Quantum Electrodynamics: Atoms, Lasers and Gravity

Contents
Preface
1. Introduction
	1.1 Accurate Numbers
	1.2 Fundamental Constants
	1.3 Overview of Chapter Contents
	1.4 Miscellaneous Remarks
	1.5 Further Thoughts
2. From Unit Systems for the Microworld to Field Quantization
	2.1 Overview
	2.2 Atoms and Field Quantization
		2.2.1 Matter Waves and Hamilton{Jacobi Formalism
		2.2.2 Field Quantization as Second Quantization
	2.3 Unit Systems Scaled to the Microworld
		2.3.1 Unit Systems and Observation Scales
		2.3.2 Natural Unit System
		2.3.3 Atomic Unit System
	2.4 Field Quantization for the Electromagnetic Field
		2.4.1 Quantization of the Free Electromagnetic Field
		2.4.2 Field Operators and Quantized Hamiltonian
	2.5 Interaction Picture and Phase Conventions
		2.5.1 Field Operators in the Schrödinger and Interaction Pictures
		2.5.2 Integration Measure and Phase Conventions
	2.6 Further Thoughts
3. Time-Ordered Perturbations
	3.1 Overview
	3.2 Time-Ordered Perturbations and Fermi's Golden Rule
		3.2.1 Derivation of Fermi's Golden Rule
		3.2.2 Fermi's Golden Rule and Nuclear Beta Decay
		3.2.3 Fermi's Golden Rule and Atomic Decay Rates
	3.3 Dynamic Stark Effect
		3.3.1 Way 1: Time-Dependent Perturbation Theory
		3.3.2 Way 2: Quantized Fields
		3.3.3 Way 3: Gell-Mann–Low–Sucher Theorem
	3.4 Static Stark Effect and Level Width
		3.4.1 Stark Shift and Large-Order Perturbation Theory
		3.4.2 Quantum Electrodynamics and Large-Order Perturbations
	3.5 Further Thoughts
4. Bound-Electron Self–Energy and Bethe Logarithm
	4.1 Overview
	4.2 Schrödinger–Coulomb Hamiltonian and Wave Functions
		4.2.1 Spectrum of the Hydrogen Atom and SO(4) Symmetry
		4.2.2 Dierential Equations and the Hydrogen Atom
		4.2.3 Schrödinger–Coulomb Bound States
		4.2.4 Schrödinger–Coulomb Virial Theorem
		4.2.5 Schrödinger–Coulomb Continuum States
		4.2.6 Continuum States for a Repulsive Potential
	4.3 Schrödinger–Coulomb Green Function in Coordinate Space
		4.3.1 Green Function and Radial Equation
		4.3.2 Solution using Whittaker Functions
		4.3.3 Solution using Laguerre Polynomials
		4.3.4 Green Function and Dynamic Polarizability
	4.4 Schrödinger–Coulomb Green Function in Momentum Space
		4.4.1 Free Green Function in Momentum Space
		4.4.2 Toward the SO(4) Symmetry
		4.4.3 Four-Dimensional Spherical Harmonics
		4.4.4 Wave Functions in Momentum Space
		4.4.5 Integral Representation
	4.5 Modern Ideas and Bound-State Self–Energy
		4.5.1 Essence of Renormalization
		4.5.2 Overlapping Parameter
	4.6 Bound-State Self–Energy: Low-Energy Part
		4.6.1 Length Gauge
		4.6.2 Velocity Gauge
		4.6.3 Calculation of the Bethe Logarithm
		4.6.4 Numerical Values of Bethe Logarithms
		4.6.5 Outline of the High-Energy Part
	4.7 Applications of the Developed Formalism
		4.7.1 Electric Dipole Decay and Imaginary Part of Self–Energy
		4.7.2 Magnetic Interactions and Decay Rates
		4.7.3 Higher-Order Terms
	4.8 Further Thoughts
5. Interatomic and Atom-Surface Interactions
	5.1 Overview
	5.2 Wigner–Brioullin Perturbation Theory
		5.2.1 Spectral Representation of the Green Function
		5.2.2 First-Order and Second-Order Perturbation Theory
		5.2.3 Higher-Order Perturbation Theory (Wigner–Brioullin)
	5.3 Application to the Finite-Size Effect
		5.3.1 Master Integrals
		5.3.2 First-Order and Second-Order Finite-Size Effect
	5.4 Interatomic Interactions
		5.4.1 Origin of van der Waals and Casimir–Polder Interactions
		5.4.2 Calculation of Interatomic Interactions
		5.4.3 Limit of Large Separation of the Two Atoms
		5.4.4 Nonretarded (van der Waals) Interatomic Interaction
		5.4.5 Interpolating Formula
	5.5 Atom-Surface Interactions
