The Quantum Theory of Fields. Volume I Foundations Steven Weinberg

Transcription

1 The Quantum Theory of Fields Volume I Foundations Steven Weinberg

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3 PREFACE NOTATION x x xxv 1 HISTORICAL INTRODUCTION Relativistic Wave Mechanics 3 De Broglie waves q Schrödinger-Klein-Gordon wave equation q Fine structure q Spin q Dirac equation q Negative energies q Exclusion principle q Positron s q Dirac equation reconsidered 1.2 The Birth of Quantum Field Theory 1 5 Born, Heisenberg, Jordan quantized field q Spontaneous emission q Anticommutators q Heisenberg-Pauli quantum field theory q Furry-Oppenheimer quantization of Dirac field q Pauli-Weisskopf quantization of scalar field q Early calculations in quantum electrodynamics q Neutrons q Meson s 1.3 The Problem of Infinities 3 1 Infinite electron energy shifts q Vacuum polarization q Scattering of light by ligh t q Infrared divergences q Search for alternatives q Renormalization q Shelter Island Conference q Lamb shift q Anomalous electron magnetic moment q Schwinger, Tomonaga, Feynman, Dyson formalisms q Why not earlier? Bibliography 3 9 References RELATIVISTIC QUANTUM MECHANICS Quantum Mechanics 4 9 Rays q Scalar products q Observables q Probabilities

4 2.2 Symmetries 5 0 Wigner's theorem q Antilinear and antiunitary operators q Observables q Grou p structure q Representations up to a phase q Superselection rules q Lie groups q Structure constants q Abelian symmetrie s 2.3 Quantum Lorentz Transformations 5 5 Lorentz transformations q Quantum operators q Inversions 2.4 The Poincare Algebra 5 8 Jµ" and Pµ q Transformation properties q Commutation relations q Conserve d and non-conserved generators q Finite translations and rotations q Inönü- Wigner contraction q Galilean algebra 2.5 One-Particle States 62 Transformation rules q Boosts q Little groups q Normalization q Massiv e particles q Massless particles q Helicity and polarization 2.6 Space Inversion and Time-Reversal 7 4 Transformation of Jµ and P µ q P is unitary and T is antiunitary q Massiv e particles q Massless particles q Kramers degeneracy q Electric dipole moment s 2.7 Projective Representations* 8 1 Two-cocyles q Central charges q Simply connected groups q No central charge s in the Lorentz group q Double connectivity of the Lorentz group q Coverin g groups q Superselection rules reconsidere d Appendix A The Symmetry Representation Theorem 9 1 Appendix B Group Operators and Homotopy Classes 9 6 Appendix C Inversions and Degenerate Multiplets 100 Problems 104 References SCATTERING THEORY `In' and `Out' States 10 7 Multi-particle states q Wave packets q Asymptotic conditions at early and late times q Lippmann- Schwinger equations q Principal value and delta function s 3.2 The S-matrix 11 3 Definition of the S-matrix q The T-matrix q Born approximation q Unitarit y of the S-matrix 3.3 Symmetries of the S-Matrix 11 6 Lorentz invariance q Sufficient conditions q Internal symmetries q Electric charge, strangeness, isospin, S U(3) q Parity conservation q Intrinsic parities q

5 Pion parity q Parity non-conservation q Time-reversal invariance q Watson' s theorem q PT non-conservation q C, CP, CPT q Neutral K-mesons q CP non - conservatio n 3.4 Rates and Cross-Sections 134 Rates in a box q Decay rates q Cross-sections q Lorentz invariance q Phase space q Dalitz plots 3.5 Perturbation Theory 14 1 Old-fashioned perturbation theory q Time-dependent perturbation theory q Time-ordered products q The Dyson series q Lorentz-invariant theories q Distorted wave Born approximation 3.6 Implications of Unitarity 14 7 Optical theorem q Diffraction peaks q CPT relations q Particle and antiparticle decay rates q Kinetic theory q Boltzmann H-theore m 3.7 Partial-Wave Expansions * 15 1 Discrete basis q Expansion in spherical harmonics q Total elastic and inelastic cross-sections q Phase shifts q Threshold behavior : exothermic, endothermic, and elastic reactions q Scattering length q High-energy elastic and inelastic scatterin g 3.8 Resonances * 15 9 Reasons for resonances: weak coupling, barriers, complexity q Energydependence q Unitarity q Breit-Wigner formula q Unresolved resonances q Phase shifts at resonance q Ramsauer-Townsend effec t Problems 165 References THE CLUSTER DECOMPOSITION PRINCIPLE Bosons and Fermions 170 Permutation phases q Bose and Fermi statistics q Normalization for identical particle s 4.2 Creation and Annihilation Operators 17 3 Creation operators q Calculating the adjoint q Derivation of commutation / anticommutation relations q Representation of general operators q Free-particle Hamiltonian q Lorentz transformation of creation and annihilation operators q C, P, T properties of creation and annihilation operator s 4.3 Cluster Decomposition and Connected Amplitudes 17 7 Decorrelation of distant experiments q Connected amplitudes q Counting delt a functions

