The Theory of Photons and Electrons The Relativistic Quantum Field Theory of Charged Particles with Spin One-half

Since the discovery of the corpuscular nature of radiation by Planck more than fifty years ago the quantum theory of radiation has gone through many stages of development which seemed to alternate between spectacular success and hopeless frustration. The most recent phase started in 1947 with the discovery of the electromagnetic level shifts and the realization that the exist­ ing theory, when properly interpreted, was perfectly adequate to explain these effects to an apparently unlimited degree of accuracy. This phase has now reached a certain conclusion: for the first time in the checkered history of this field of research it has become possible to give a unified and consistent presen­ tation of radiation theory in full conformity with the principles of relativity and quantum mechanics. To this task the present book is devoted. The plan for a book of this type was conceived during the year 1951 while the first-named author (J. M. J. ) held a Fulbright research scholarship at Cambridge University. During this year of freedom from teaching and other duties he had the opportunity of conferring with physicists in many different countries on the recent developments in radiation theory. The comments seemed to be almost unanimous that a book on quantum electrodynamics at the present time would be of inestimable value to physicists in many parts of the world. However, it was not until the spring of 1952 that work on the book began in earnest.

1 General Principles.- 1-1 The natural unit system.- 1-2 Some fundamental notions of the special theory of relativity.- 1-3 Some basic notions of quantum mechanics.- 1-4 Localizability.- 1-5 Observables of a field.- 1-6 Canonical transformations.- 1-7 Lorentz transformations as canonical transformations.- 1-8 The action principle.- 1-9 The equation of motion.- 1-10 Momentum operators.- 1-11 Conservation laws.- 1-12 Commutation rules.- 2 the Radiation Field.- 2-1 The classical field equations.- 2-2 The associated boundary value problem.- 2-3 A Lagrangian for the radiation field.- 2-4 Quantization of the radiation field.- 2-5 Momentum operators for the radiation field.- 2-6 Plane wave decomposition of the radiation field.- 2-7 Explicit representations of the field operators.- 2-8 The spin of the photon.- 2-9 Definition of the vacuum.- 3 Relativistic Theory of Free Electrons.- 3-1 The field equations for the one-particle problem.- 3-2 The associated boundary value problem.- 3-3 Relativistic invariance of the field equations.- 3-4 The bilinear covariants.- 3-5 A Lagrangian for the spinor field.- 3-6 Quantization.- 3-7 Momentum operators.- 3-8 Plane wave decomposition.- 3-9 Explicit representation of the field operators.- 3-10 The definition of the vacuum.- 4 Interaction of Radiation with Electrons.- 4-1 The field equations.- 4-2 Commutation rules for the interacting fields.- 4-3 The interaction picture.- 4-4 Measurability of the fields.- 5 Invariance Properties of the Coupled Fields.- 5-1 Proper Lorentz transformations.- 5-2 Gauge transformations.- 5-3 Space inversion.- 5-4 Time inversion.- 5-5 Charge conjugation.- 5-6 Scale transformations.- 6 Subsidiary Condition and Longitudinal Field.- 6-1 The covariant Coulomb interaction.- 6-2 The subsidiary condition and the construction of the state vector.- 6-3 The Gupta method.- 6-4 Gauge-independent interaction.- 6-5 Radiation fields with finite mass.- 7 the S-Matrix.- 7-1 Preliminary definition of the S-matrix.- 7-2 The wave matrix.- 7-3 The wave operator.- 7-4 Integral representation of the wave operator.- 7-5 Definition of the S-matrix.- 7-6 Invariance properties of the S-matrix.- 8 Evaluation of the S-Matrix.- 8-1 The iteration solution.- 8-2 The Feynman-Dyson diagrams.- 8-3 Diagrams in momentum space.- 8-4 Closed loops.- 8-5 The substitution law.- 8-6 Lifetimes and cross sections.- 8-7 Evaluation of the S-matrix in the Heisenberg picture.- 9 the Divergences in the Iteration Solution.- 9-1 Historical background.- 9-2 Classification of divergences.- 9-3 The vacuum fluctuations.- 9-4 The self-energy of the electron.