		5.5.1 Perfectly Conducting Wall, Long Range
		5.5.2 Dipole Interaction
		5.5.3 Multipole Interactions
		5.5.4 Interactions with a Dielectric Surface
	5.6 Further Thoughts
6. Racah–Wigner Algebra
	6.1 Overview
	6.2 Clebsch–Gordan Coefficients
		6.2.1 Expansions and Clebsch–Gordan Coefficients
		6.2.2 Matrix Elements and Clebsch–Gordan Coefficients
	6.3 Coefficients and Rotations
		6.3.1 Vector Addition or Clebsch–Gordan Coefficients
		6.3.2 Wigner 3j, 6j and 9j Symbols
		6.3.3 Gaunt Coefficients
		6.3.4 Representation of Finite Rotations
	6.4 Composed Tensors of Higher Order
		6.4.1 Construction of the Spin-Angular Function
		6.4.2 Construction of the Vector Spherical Harmonic
		6.4.3 Spherical Biharmonic
	6.5 Applications of Racah–Wigner Algebra
		6.5.1 Tensor Decomposition of the Light Shift
		6.5.2 Tensorial Decomposition of a Dipole Transition
	6.6 Rydberg Electron and Hydrogenlike Core
		6.6.1 Physical Foundation
		6.6.2 Angular Algebra
	6.7 Further Thoughts
7. Free Dirac Equation
	7.1 Overview
	7.2 Properties of the Free Dirac Equation
		7.2.1 Dirac Equation as the Linearized Klein–Gordon Equation
		7.2.2 Spinor Lorentz Transformation
		7.2.3 Discrete Symmetries
		7.2.4 Overview of the Symmetry Properties
	7.3 Solutions of the Free Dirac Equation
		7.3.1 Plane-Wave Solutions of the Dirac Equation
		7.3.2 Dirac Angular Quantum Number
		7.3.3 Angular Momenta and Massless Dirac Equation
		7.3.4 Angular Momenta and Free Dirac Equation
	7.4 Quantized Dirac Field and Propagators
		7.4.1 Free Dirac Progator in Feynman's Formulation
		7.4.2 Feynman Propagator and Green Function
		7.4.3 Free Dirac Propagator in the Angular Momentum Basis
	7.5 Further Thoughts
8. Dirac Equation for Bound States, Lasers and Gravity
	8.1 Overview
	8.2 Dirac Equation and Coulomb Field
		8.2.1 Electromagnetic Covariant Derivative
		8.2.2 Dirac–Coulomb Bound-State Wave Functions
		8.2.3 Dirac–Coulomb Continuum-State Wave Functions
		8.2.4 Dirac–Coulomb Virial Theorem
		8.2.5 Dirac–Coulomb Propagator
	8.3 Dirac–Volkov Equation for Laser Fields
		8.3.1 Dirac–Volkov Solutions for Laser Fields
		8.3.2 Dirac–Volkov Propagator
	8.4 Dirac Equation with Coupling to Gravitational Fields
		8.4.1 Metric and Covariant Derivative
		8.4.2 Tetrad Basis and Affine Connection Matrix
		8.4.3 Ricci Rotation Coefficients
		8.4.4 Covariant Derivative of a Spinor
		8.4.5 Covariant Derivative of the Dirac Matrices
		8.4.6 Spin Connection
		8.4.7 Transformation Properties of Rotation Coefficients
	8.5 Applications of Gravitational Coupling
		8.5.1 Dirac–Schwarzschild Hamiltonian
		8.5.2 Dirac Adjoint for Curved Space-Times
		8.5.3 Lagrangian and Charge Conjugation
	8.6 Further Thoughts
9. Electromagnetic Field and Photon Propagators
	9.1 Overview
	9.2 Time Orderings, Field Commutators and Green Functions
		9.2.1 Miscellaneous Fundamental Relations for Green Functions
		9.2.2 Distributions and Fourier Transforms
	9.3 Photon Propagator in Coulomb Gauge
		9.3.1 Legendre Transformation and Hamiltonian
		9.3.2 Matching and Photon Propagator
	9.4 Photon Propagator in Lorenz Gauge
		9.4.1 Gauge Invariance and Mass of Photon
		9.4.2 Gauge-Fixing Term and Quantization
		9.4.3 Representations of the Photon Propagator
	9.5 Photon Propagator in General Gauges
		9.5.1 Construction Principle
		9.5.2 Most General Form and Weyl Gauge
		9.5.3 Gupta–Bleuler Condition
	9.6 Wick Theorem and Applications
		9.6.1 Time Ordering and Wick Theorem
		9.6.2 Field Commutators and Current Distributions
	9.7 Further Thoughts
10. Tree-Level and Loop Diagrams, and Renormalization
	10.1 Overview
	10.2 Tree-Level
		10.2.1 Rutherford Scattering
		10.2.2 Feynman Rules
