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The quantum theory of fields: 1 Foundations

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Bibliographische Detailangaben
Beteilige Person:	Weinberg, Steven 1933-2021 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Cambridge [u.a.] Cambridge Univ. Press 2007
Ausgabe:	Paperback ed., repr.
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016258084&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XXVI, 609 S. graph. Darst.
ISBN:	9780521670531

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Datensatz im Suchindex

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adam_text	Contents Sections marked with an asterisk are somewhat out of the book s main line of development and may be omitted in a first reading. PREFACE xx NOTATION xxv 1 HISTORICAL INTRODUCTION 1 1.1 Relativistic Wave Mechanics 3 De Broglie waves G Schrödinger-Klein-Gordon wave equation G Fine structure G Spin Q Dirac equation G Negative energies G Exclusion principle □ Positrons G Dirac equation reconsidered 1.2 The Birth of Quantum Field Theory 15 Born, Heisenberg, Jordan quantized field G Spontaneous emission G Anticom- mutators G Heisenberg-Pauli quantum field theory G Furry-Oppenheimer quan¬ tization of Dirac field G Pauli-Weisskopf quantization of scalar field Π Early calculations in quantum electrodynamics G Neutrons G Mesons 1.3 The Problem of Infinities 31 Infinite electron energy shifts D Vacuum polarization G Scattering of light by light G Infrared divergences G Search for alternatives G Renormalization G Shelter Island Conference □ Lamb shift G Anomalous electron magnetic moment G Schwinger, Tomonaga, Feynman, Dyson formalisms G Why not earlier? Bibliography 39 References 40 2 RELATIVISTIC QUANTUM MECHANICS 49 2.1 Quantum Mechanics 49 Rays Q Scalar products о Observables G Probabilities ix χ Contents 2.2 Symmetries 50 Wigner s theorem □ Antilinear and antiunitary operators D Observables α Group structure G Representations up to a phase Π Superselection rules □ Lie groups D Structure constants □ Abelian symmetries 2.3 Quantum Lorentz Transformations 55 Lorentz transformations D Quantum operators D Inversions 2.4 The Poincaré Algebra 58 J* 1 and Рџ α Transformation properties □ Commutation relations □ Conserved and non-conserved generators D Finite translations and rotations □ Inönü- Wigner contraction □ Galilean algebra 2.5 One-Particle States 62 Transformation rules □ Boosts □ Little groups D Normalization □ Massive particles G Massless particles G Helicity and polarization 2.6 Space Inversion and Time-Reversal 74 Transformation of Јџу and Рџ O P is unitary and T is antiunitary □ Massive particles О Massless particles □ Kramers degeneracy □ Electric dipole moments 2.7 Projective Representations* 81 Two-cocyles ü Central charges α Simply connected groups □ No central charges in the Lorentz group G Double connectivity of the Lorentz group □ Covering groups G Superselection rules reconsidered Appendix A The Symmetry Representation Theorem 91 Appendix В Group Operators and Homotopy Classes 96 Appendix С Inversions and Degenerate Multiplets 100 Problems 104 References 105 3 SCATTERING THEORY 107 3.1 In and Out States 107 Multi-particle states D Wave packets □ Asymptotic conditions at early and late times G Lippmann- Schwinger equations α Principal value and delta functions 3.2 TheS-matrix 113 Definition of the S -matrix G The Т -matrix α Born approximation G Unitarity of the S-matrix 3.3 Symmetries of the S-Matrix 116 Lorentz invariance □ Sufficient conditions α Internal symmetries □ Electric charge, strangeness, isospin, SU(3) G Parity conservation α Intrinsic parities □ Contents xi Pion parity Q Parity non-conservation □ Time-reversal invariance G Watson s theorem Π PT non-conservation α CCP.CPT □ Neutral Χ -mesons □ CP non- conservation 3.4 Rates and Cross-Sections 134 Rates in a box □ Decay rates G