6 4.4 Structure of the Interaction 182 Condition for cluster decomposition q Graphical analysis q Two-body scatterin g implies three-body scattering Problems 18 9 References QUANTUM FIELDS AND ANTIPARTICLES Free Fields 19 1 Creation and annihilation fields q Lorentz transformation of the coefficient functions q Construction of the coefficient functions q Implementing cluster decomposition q Lorentz invariance requires causality q Causality requires antiparticle s q Field equations q Normal orderin g 5.2 Causal Scalar Fields 20 1 ~ Creation and annihilation fields q Satisfying causality q Scalar fields describ e bosons q Antiparticles q P, C, T transformations q 7T 5.3 Causal Vector Fields 20 7 Creation and annihilation fields q Spin zero or spin one q Vector fields describ e bosons q Polarization vectors q Satisfying causality q Antiparticles q Mass zero limit q P, C, T transformation s 5.4 The Dirac Formalism 21 3 Clifford representations of the Poincare algebra q Transformation of Dirac matrices q Dimensionality of Dirac matrices q Explicit matrices q y 5 q Pseudounitarit y q Complex conjugate and transpos e 5.5 Causal Dirac Fields 219 Creation and annihilation fields q Dirac spinors q Satisfying causality q Dirac fields describe fermions q Antiparticles q Space inversion q Intrinsic parity o f particle-antiparticle pairs q Charge-conjugation q Intrinsic C-phase of particleantiparticle pairs q Majorana fermions q Time-reversal q Bilinear covariants q Beta decay interaction s 5.6 General Irreducible Representations of the Homogeneous Lorent z Group* 229 Isomorphism with S U(2) O S U(2) q (A, B) representation of familiar fields q Rarita-Schwinger field q Space inversion 5.7 General Causal Fields * 23 3 Constructing the coefficient functions q Scalar Hamiltonian densities O Satisfyin g causality q Antiparticles q General spin-statistics connection q Equivalence of different field types q Space inversion q Intrinsic parity of general particleantiparticle pairs 0 Charge-conjugation q Intrinsic C-phase of antiparticles q

7 Self-charge-conjugate particles and reality relations q Time-reversal q Problems for higher spin? 5.8 The CPT Theorem 24 4 CPT transformation of scalar, vector, and Dirac fields q CPT transformation o f scalar interaction density q CPT transformation of general irreducible fields q CPT invariance of Hamiltonian 5.9 Massless Particle Fields 24 6 Constructing the coefficient functions q No vector fields for helicity ±1 q Need for gauge invariance q Antisymmetric tensor fields for helicity ±1 q Sums over helicity q Constructing causal fields for helicity ±1 q Gravitons q Spin > 3 q General irreducible massless fields q Unique helicity for (A, B) fields Problems 25 5 References THE FEYNMAN RULES Derivation of the Rules 25 9 Pairings q Wick's theorem q Coordinate space rules q Combinatoric factors q Sign factors q Example s 6.2 Calculation of the Propagator 274 Numerator polynomial q Feynman propagator for scalar fields q Dirac fields q General irreducible fields q Covariant propagators q Non-covariant terms in time-ordered products 6.3 Momentum Space Rules 28 0 Conversion to momentum space q Feynman rules q Counting independen t momenta q Examples q Loop suppression factor s 6.4 Off the Mass Shell 28 6 Currents q Off-shell amplitudes are exact matrix elements of Heisenberg-pictur e operators q Proof of the theorem Problems 29 0 References THE CANONICAL FORMALISM Canonical Variables 29 3 Canonical commutation relations q Examples : real scalars, complex scalars, vector fields, Dirac fields q Free-particle Hamiltonians q Free-field Lagrangian q Canonical formalism for interacting fields