- 9-5 The self-energy of the photon.- 9-6 The vertex part.- 10 Renormalization.- 10-1 The primitive divergences.- 10-2 Irreducible and proper diagrams.- 10-3 Separation of divergences in irreducible parts.- 10-4 Separation of divergences in reducible parts.- 10-5 Mass renormalization.- 10-6 Charge renormalization.- 10-7 Wave function renormalization.- 10-8 Sufficiency proof.- 10-9 Regulators.- 11 the Photon-Electron System.- 11-1 Compton scattering.- 11-2 Double Compton scattering.- 11-3 Radiative corrections to Compton scattering.- 11-4 Pair production in photon-electron collisions.- 12 the Electron-Electron System.- 12-1 Møller scattering.- 12-2 Bhabha scattering.- 12-3 Bremsstrahlung in electron-electron collisions.- 12-4 Annihilation of free negaton-positon pairs.- 12-5 Positronium; selection rules.- 12-6 Positronium annihilation.- 13 the Photon-Photon System.- 13-1 Photon-photon scattering as part of a diagram.- 13-2 Photon-photon scattering cross sections.- 13-3 Pair production in photon-photon collision.- 14 Theory of the External Field.- 14-1 The external field approximation.- 14-2 The bound interaction picture.- 14-3 Commutation rules.- 14-4 The electron propagation function.- 14-5 The S-matrix in the external field approximation.- 14-6 Renormalization.- 14-7 Cross sections and energy levels.- 15 External Field Problems.- 15-1 Coulomb scattering.- 15-2 Radiative corrections to Coulomb scattering.- 15-3 The magnetic moment of the election.- 15-4 Energy levels in hydrogen-like atoms.- 15-5 Radiative transitions between bound states.- 15-6 Bremsstrahlung.- 15-7 Pair production and annihilation.- 15-8 Delbrück and Rayleigh scattering.- 16 Special Problems.- 16-1 The infrared divergences.- 16-2 Radiation damping in collision processes.- 16-3 The natural line width of stationary states.- 16-4 The self-stress of the electron.- 16-5 Outlook.- Mathematical Appendix.- Appendix A1 the Invariant Functions.- A1-1 The homogeneous delta-functions.- A1-2 The inhomogeneous delta-functions.- A1-3 Relations between the ?-functions.- A1-4 Integral representations.- A1-5 Explicit expressions.- A1-6 The S-functions.- Appendix A2 the Gamma-Matrices.- A2-1 Various representations.- A2-3 The amplitudes of the plane wave solutions.- A2-5 Spin sums.- A2-6 Polarization sums.- Appendix A3 a Theorem on the Representation of the Extended.- Lorentz Group by Irreducible Tensors.- Appendix A4 the Ordering Theorem.- A4-1 The ordering theorem for commuting fields.- A4-2 The ordering theorem for anticommuting fields.- A4-3 A generalization of the ordering theorem.- A4-4 The ordering of chronological products.- Appendix A5 on the Evaluation of Certain Integrals.- A5-1 Convergent integrals.- A5-2 Divergent integrals.- A5-3 The integral for the photon self-energy part.- A5-4 The integral for the electron self-energy part.- Appendix A6 a Limiting Relation for the ?-function.- Appendix A7 the Method of Analytic Continuation.- A7-1 The Bohr-Peierls-Placzek relation.- A7-2 The principle of limiting distance.- A7-3 The fundamental theorem on analytic continuation.- A7-4 Applications.- Appendix A8 Notation.- Supplement for the Second Edition.- Supplement S1 Formulations of Quantum Electrodynamics.- S1-1 Lagrangian QFT.- S1-2 Axiomatic QFT.- S1-3 Locality, covariance, and indefinite metric.- S1-4 Lehmann-Symanzik-Zimmermann and related formalisms.- S1-5 Null plane QED.- References.- Supplement S2 Renormalization.- S2-1 Dyson-Salam-Ward renormalization.- S2-2 Bogoüubov-Parasiuk-Hepp-Zimmermann renormalization.- S2-3 Analytic renormalization.- References.- Supplement S3 Coherent States.- S3-1 A finite number of degrees of freedom.- S3-2 Coherent states of the radiation field.- S3-3 Application to ordering theorems.- References.- Supplement S4 Infrared Divergences.- S4-1 Dollard’s discovery.- S4-2 A new picture.- S4-3 The asymptotically modified fields.- References.- Supplement S5 Predictions and Precision Experiments.- S5-1 The anomalous magnetic moment.- S5-2 The hyperfine structure of the hydrogen ground state.- S5-3 The Lamb-Retherford shift in hydrogen.- S5-4 Energy levels in positronium.- S5-5 Muonium hyperfine structure.- References.- Author Index.