Cross-sections □ Lorentz invariance D Phase space α Dalitz plots 3.5 Perturbation Theory 141 Old-fashioned perturbation theory □ Time-dependent perturbation theory □ Time-ordered products О The Dyson series О Lorentz-invariant theories D Dis¬ torted wave Born approximation 3.6 Implications of Unitarity 147 Optical theorem □ Diffraction peaks □ CPT relations □ Particle and antiparticle decay rates □ Kinetic theory О Boltzmann Я -theorem 3.7 Partial-Wave Expansions* 151 Discrete basis G Expansion in spherical harmonics D Total elastic and inelastic cross-sections G Phase shifts G Threshold behavior: exothermic, endothermic, and elastic reactions G Scattering length G High-energy elastic and inelastic scattering 3.8 Resonances* 159 Reasons for resonances: weak coupling, barriers, complexity Q Energy- dependence G Unitarity G Breit-Wigner formula G Unresolved resonances G Phase shifts at resonance G Ramsauer-Townsend effect Problems 165 References 166 4 THE CLUSTER DECOMPOSITION PRINCIPLE 169 4.1 Bosons and Fermions 170 Permutation phases Q Bose and Fermi statistics Q Normalization for identical particles 4.2 Creation and Annihilation Operators 173 Creation operators G Calculating the adjoint α Derivation of commutation/ anticommutation relations G Representation of general operators α Free-particle Hamiltonian D Lorentz transformation of creation and annihilation operators α C, P, T properties of creation and annihilation operators 43 Cluster Decomposition and Connected Amplitudes 177 Decorrelation of distant experiments G Connected amplitudes G Counting delta functions xii Contents 4.4 Structure of the Interaction 182 Condition for cluster decomposition □ Graphical analysis α Two-body scattering implies three-body scattering Problems 189 References 189 5 QUANTUM FIELDS AND ANTIPARTICLES 191 5.1 Free Fields 191 Creation and annihilation fields D Lorentz transformation of the coefficient func¬ tions G Construction of the coefficient functions D Implementing cluster decom¬ position □ Lorentz invariance requires causality □ Causality requires antiparticles ü Field equations □ Normal ordering 5.2 Causal Scalar Fields 201 Creation and annihilation fields О Satisfying causality α Scalar fields describe bosons D Antiparticles π Ρ, C, T transformations ü π° 5.3 Causal Vector Fields 207 Creation and annihilation fields о Spin zero or spin one D Vector fields describe bosons D Polarization vectors D Satisfying causality □ Antiparticles □ Mass zero limit o P, C, T transformations 5.4 The Dirac Formalism 213 Clifford representations of the Poincaré algebra ГЈ Transformation of Dirac matri¬ ces G Dimensionality of Dirac matrices G Explicit matrices α y5 Q Pseudounitarity D Complex conjugate and transpose 5.5 Causal Dirac Fields 219 Creation and annihilation fields G Dirac spinors G Satisfying causality G Dirac fields describe fermions α Antiparticles D Space inversion α Intrinsic parity of particle-antiparticle pairs ü Charge-conjugation π Intrinsic С -phase of particle- antiparticle pairs G Majorana fermions G Time-reversal Q Bilinear covariante G Beta decay interactions 5.6 General Irreducible Representations of the Homogeneous Lorentz Group* 229 Isomorphism with S 1/(2) ® SU{2) G (А,В) representation of familiar fields G Rarita-Schwinger field G Space inversion 5.7 General Causal Fields* 233 Constructing the coefficient functions α Scalar Hamiltonian densities α Satisfying causality G Antiparticles α General spin-statistics connection α Equivalence of different field types D Space inversion G Intrinsic parity of general particle- antiparticle pairs G Charge-conjugation G Intrinsic С -phase of antiparticles G Contents xiii Self-charge-conjugate particles and reality relations G Time-reversal О Problems for higher spin? 5.8 The CPT Theorem 244 CPT transformation of scalar, vector, and Dirac fields □ CPT transformation of scalar interaction density D CPT transformation