	10.3 Vertex Correction
		10.3.1 Vertex and Pauli–Villars Regularization
		10.3.2 Vertex and Form Factors
		10.3.3 Vertex and Renormalization
		10.3.4 Detour on Dimensional Regularization
		10.3.5 Detour on Feynman Parameterization
		10.3.6 Vertex and Dimensional Regularization
	10.4 Vacuum Polarization
		10.4.1 Initial Considerations
		10.4.2 Vacuum Polarization and Dimensional Regularization
		10.4.3 Vacuum Polarization and Coulomb Potential
		10.4.4 Vacuum Polarization and Asymptotics
	10.5 Self–Energy Operator
		10.5.1 Self–Energy and Pauli–Villars Regularization
		10.5.2 Self–Energy and Dimensional Regularization
	10.6 Renormalization of QED
		10.6.1 Bare and Renormalized Lagrangian
		10.6.2 Renormalization of Vertex and Self–Energy
		10.6.3 Renormalization of Vacuum Polarization
		10.6.4 Compilation of Renormalization Constants
		10.6.5 Forest Formula
	10.7 Further Thoughts
11. Foldy–Wouthuysen Transformation and Lamb Shift
	11.1 Overview
	11.2 Leading Relativistic Corrections
		11.2.1 Unitary Transformation and Hamiltonian
		11.2.2 Free Dirac Particle
		11.2.3 Transformation in the General Case
		11.2.4 Radiatively Corrected Dirac Hamiltonian
		11.2.5 General Electromagnetic Coupling
		11.2.6 General Particle Hamiltonians
	11.3 Applications
		11.3.1 Coulomb Field Coupling
		11.3.2 Magnetic Field Coupling and Electron Moment
		11.3.3 Magnetic Fields and Electrostatic Potentials
		11.3.4 Transition Current
		11.3.5 Nonrelativistic Transition Current and Multipoles
	11.4 Dirac Form Factor and Bound-State Radiative Energy Shifts
		11.4.1 Foldy–Wouthuysen Transformation and Form Factors
		11.4.2 High-Energy Part in Photon Mass Regularization
		11.4.3 High-Energy Part in Photon Energy Regularization
		11.4.4 Self–Energy in Dimensional Regularization
		11.4.5 Vacuum Polarization
	11.5 Foldy–Wouthuysen Transformation and Gravity
	11.6 Further Thoughts
12. Relativistic Interactions for Many-Particle and Compound Systems
	12.1 Overview
	12.2 Interatomic Interactions in Covariant Formalism
		12.2.1 General Paradigm of the Matching
		12.2.2 Scattering Amplitudes and Interatomic Interactions
	12.3 From the One-Body to the Two-Body Hamiltonian
		12.3.1 Relevant Hamiltonians
		12.3.2 Matching the Two-Particle Interaction Hamiltonian
		12.3.3 Photon Exchange and Breit Hamiltonian
		12.3.4 Breit Hamiltonian for Unequal Particles
	12.4 Applications and Generalizations for Many-Particle Systems
		12.4.1 Application to Many-Electron Systems
		12.4.2 Single-Particle Hamiltonian for Arbitrary Spin
		12.4.3 Interaction Hamiltonian for General Spin
	12.5 Application to Two-Body Bound Systems
		12.5.1 General Aspects
		12.5.2 Definition of the Lamb Shift
		12.5.3 Application to Hydrogenlike Ions
	12.6 Compound System in a Homogeneous Magnetic Field
		12.6.1 Power–Zienau Transformation for a Magnetic Field
		12.6.2 Reduced-Mass Corrections to the Zeeman Effect
		12.6.3 Nuclear Magnetic Shielding
	12.7 Further Thoughts
13. Fully Correlated Basis Sets and Helium
	13.1 Overview
	13.2 Essential Ingredients for Many-Electron Systems
		13.2.1 Spin Wave Functions
		13.2.2 Two-Body Problem: Center-of-Mass Coordinates
		13.2.3 Three-Body Problem: Mass Polarization
	13.3 Relativistic and Radiative Corrections
		13.3.1 Relativistic Corrections
		13.3.2 Derivation of the Helium Lamb Shift
	13.4 Numerical Calculation of the Helium Spectrum
		13.4.1 Fully Correlated Basis Sets
		13.4.2 Matrix Elements and Nonorthogonal Basis
		13.4.3 Numerical Results and Technical Issues
	13.5 Further Thoughts
14. Relativistic Many-Particle Calculations
	14.1 Overview
	14.2 Many-Particle Systems and Hartree–Fock Approximation