8 7.2 The Lagrangian Formalism 29 8 Lagrangian equations of motion q Action q Lagrangian density q Euler- Lagrange equations q Reality of the action q From Lagrangians to Hamiltonian s q Scalar fields revisited q From Heisenberg to interaction picture q Auxiliary fields q Integrating by parts in the actio n 7.3 Global Symmetries 30 6 Noether's theorem q Explicit formula for conserved quantities q Explicit formula for conserved currents q Quantum symmetry generators q Energy-momentum tensor q Momentum q Internal symmetries q Current commutation relation s 7.4 Lorentz Invariance 31 4 Currents JZ Pµv q Generators Jµ" q Belinfante tensor q Lorentz invariance o f S-matrix 7.5 Transition to Interaction Picture : Examples 31 8 Scalar field with derivative coupling q Vector field q Dirac fiel d 7.6 Constraints and Dirac Brackets 32 5 Primary and secondary constraints q Poisson brackets q First and second clas s constraints q Dirac brackets q Example : real vector field 7.7 Field Redefinitions and Redundant Couplings* 33 1 Redundant parameters q Field redefinitions q Example : real scalar fiel d Appendix Dirac Brackets from Canonical Commutators 33 2 Problems 33 7 References ELECTRODYNAMICS Gauge Invariance 33 9 Need for coupling to conserved current q Charge operator q Local symmetry q Photon action q Field equations q Gauge-invariant derivative s 8.2 Constraints and Gauge Conditions 34 3 Primary and secondary constraints q Constraints are first class q Gauge fixing q Coulomb gauge q Solution for A 8.3 Quantization in Coulomb Gauge 34 6 Remaining constraints are second class q Calculation of Dirac brackets i n Coulomb gauge q Construction of Hamiltonian q Coulomb interactio n 8.4 Electrodynamics in the Interaction Picture 350 Free-field and interaction Hamiltonians q Interaction picture operators q Normal mode decomposition

9 8.5 The Photon Propagator 35 3 Numerator polynomial q Separation of non-covariant terms q Cancellation o f non-covariant terms 8.6 Feynman Rules for Spinor Electrodynamics 35 5 Feynman graphs q Vertices q External lines q Internal lines q Expansion i n a/4n q Circular, linear, and elliptic polarization q Polarization and spin sum s 8.7 Compton Scattering 36 2 S-matrix q Differential cross-section q Kinematics q Spin sums q Traces q Klein-Nishina formula q Polarization by Thomson scattering q Total cross - sectio n 8.8 Generalization : p-form Gauge Fields* 369 Motivation q p-forms q Exterior derivatives q Closed and exact p-forms q p-form gauge fields q Dual fields and currents in D spacetime dimensions q p-form gauge fields equivalent to (D - p - 2)-form gauge fields q Nothing new i n four spacetime dimensions Appendix Traces 37 2 Problems 374 References PATH-INTEGRAL METHODS The General Path-Integral Formula 37 8 Transition amplitudes for infinitesimal intervals q Transition amplitudes for finite intervals q Interpolating functions q Matrix elements of time-ordered product s q Equations of motion 9.2 Transition to the S-Matrix 38 5 Wave function of vacuum q ie terms 9.3 Lagrangian Version of the Path-Integral Formula 38 9 Integrating out the `momenta ' q Derivatively coupled scalars q Non-linear sigm a model q Vector field 9.4 Path-Integral Derivation of Feynman Rules 39 5 Separation of free-field action q Gaussian integration q Propagators : scalar fields, vector fields, derivative coupling 9.5 Path Integrals for Fermions 39 9 Anticommuting c-numbers q Eigenvectors of canonical operators q Summing states by Berezin integration q Changes of variables q Transition amplitudes for infinitesimal intervals q Transition amplitudes for finite intervals q Derivatio n of Feynman rules q Fermion propagator q Vacuum amplitudes as determinants

10 9.6 Path-Integral Formulation of Quantum Electrodynamics 41 3 Path integral in Coulomb gauge q Reintroduction of a q Transition to covariant gauge s 9.7 Varieties of Statistics* 41 8 Preparing `in' and `out ' states q Composition rules o Only bosons and fermion s in > 3 dimensions q Anyons in two dimensions Appendix Gaussian Multiple Integrals 420 Problems 423 References NON-PERTURBATIVE METHODS Symmetries 425 Translations n Charge conservation q Furry's theorem 10.2 Polology 42 8 Pole formula for general amplitudes q Derivation of the pole formula q Pio n exchang e 10.3 Field and Mass Renormalization 436 LSZ reduction formula q Renormalized fields q Propagator poles q No radiativ e corrections in external lines q Counterterms in self-energy part s 10.4 Renormalized Charge and Ward Identities 442 Charge operator q Electromagnetic field renormalization q Charge renormalization q Ward -Takahashi identity q Ward identity 10.5 Gauge Invariance 44 8 Transversality of multi-photon amplitudes q Schwinger terms q Gauge terms in photon propagator q Structure of photon propagator q Zero photon renormalized mass q Calculation of Z 3 q Radiative corrections to choice of gaug e 10.6 Electromagnetic Form Factors and Magnetic Moment 45 2 Matrix elements of J 0 q Form factors of JP : spin zero q Form factors of J P : spin z q Magnetic moment of a spin i particle q Measuring the form factor s 10.7 The Killen-Lehmann Representation * 45 7 Spectral functions q Causality relations q Spectral representation q Asymptotic behavior of propagators q Poles q Bound on field renormalization constant q Z = 0 for composite particle s 10.8 Dispersion Relations * 46 2 History q Analytic properties of massless boson forward scattering amplitude

