Classical and Quantum Description of Plasma and Radiation in Strong Fields

Supervisor’s Foreword
Acknowledgements
Contents
List of Symbols
Conventions and Units
1 Introduction
	1.1 Strong-Field Regime of Quantum Electrodynamics
	1.2 Schwinger-Sauter Mechanism
	1.3 Strong-Field QED Processes
	1.4 Sources of Strong Fields and High-Energy Particles in Nature and in the Laboratory
	1.5 Laser-Plasma Interaction at Ultra-High Intensity
	1.6 Presentation of This Work
	References
Theoretical Background
2 Classical Electrodynamics
	2.1 Outline
	2.2 Classical Field Theory
		2.2.1 Euler-Lagrange Equations
		2.2.2 Noether\'s Theorem
		2.2.3 Poincaré Invariance
		2.2.4 Energy-Momentum Tensor
	2.3 Classical Electrodynamics
		2.3.1 Lagrangian
		2.3.2 Equations of Motion
		2.3.3 Energy-Momentum
	2.4 Radiation by Moving Charges
		2.4.1 Physical Picture
		2.4.2 Retarded and Advanced Propagators
		2.4.3 Liénard-Wiechert Potentials
		2.4.4 Radiated Electromagnetic Field
		2.4.5 Larmor Formula
		2.4.6 Radiated Spectrum
		2.4.7 Charged Particle in a Plane-Wave Field
		2.4.8 Linear and Nonlinear Thomson Scattering
		2.4.9 Radiation by an Ultra-Relativistic Particle in Arbitrary Motion
	2.5 Classical Radiation-Reaction
		2.5.1 The Lorentz-Abraham-Dirac Equation
		2.5.2 The Landau-Lifshitz Equation
		2.5.3 Charged Particle in a Constant-Uniform Magnetic Field with RR
		2.5.4 Charged Particle in a Plane-Wave Field with RR
		2.5.5 Limit of Applicability of CED and Classical Radiation Dominated Regime (CRDR)
	References
3 Quantum Electrodynamics
	3.1 Outline
	3.2 Quantum Field Theory
		3.2.1 Canonical Quantization of Classical Fields
		3.2.2 Fock Space
		3.2.3 The S-Matrix
	3.3 Quantum Electrodynamics
		3.3.1 Lagrangian
		3.3.2 Equations of Motion
	3.4 Strong Field QED
		3.4.1 Coherent States
		3.4.2 Furry Picture
		3.4.3 Volkov States
		3.4.4 Position Space SFQED Feynman Rules
	3.5 Non-linear Compton Scattering
	3.6 Non-linear Breit-Wheeler Pair Production
	3.7 Quantum Radiation-Reaction
	References
4 Kinetic Description
	4.1 Outline
	4.2 Kinetic Description
		4.2.1 The Klimontovich Equation
		4.2.2 The Vlasov Equation
	4.3 Kinetic Description of Radiation-Reaction
		4.3.1 Motivation
		4.3.2 Vlasov Equation with Classical RR
		4.3.3 Vlasov Equation with Quantum RR
		4.3.4 Vlasov Equation with Pair-Production
	4.4 Extended PIC Method for SFQED
		4.4.1 Classical RR Pusher
		4.4.2 Monte-Carlo Module
	References
From Quantum to Classical Radiation Reaction
5 Effect of RR on the Electron Distribution Function
	5.1 Outline
	5.2 Dynamics of a Radiating Electron in Classical and Quantum Electrodynamics
		5.2.1 Dynamics of a Classical Radiating Electron
		5.2.2 Radiation Friction Force Acting on an Ultra-Relativistic Electron
		5.2.3 Dynamics of a Quantum Radiating Electron
	5.3 From Quantum to Classical Radiation Reaction for Ultra-relativistic Electrons
		5.3.1 Kinetic Point of View: The Linear Boltzmann Equation
		5.3.2 Toward the Classical Limit: The Fokker-Planck Approach
		5.3.3 Domain of Validity of the Fokker-Planck and Quantum-Corrected Landau-Lifshitz Descriptions
	5.4 Local Temporal Evolution of Integrated Quantities
		5.4.1 Local Energy Moments of the Collision Operators
		5.4.2 Equations of Continuity for the Electrons and Photons
		5.4.3 Equation of Evolution of the Mean Momentum
	5.5 Global Temporal Evolution of Average Quantities
		5.5.1 Conservation of the Number of Electrons and Total Energy, and Photon Production Rate
		5.5.2 Equations of Evolution
		5.5.3 The Perturbative Expansion
		5.5.4 Electron Mean Energy
		5.5.5 Variance in Energy: Radiative Cooling vs. Energy Spreading
		5.5.6 Third Order Moment and Link to the Quenching of Radiation Losses
	5.6 Conclusion
	References
6 Domain of Validity of the Different Descriptions and Numerical Simulations
	6.1 Outline