of general irreducible fields о CPT invariance of Hamiltonian 5.9 Massless Particle Fields 246 Constructing the coefficient functions D No vector fields for helicity +1 □ Need for gauge invariance π Antisymmetric tensor fields for helicity ±1 □ Sums over helicity Π Constructing causal fields for helicity ±1 □ Gravitons □ Spin > 3 □ General irreducible massless fields □ Unique helicity for {A,B) fields Problems 255 References 256 6 THE FEYNMAN RULES 259 6.1 Derivation of the Rules 259 Pairings □ Wick s theorem D Coordinate space rules о Combinatoric factors О Sign factors □ Examples 6.2 Calculation of the Propagator 274 Numerator polynomial □ Feynman propagator for scalar fields □ Dirac fields D General irreducible fields о Covariant propagators D Non-covariant terms in time-ordered products 6.3 Momentum Space Rules 280 Conversion to momentum space □ Feynman rules ГЈ Counting independent momenta Π Examples D Loop suppression factors 6.4 Off the Mass Shell 286 Currents D Off-shell amplitudes are exact matrix elements of Heisenberg-picture operators □ Proof of the theorem Problems 290 References 291 7 THE CANONICAL FORMALISM 292 7.1 Canonical Variables 293 Canonical commutation relations о Examples: real scalars, complex scalars, vector fields, Dirac fields □ Free-particle Hamiltonians D Free-field Lagrangian D Canonical formalism for interacting fields xiv Contents 7.2 The Lagrangian Formalism 298 Lagrangian equations of motion □ Action G Lagrangian density Π Euler- Lagrange equations □ Reality of the action □ From Lagrangians to Haroiltonians D Scalar fields revisited О From Heisenberg to interaction picture □ Auxiliary fields ü Integrating by parts in the action 7.3 Global Symmetries 306 Noether s theorem □ Explicit formula for conserved quantities □ Explicit formula for conserved currents π Quantum symmetry generators G Energy-momentum tensor □ Momentum π Internal symmetries D Current commutation relations 7.4 Lorentz Invariance 314 Currents Лрџ> α Generators 3μν G Belinfante tensor Q Lorentz invariance of S -matrix 7.5 Transition to Interaction Picture: Examples 318 Scalar field with derivative coupling α Vector field π Dirac field 7.6 Constraints and Dirac Brackets 325 Primary and secondary constraints □ Poisson brackets Π First and second class constraints D Dirac brackets D Example : real vector field 7.7 Field Redefinitions and Redundant Couplings* 331 Redundant parameters G Field redefinitions α Example: real scalar field Appendix Dirac Brackets from Canonical Commutators 332 Problems 337 References 338 8 ELECTRODYNAMICS 339 8.1 Gauge Invariance 339 Need for coupling to conserved current G Charge operator α Local symmetry D Photon action G Field equations G Gauge-invariant derivatives 8.2 Constraints and Gauge Conditions 343 Primary and secondary constraints α Constraints are first class G Gauge fixing α Coulomb gauge D Solution for A° 8.3 Quantization in Coulomb Gauge 346 Remaining constraints are second class D Calculation of Dirac brackets in Coulomb gauge □ Construction of Hamiltonian D Coulomb interaction 8.4 Electrodynamics in the Interaction Picture 350 Free-field and interaction Hamiltonians G Interaction picture operators G Normal mode decomposition Contents xv 8.5 The Photon Propagator 353 Numerator polynomial D Separation of non-covariant terms □ Cancellation of non-covariant terms 8.6 Feynman Rules for Spinor Electrodynamics 355 Feynman graphs Q Vertices □ External lines π Internal lines □ Expansion in α/4π □ Circular, linear, and elliptic polarization □ Polarization and spin sums 8.7 Compton Scattering 362 S-matrix D Differential cross-section □ Kinematics □ Spin sums D Traces G Klein-Nishina formula □ Polarization by Thomson scattering Π Total cross- section 8.8 Generalization : p-form Gauge Fields* 369 Motivation □ p-forms □ Exterior derivatives π Closed and exact p-forms D p-form gauge fields О Dual fields and currents