11 q Subtractions q Dispersion relation q Crossing symmetry q theorem q Regge asymptotic behavior q Photon scatterin g Pomeranchuk' s Problems 469 References ONE-LOOP RADIATIVE CORRECTIONS IN QUANTU M ELECTRODYNAMICS Counterterms 472 Field, charge, and mass renormalization q Lagrangian counterterm s 11.2 Vacuum Polarization 47 3 One-loop integral for photon self-energy part q Feynman parameters q Wick rotation q Dimensional regularization q Gauge invariance q Calculation of Z 3 q Cancellation of divergences q Vacuum polarization in charged particle scatterin g q Uehling effect q Muonic atoms 11.3 Anomalous Magnetic Moments and Charge Radii 48 5 One-loop formula for vertex function q Calculation of form factors q Anomalou s lepton magnetic moments to order a q Anomalous muon magnetic moment to order a 2 ln(mµ/me) q Charge radius of lepton s 11.4 Electron Self-Energy 49 3 One-loop formula for electron self-energy part q Electron mass renormalization q Cancellation of ultraviolet divergence s Appendix Assorted Integrals 49 7 Problems 49 8 References GENERAL RENORMALIZATION THEORY Degrees of Divergence 50 0 Superficial degree of divergence q Dimensional analysis q Renormalizability q Criterion for actual convergence 12,2 Cancellation of Divergences 50 5 Subtraction by differentiation q Renormalization program q Renormalizable theories 0 Example : quantum electrodynamics q Overlapping divergences q BPHZ renormalization prescription q Changing the renormalization point : 0 4 theory 12.3 Is Renormalizability Necessary? 516

12 Renormalizable interactions cataloged q No renormalizable theories of gravitation q Cancellation of divergences in non-renormalizable theories q Suppression of non-renormalizable interactions q Limits on new mass scales q Problems with higher derivatives? q Detection of non-renormalizable interactions q Low-energy expansions in non-renormalizable theories q Example : scalar with only derivativ e coupling q Saturation or new physics? q Effective field theorie s 12.4 The Floating Cutor 52 5 Wilson's approach q Renormalization group equation q Polchinski's theorem q Attraction to a stable surface q Floating cutoff vs renormalization 12.5 Accidental Symmetries* 52 9 General renormalizable theory of charged leptons q Redefinition of the lepton fields q Accidental conservation of lepton flavors, P, C, and T Problems 53 1 References INFRARED EFFECTS Soft Photon Amplitudes 534 Single photon emission q Negligible emission from internal lines q Lorentz invariance implies charge conservation q Single graviton emission q Lorentz invariance implies equivalence principle q Multi-photon emission q Factorizatio n 13.2 Virtual Soft Photons 539 Effect of soft virtual photons q Radiative corrections on internal lines 13.3 Real Soft Photons; Cancellation of Divergences 544 Sum over helicities q Integration over energies q Sum over photon number q Cancellation of infrared cutoff factors q Likewise for gravitation 13.4 General Infrared Divergences 548 Massless charged particles q Infrared divergences in general q Jets q Lee-Nauenberg theorem 13.5 Soft Photon Scattering* 55 3 Poles in the amplitude q Conservation relations q Universality of the low-energy limit 13.6 The External Field Approximation* 556 Sums over photon vertex permutations q Non-relativistic limit q Crossed ladder exchang e Problems 56 2 References 562

13 14 BOUND STATES IN EXTERNAL FIELDS The Dirac Equation 56 5 Dirac wave functions as field matrix elements q Anticommutators and completeness q Energy eigenstates q Negative energy wave functions q Orthonormalization q `Large' and ` small' components q Parity q Spin- and angle-dependence q Radial wave equations q Energies q Fine structure q Non-relativistic approximation s 14.2 Radiative Corrections in External Fields 57 2 Electron propagator in an external field q Inhomogeneous Dirac equation q Effects of radiative corrections q Energy shift s 14.3 The Lamb Shift in Light Atoms 57 8 Separating high and low energies q High-energy term q Low-energy term q Effect of mass renormalization q Total energy shift q e = 0 q t # 0 Numerical results q Theory vs experiment for classic Lamb shift q Theory vs experiment for is energy shift Problems 594 References 59 5 AUTHOR INDEX 597 SUBJECT INDEX 602