Quantizing Spaces of Functions -- 4 The Lorentz Group and the Poincaré Group -- 4.1 Notation and Basics -- 4.2 Rotations -- 4.3 Pure Boosts -- 4.4 The Mass Shell and Its Invariant Measure -- 4.5 More about Unitary Representations -- 4.6 Group Actions and Representations -- 4.7 Quantum Mechanics, Special Relativity and the Poincaré Group -- 4.8 A Fundamental Representation of the Poincaré Group.
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4.9 Particles and Representations -- 4.10 The States |p) and |p) -- 4.11 The Physicists' Way -- 5 The Massive Scalar Free Field -- 5.1 Intrinsic Definition -- 5.2 Explicit Formulas -- 5.3 Time-evolution -- 5.4 Lorentz Invariant Formulas -- 6 Quantization -- 6.1 The Klein-Gordon Equation -- 6.2 Naive Quantization of the Klein-Gordon Field -- 6.3 Road Map -- 6.4 Lagrangian Mechanics -- 6.5 From Lagrangian Mechanics to Hamiltonian Mechanics -- 6.6 Canonical Quantization and Quadratic Potentials -- 6.7 Quantization through the Hamiltonian -- 6.8 Ultraviolet Divergences -- 6.9 Quantization through Equal-time Commutation Relations -- 6.10 Caveat -- 6.11 Hamiltonian -- 7 The Casimir Effect -- 7.1 Vacuum Energy -- 7.2 Regularization -- Part II Spin -- 8 Representations of the Orthogonal and the Lorentz Group -- 8.1 The Groups SU(2) and SL(2,C) -- 8.2 A Fundamental Family of Representations of SU(2) -- 8.3 Tensor Products of Representations -- 8.4 SL(2, C) as a Universal Cover of the Lorentz Group -- 8.5 An Intrinsically Projective Representation -- 8.6 Deprojectivization -- 8.7 A Brief Introduction to Spin -- 8.8 Spin as an Observable -- 8.9 Parity and the Double Cover SL[sup(+)](2,C) of O[sup(+)](1,3) -- 8.10 The Parity Operator and the Dirac Matrices -- 9 Representations of the Poincaré Group -- 9.1 The Physicists' Way -- 9.2 The Group P* -- 9.3 Road Map -- 9.3.1 How to Construct Representations? -- 9.3.2 Surviving the Formulas -- 9.3.3 Classifying the Representations -- 9.3.4 Massive Particles -- 9.3.5 Massless Particles -- 9.3.6 Massless Particles and Parity -- 9.4 Elementary Construction of Induced Representations -- 9.5 Variegated Formulas -- 9.6 Fundamental Representations -- 9.6.1 Massive Particles -- 9.6.2 Massless Particles -- 9.7 Particles, Spin, Representations -- 9.8 Abstract Presentation of Induced Representations -- 9.9 Particles and Parity.
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9.10 Dirac Equation -- 9.11 History of the Dirac Equation -- 9.12 Parity and Massless Particles -- 9.13 Photons -- 10 Basic Free Fields -- 10.1 Charged Particles and Anti-particles -- 10.2 Lorentz Covariant Families of Fields -- 10.3 Road Map I -- 10.4 Form of the Annihilation Part of the Fields -- 10.5 Explicit Formulas -- 10.6 Creation Part of the Fields -- 10.7 Microcausality -- 10.8 Road Map II -- 10.9 The Simplest Case (N = 1) -- 10.10 A Very Simple Case (N = 4) -- 10.11 The Massive Vector Field (N = 4) -- 10.12 The Classical Massive Vector Field -- 10.13 Massive Weyl Spinors, First Attempt (N = 2) -- 10.14 Fermion Fock Space -- 10.15 Massive Weyl Spinors, Second Attempt -- 10.16 Equation of Motion for the Massive Weyl Spinor -- 10.17 Massless Weyl Spinors -- 10.18 Parity -- 10.19 Dirac Field -- 10.20 Dirac Field and Classical Mechanics -- 10.21 Majorana Field -- 10.22 Lack of a Suitable Field for Photons -- Part III Interactions -- 11 Perturbation Theory -- 11.1 Time-independent Perturbation Theory -- 11.2 Time-dependent Perturbation Theory and the Interaction Picture -- 11.3 Transition Rates -- 11.4 A Side Story: Oscillating Interactions -- 11.5 Interaction of a Particle with a Field: A Toy Model -- 12 Scattering, the Scattering Matrix and Cross-Sections -- 12.1 Heuristics in a Simple Case of Classical Mechanics -- 12.2 Non-relativistic Quantum Scattering by a Potential -- 12.3 The Scattering Matrix in Non-relativistic Quantum Scattering -- 12.4 The Scattering Matrix and Cross-Sections, I -- 12.5 Scattering Matrix in Quantum Field Theory -- 12.6 Scattering Matrix and Cross-Sections, II -- 13 The Scattering Matrix in Perturbation Theory -- 13.1 The Scattering Matrix and the Dyson Series -- 13.2 Prologue: The Born Approximation in Scattering by a Potential -- 13.3 Interaction Terms in Hamiltonians -- 13.4 Prickliness of the Interaction Picture.