in D spacetime dimensions □ p-form gauge fields equivalent to (D—p — 2)-form gauge fields D Nothing new in four spacetime dimensions Appendix Traces 372 Problems 374 References 375 9 PATH-INTEGRAL METHODS 376 9.1 The General Path-Integral Formula 378 Transition amplitudes for infinitesimal intervals D Transition amplitudes for finite intervals G Interpolating functions □ Matrix elements of time-ordered products О Equations of motion 9.2 Transition to the S-Matrix 385 Wave function of vacuum O ie terms 9.3 Lagrangian Version of the Path-Integral Formula 389 Integrating out the momenta □ Derivatively coupled scalars O Non-linear sigma model □ Vector field 9.4 Path-Integral Derivation of Feynman Rules 395 Separation of free-field action □ Gaussian integration о Propagators: scalar fields, vector fields, derivative coupling 9.5 Path Integrals for Fermions 399 Anticommuting с -numbers Q Eigenvectors of canonical operators □ Summing states by Berezin integration □ Changes of variables □ Transition amplitudes for infinitesimal intervals □ Transition amplitudes for finite intervals о Derivation of Feynman rales □ Fermion propagator □ Vacuum amplitudes as determinants xvi Contents 9.6 Path-Integral Formulation of Quantum Electrodynamics 413 Path integral in Coulomb gauge □ Reintroduction of α° Ο Transition to covariant gauges 9.7 Varieties of Statistics* 418 Preparing in and out states Π Composition rules □ Only bosons and fermions in > 3 dimensions □ Anyons in two dimensions Appendix Gaussian Multiple Integrals 420 Problems 423 References 423 10 NON-PERTURBATIVE METHODS 425 10.1 Symmetries 425 Translations □ Charge conservation □ Furry s theorem 10.2 Polology 428 Pole formula for general amplitudes D Derivation of the pole formula □ Pion exchange 10.3 Field and Mass Renormalization 436 LSZ reduction formula □ Renormalized fields π Propagator poles □ No radiative corrections in external lines D Counterterms in self-energy parts 10.4 Renormalized Charge and Ward Identities 442 Charge operator D Electromagnetic field renormalization □ Charge renormaliz¬ ation D Ward-Takahashi identity D Ward identity 10.5 Gauge Invariance 448 Transversality of multi-photon amplitudes Q Schwinger terms D Gauge terms in photon propagator □ Structure of photon propagator □ Zero photon renormal¬ ized mass □ Calculation of Z3 D Radiative corrections to choice of gauge 10.6 Electromagnetic Form Factors and Magnetic Moment 452 Matrix elements of J° □ Form factors of Јџ : spin zero О Form factors of Јџ : spin 5 Π Magnetic moment of a spin particle α Measuring the form factors 10.7 The Källen-Lehmann Representation* 457 Spectral functions о Causality relations □ Spectral representation D Asymptotic behavior of propagators □ Poles π Bound on field renormalization constant О Ζ = 0 for composite particles 10.8 Dispersion Relations* 462 History □ Analytic properties of massless boson forward scattering amplitude Contents xvii Π Subtractions □ Dispersion relation □ Crossing symmetry π Pomeranchuk s theorem D Regge asymptotic behavior □ Photon scattering Problems 469 References 470 11 ONE-LOOP RADIATIVE CORRECTIONS IN QUANTUM ELECTRODYNAMICS 472 11.1 Counterterms 472 Field, charge, and mass renormalization O Lagrangian counterterms 11.2 Vacuum Polarization 473 One-loop integral for photon self-energy part □ Feynman parameters О Wick rotation □ Dimensional regularization □ Gauge invariance D Calculation of Z-¡ G Cancellation of divergences □ Vacuum polarization in charged particle scattering □ Uehling effect □ Muonic atoms 11.3 Anomalous Magnetic Moments and Charge Radii 485 One-loop formula for vertex function □ Calculation of form factors Π Anomalous lepton magnetic moments to order α D Anomalous muon magnetic moment to order a2 1п(тџ/те) П Charge radius of leptons 11.4 Electron Self-Energy 493 One-loop formula for electron self-energy part D Electron mass renormalization ü Cancellation of ultraviolet divergences Appendix Assorted Integrals 497 Problems 498 References 498 12 GENERAL RENORMALIZATION THEORY 499 12.1 Degrees of Divergence 500 Superficial degree of divergence □ Dimensional analysis D Renormalizability О Criterion for actual convergence 12.2 Cancellation of Divergences 505 Subtraction by differentiation Π Renormalization program D Renormalizable theories □ Example: quantum electrodynamics □ Overlapping divergences D BPHZ renormalization prescription D Changing the renormalization point: φ4 theory 12.3 Is Renormalizability Necessary? 