		14.2.1 Nonrelativistic Hartree–Fock Approximation
		14.2.2 Implementation and Practical Aspects
	14.3 Relativistic Hartree–Fock Methods
		14.3.1 Brown–Ravenhall Disease
		14.3.2 Relativistic Frequency-Dependent Breit Interaction
		14.3.3 Relativistic Hartree–Fock Approximation
		14.3.4 Relativistic Multi–Configurational Approach
		14.3.5 Relativistic Breit Interaction and Spinors
		14.3.6 Subtleties of Nuclear Eects, QED and MCDHF
		14.3.7 Software Packages and Applications
	14.4 Systems with a Large Number of Electrons
	14.5 Further Thoughts
15. Beyond Breit Hamiltonian and On-Shell Form Factors
	15.1 Overview
	15.2 Furry Picture, Scattering Amplitude and Self–Energy
		15.2.1 Furry Picture
		15.2.2 Matching for the Bound-State Self–Energy
	15.3 Binding Correction to the Lamb Shift
		15.3.1 Two-Coulomb-Vertex Scattering Amplitude
		15.3.2 Forward Scattering Amplitudes
		15.3.3 Dispersion Relation and Subtractions
		15.3.4 Example Integral
		15.3.5 Final Integration and A50 Coefficient
	15.4 Higher-Order Terms
	15.5 Relativistic Recoil Correction
		15.5.1 Retardation and Salpeter Correction
		15.5.2 Low-Energy Part and Dipole Approximation
		15.5.3 Middle-Energy Part and Araki–Sucher Distribution
		15.5.4 Seagull Part
		15.5.5 High-Energy Part
		15.5.6 Final Result
	15.6 Further Thoughts
16. Bethe–Salpeter Equation
	16.1 Overview
	16.2 General Ideas
		16.2.1 Two-Body Bound States and Green Functions
		16.2.2 Questions of Gauge and Transformations
		16.2.3 Reference Kernels and Exact Solutions
		16.2.4 Bound-State Perturbation Theory
		16.2.5 Energy Levels at O(Zα)4 (BSEQ Approach)
	16.3 Relativistic Recoil Correction (BSEQ Approach)
		16.3.1 Exploring Integration Regions
		16.3.2 Method of Regions
		16.3.3 Contribution from the Hard Region
		16.3.4 Soft Contribution from Two Transverse Photons
		16.3.5 Soft Contribution from a Single Transverse Photon
		16.3.6 Ultrasoft Contribution from a Single Transverse Photon
		16.3.7 Sum of the Contributions
	16.4 Hyperne Structure of S States (BSEQ Approach)
	16.5 Further Thoughts
17. NRQED: An Effective Field Theory for Atomic Physics
	17.1 Overview
	17.2 Basics of NRQED
		17.2.1 NRQED Lagrangian and Feynman Rules
		17.2.2 Matching Coefficients
		17.2.3 Bethe–Salpeter Equation of NRQED
	17.3 Applications of NRQED
		17.3.1 Energy Levels at O(Zα)4 (NRQED Approach)
		17.3.2 Relativistic Recoil Correction (NRQED Approach)
		17.3.3 Hyperfine Structure of S States (NRQED Approach)
	17.4 Discussion
	17.5 Further Thoughts
18. Fermionic Determinants and Effective Lagrangians
	18.1 Overview
	18.2 Derivation of the Heisenberg–Euler Effective Lagrangian
		18.2.1 Heisenberg–Euler Lagrangian for Electric Fields
		18.2.2 Heisenberg–Euler Lagrangian for General Fields
	18.3 Application of the Heisenberg–Euler Lagrangian
		18.3.1 Modification of the Speed of Light in Background Fields
		18.3.2 Effective Lagrangian and Wichmann–Kroll Potential
	18.4 Further Thoughts
19. Renormalization-Group Equations
	19.1 Orientation and Motivation
		19.1.1 Renormalization on Different Scales
		19.1.2 Renormalization-Group Equations
	19.2 Scale Transformations
		19.2.1 From Poor Man's Scaling to Nontrivial Scaling
		19.2.2 Scale Transformation and Schrödinger Hamiltonian
	19.3 Renormalization Group Transformations
		19.3.1 Functional Equations in the Asymptotic Regime
		19.3.2 Renormalization Group of Callan and Symanzik
		19.3.3 Connection of Callan–Symanzik and Gell-Mann–Low
	19.4 RG Equations and Applications
		19.4.1 Effective Action and Renormalization Group
		19.4.2 Brodsky–Lepage–Mackenzie Scale Setting
	19.5 Further Thoughts
Bibliography
Index