Quantum Electrodynamics

1 Introduction.- 1.1 General Introduction and Definition of Notation.- 1.1.1 Chronological Review.- 1.1.2 Basic Notation.- 1.2 Quantum Electrodynamics with an External Field.- 2 Scattering Processes in Which Interactions with the External Field are Taken into Account Exactly.- 2.1 The Quantized Charged Field in an External Electromagnetic Field.- 2.1.1 Spinor Field.- 2.1.2 Scalar Field.- 2.2 Perturbation Expansion in the Radiative Interaction in QED with an External Field.- 2.3 Green's Functions. The Generating Functional.- 2.4 Appendix. Extension of the Normal-Ordering Process to Theories with an Unstable Vacuum.- 3 Expectation Values.- 3.1 Green's Functions for the Calculation of Expectation Values.- 3.2 Perturbations in Powers of Radiative Interaction for Expectation Values.- 3.3 Equation for the Average Electromagnetic Field. Effective Action.- 3.4 Density Matrix of Particles Created by an External Field.- 4 Total Probabilities of Radiative Processes in an External Field.- 4.1 Total Irradiation Probability.- 4.1.1 Total Probability of Irradiation from the Vacuum Accompanied by Pair Creation.- 4.1.2 Differential Probability for Photon Irradiation from the Vacuum Accompanied by Pair Creation.- 4.1.3 Probability of Photon Irradiation from the Vacuum Accompanied by the Creation of a Single Pair.- 4.1.4 Total Probability of Irradiation from a One-Electron State Accompanied by Pair Creation.- 4.1.5 Differential Probability of a One-Photon Emission from a One-Electron State Accompanied by Pair Creation.- 4.1.6 Probability of One-Photon Emission from a One-Electron State Without Pair Creation.- 4.1.7 Radiative Processes with a Photon in the Initial State.- 4.2 Unitarity Relation and the Optical Theorem.- 4.3 Generating Functional for the Total Probabilities of Radiation Processes.- 4.4 Decay Probability of States.- 4.4.1 Vacuum Decay Probability.- 4.4.2 Decay Probability of a One-Electron State.- 4.4.3 Decay Probability of a One-Photon State.- 5 Calculations of Zero-Order Processes in External Electromagnetic Fields.- 5.1 Processes in an Electric Field.- 5.1.1 Constant Electric Field.- 5.1.2 Alternating Electric Field.- 5.2 A Constant Field Combined with that of a Plane Wave.- 5.2.1 Solutions of the Klein-Gordon Equation.- 5.2.2 Solutions of the Dirac Equation.- 5.2.3 Calculation of Zero-Order Processes. Spinor QED.- 5.2.4 Calculation of Zero-Order Processes. Scalar QED.- 5.3 Creation of Particles from the Vacuum in Coherent States.- 5.4 Calculation of the Density Matrix for Particles Created in an External Field.- 6 Propagators of Particles in External Electromagnetic Fields.- 6.1 Introduction.- 6.2 Determination of Propagators by Summing Solutions of the Relativistic Wave Equations.- 6.2.1 Constant Electric Field.- 6.2.2 A Constant Field Combined with that of a Plane Wave.- 6.2.3 Eigenfunction Method.- 6.3 Schwinger's Proper-Time Method.- 6.4 Calculation of the Green's Functions by the Functional Integration Method.- 6.4.1 Path Integral Representation of the Green's Functions.- 6.4.2 Combination of a Constant Field with a Plane Wave.- 6.4.3 Stationary Phase Method.- 7 Calculations of Radiative Processes in External Electromagnetic Fields.- 7.1 Effective Action in the One-Loop Approximation.- 7.2 Vacuum Processes.- 7.2.1 Mean Current of Created Particles.- 7.2.2 Probability of Photon Emission from the Vacuum Accompanied by Pair Creation.- 7.2.3 Total Probability of Photon Emission from the Vacuum Accompanied by Pair Creation.- 7.2.4 Probability of a Photon Emission from the Vacuum Accompanied by the Creation of a Single Pair. Vacuum Decay Probability.- 7.3 Processes with an Electron in the Initial State. Mass Operator.- 7.3.1 Probability of Transition from a One-Electron State with the Emission of a Photon and the Creation of Pairs.- 7.3.2 Total Transition Probability from a One-Electron State with Photon Emission and Pair Creation.- 7.3.3 Probability of Photon Emission from a One-Electron State without Pair Creation. Decay Probability of a One-Electron State.- 7.4 Radiative Processes with a Photon in the Initial State. Polarization Operator.- 8 Green's Function in Non-Abelian Theories.- 8.1 Introduction.- 8.2 Calculation of Green's Functions by the Functional Integration Method.- 8.3 Calculation of Green's Functions in an Abelian-like External Field.- References.