516 xviii Contents Renormalizable interactions cataloged So rcnormali/able theories of gravita¬ tion О Cancellation of divergences in non-renormalizable theories і ; Suppression of non-renormalizabie interactions С Limits on new mass scales Π Problems with higher derivatives? □ Detection of non-renormalizable interactions Π Low-energy expansions in non-renormalizabie theories С Example: scalar with only derivative coupling π Saturation or new physics? С Effective field theories 12.4 The Floating Cutoff* 525 Wilson s approach о Renormalization group equation □ Polchinskľs theorem □ Attraction to a stable surface Π Floating cutoff vs renormalization 12.5 Accidental Symmetries* 529 General renormalizable theory of charged leptons G Redefinition of the lepton fields о Accidental conservation of lepton flavors, P, C, and Τ Problems 531 References 532 13 INFRARED EFFECTS 534 13.1 Soft Photon Amplitudes 534 Single photon emission Q Negligible emission from internal lines □ Lorentz invariance implies charge conservation D Single graviten emission □ Lorentz invariance implies equivalence principle □ Multi-photon emission О Factorization 13.2 Virtual Soft Photons 539 Effect of soft virtual photons D Radiative corrections on internal lines 13.3 Real Soft Photons; Cancellation of Divergences 544 Sum over helicities О Integration over energies □ Sum over photon number D Cancellation of infrared cutoff factors D Likewise for gravitation 13.4 General Infrared Divergences 548 Massless charged particles D Infrared divergences in general Π Jets □ Lee- Nau- enberg theorem 13.5 Soft Photon Scattering 553 Poles in the amplitude α Conservation relations α Universality of the low-energy limit 13.6 The External Field Approximation* 556 Sums over photon vertex permutations D Non-relativistic limit α Crossed ladder exchange Problems 562 References 562 Contents xix 14 BOUND STATES IN EXTERNAL FIELDS 564 14.1 The Dirac Equation 565 Dirac wave functions as field matrix elements □ Anticommutators and complete¬ ness О Energy eigenstates □ Negative energy wave functions □ Orthonormaliza- tion D Large and small components О Parity α Spin- and angle-dependence О Radial wave equations □ Energies □ Fine structure □ Non-relativistic approxi¬ mations 14.2 Radiative Corrections in External Fields 572 Electron propagator in an external field □ Inhomogeneous Dirac equation О Effects of radiative corrections □ Energy shifts 14.3 The Lamb Shift in Light Atoms 578 Separating high and low energies D High-energy term □ Low-energy term D Effect of mass renormalization □ Total energy shift D^Oö/^0 Numerical results □ Theory vs experiment for classic Lamb shift D Theory vs experiment for Is energy shift Problems 594 References 595 AUTHOR INDEX 597 SUBJECT INDEX 602 OUTLINE OF VOLUME II 15 NON-ABELIAN GAUGE THEORIES 16 EXTERNAL FIELD METHODS 17 RENORMALIZATION OF GAUGE THEORIES 18 RENORMALIZATION GROUP METHODS 19 SPONTANEOUSLY BROKEN GLOBAL SYMMETRIES 20 OPERATOR PRODUCT EXPANSIONS 21 SPONTANEOUSLY BROKEN LOCAL SYMMETRIES 22 ANOMALIES 23 EXTENDED FIELD CONFIGURATIONS
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title_full	The quantum theory of fields 1 Foundations Steven Weinberg
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title_sort	the quantum theory of fields foundations
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