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13.5 Admissible Hamiltonian Densities -- 13.6 Simple Models for Interacting Particles -- 13.7 A Computation at the First Order -- 13.8 Wick's Theorem -- 13.9 Interlude: Summing the Dyson Series -- 13.10 The Feynman Propagator -- 13.11 Redefining the Incoming and Outgoing States -- 13.12 A Computation at Order Two with Trees -- 13.13 Feynman Diagrams and Symmetry Factors -- 13.14 The φ[sup(4)] Model -- 13.15 A Closer Look at Symmetry Factors -- 13.16 A Computation at Order Two with One Loop -- 13.17 One Loop: A Simple Case of Renormalization -- 13.18 Wick Rotation and Feynman Parameters -- 13.19 Explicit Formulas -- 13.20 Counter-terms, I -- 13.21 Two Loops: Toward the Central Issues -- 13.22 Analysis of Diagrams -- 13.23 Cancellation of Infinities -- 13.24 Counter-terms, II -- 14 Interacting Quantum Fields -- 14.1 Interacting Quantum Fields and Particles -- 14.2 Road Map I -- 14.3 The Gell-Mann-Low Formula and Theorem -- 14.4 Adiabatic Switching of the Interaction -- 14.5 Diagrammatic Interpretation of the Gell-Mann-Low Theorem -- 14.6 Road Map II -- 14.7 Green Functions and S-matrix -- 14.8 The Dressed Propagator in the Källén-Lehmann Representation -- 14.9 Diagrammatic Computation of the Dressed Propagator -- 14.10 Mass Renormalization -- 14.11 Difficult Reconciliation -- 14.12 Field Renormalization -- 14.13 Putting It All Together -- 14.14 Conclusions -- Part IV Renormalization -- 15 Prologue: Power Counting -- 15.1 What Is Power Counting? -- 15.2 Weinberg's Power Counting Theorem -- 15.3 The Fundamental Space ker L -- 15.4 Power Counting in Feynman Diagrams -- 15.5 Proof of Theorem 15.3.1 -- 15.6 A Side Story: Loops -- 15.7 Parameterization of Diagram Integrals -- 15.8 Parameterization of Diagram Integrals by Loops -- 16 The Bogoliubov-Parasiuk-Hepp-Zimmermann Scheme -- 16.1 Overall Approach -- 16.2 Simple Examples.
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16.3 Canonical Flow and the Taylor Operation -- 16.4 Subdiagrams -- 16.5 Forests -- 16.6 Renormalizing the Integrand: The Forest Formula -- 16.7 Diagrams That Need Not Be 1-PI -- 16.8 Interpretation -- 16.9 Specificity of the Parameterization -- 17 Counter-terms -- 17.1 What Is the Counter-term Method? -- 17.2 A Very Simple Case: Coupling Constant Renormalization -- 17.3 Mass and Field Renormalization: Diagrammatics -- 17.4 The BPHZ Renormalization Prescription -- 17.5 Cancelling Divergences with Counter-terms -- 17.6 Determining the Counter-terms from BPHZ -- 17.7 From BPHZ to the Counter-term Method -- 17.8 What Happened to Subdiagrams? -- 17.9 Field Renormalization, II -- 18 Controlling Singularities -- 18.1 Basic Principle -- 18.2 Zimmermann's Theorem -- 18.3 Proof of Proposition 18.2.12 -- 18.4 A Side Story: Feynman Diagrams and Wick Rotations -- 19 Proof of Convergence of the BPHZ Scheme -- 19.1 Proof of Theorem 16.1.1 -- 19.2 Simple Facts -- 19.3 Grouping the Terms -- 19.4 Bringing Forward Cancellation -- 19.5 Regular Rational Functions -- 19.6 Controlling the Degree -- Part V Complements -- Appendix A Complements on Representations -- A.1 Projective Unitary Representations of R -- A.2 Continuous Projective Unitary Representations -- A.3 Projective Finite-dimensional Representations -- A.4 Induced Representations for Finite Groups -- A.5 Representations of Finite Semidirect Products -- A.5.1 Semidirect Products -- A.5.2 Finding All Irreducible Unitary Representations of Finite Semidirect Products -- A.5.3 Classifying All Unitary Representations of Finite Semidirect Products -- A.5.4 Formal Application to the Poincaré Group. -- A.6 Representations of Compact Groups -- Appendix B End of Proof of Stone's Theorem -- Appendix C Canonical Commutation Relations -- C.1 First Manipulations -- C.2 Coherent States for the Harmonic Oscillator.
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C.3 The Stone-von Neumann Theorem.
Weitere Ausg.:
ISBN 9781316510278
Weitere Ausg.:
ISBN 1-316-51027-1
Sprache:
Englisch
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