	6.2 Domain of Validity of the Three Descriptions
		6.2.1 Case of an Initially Symmetric Energy Distribution
		6.2.2 Case of an Initially Non-symmetric Energy Distribution
		6.2.3 Physical Implications
	6.3 Stochastic (Fokker-Planck) Pusher
	6.4 Numerical Results
		6.4.1 Constant-Uniform Magnetic Field
		6.4.2 Linearly Polarized Plane-Wave
		6.4.3 Electron Population with a Broad Energy Dispersion
	6.5 Interpretation of the ``Vlasov Terms\'\'
	6.6 Conclusion
	References
7 Photon Distribution Function
	7.1 Outline
	7.2 High-Energy Photon Emission and Its Back Reaction
		7.2.1 Photon Emission Rate and Energy Spectrum
		7.2.2 Radiation Reaction
		7.2.3 High Energy Photon Spectrum Accounting for Radiation Reaction
	7.3 Numerical Simulations
		7.3.1 Method
		7.3.2 Results
		7.3.3 Electron Beam Head-On Collision with an UHI Plane-Wave
		7.3.4 Hot (Maxwell-Jüttner) Electron Population Radiating in a Constant Magnetic Field
	7.4 Conclusions
	References
8 QED Cascades in Counterpropagating Plane-Waves
	8.1 Outline
	8.2 Motion of a Single Electron in Two Counter-Propagating Plane-Waves
		8.2.1 Without RR
		8.2.2 With RR
	8.3 Evolution of the Electron Distribution Function …
		8.3.1 Equations of Evolution of the Energy Moments of the Electron and Photon Distribution Functions in the Presence of a Source
		8.3.2 Asymptotic Energy Spread
		8.3.3 Asymptotic Third Order Moment
		8.3.4 Average Energy
		8.3.5 Evolution of an Initial Cold Electron Distribution Function
		8.3.6 Interaction of a Hot Maxwell-Jüttner Distribution Function with a Uniformly Rotating Electric Field
	8.4 Evolution of the Electron Distribution Function …
		8.4.1 Equations of Evolution of the Energy Moments of the Electron and Photon Distribution Functions in the Presence of a Source and with Pair Production
		8.4.2 Number of Particles
		8.4.3 Average Energy
		8.4.4 Asymptotic Energy Spread
	8.5 Conclusion
	References
Nonlinear Breit-Wheeler Pair Production with Laguerre-Gauss Beams
9 Orbital Angular Momentum of Light: A State of the Art
	9.1 Introduction
	9.2 Paraxial Optics
		9.2.1 The Helmholtz Equation
		9.2.2 The Paraxial Wave Equation
	9.3 Linear and Angular Momentum of Light
		9.3.1 Linear Momentum of Light
		9.3.2 Angular Momentum of Light
	9.4 Gaussian Beam
	9.5 Higher-Order Beams
		9.5.1 Laguerre-Gaussian Beams
	9.6 Angular Momentum of LG Beams
	References
10 Soft Pair Showers in the Collision of Gamma Rays with Laguerre-Gauss Beams
	10.1 Outline
	10.2 The Nonlinear Breit-Wheeler Process
	10.3 Characteristics of the LG Beams
	10.4 Nonlinear Breit-Wheeler Pair Production in an External LG Beam
	10.5 Conclusion
	References
11 Conclusion and Perspectives
Appendix A Classical Electrodynamics
	A.1 Bessel Functions
	A.1.1. Bessel Functions of the First Kind
	A.1.2 Modified Bessel Functions of the Second Kind
	A.2 Airy Functions
	A.3 Tensor Identities
	A.3.1 Levi-Civita Symbols
	A.3.2 Constant Field Tensor Identities
	A.4 Light-Cone Coordinates
	A.5 Solution of the Equation of Motion of a Charged Particle in a Plane-Wave Electromagnetic Field Without RR
	A.6 Non-linear Thomson Scattering
	A.6.1 Circularly Polarized Plane-Wave
	A.6.2 Linearly Polarized Plane-Wave
	A.7 Regularized Self-field
	A.8 Solution of the Equation of Motion of a Charged Particle in a Plane-Wave Electromagnetic Field with RR
Appendix B Quantum Electrodynamics
	B.1 Gamma Matrix Algebra
	Product of Gamma Matrices
	Trace of Gamma Matrices
	B.2 Useful Identities for QED
	Trace Identities
	Slashed Quantities Identities
	B.3 The Dirac Spinors
	B.4 The Dirac-Volkov States
	B.5 Nonlinear Compton Scattering
Appendix C Electron Distribution Function
	C.1 Derivation of the Master Equation (5.31摥映數爠eflinkeq:Master55.315)
	C.2 Exact and Approximate Expressions of an(χ) Functions
	C.3 Exact and Approximate Expressions of bn(χ) Functions