Copyright by Jeffrey Michael Heninger 2020 The Dissertation Committee for Jeffrey Michael Heninger certifies that this is the approved version of the following dissertation: Hamiltonian Magnetic Monopoles and An Integral Transform for Kinetic Plasmas Committee: Philip J. Morrison, Supervisor Boris Breizman Irene M. Gamba Anna Tenerani Hamiltonian Magnetic Monopoles and An Integral Transform for Kinetic Plasmas by Jeffrey Michael Heninger DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN August 2020 To our Heavenly Parents. Acknowledgments I have had the privilege of collaborating with many people during my research: Predrag Cvitanovi´cand Domenico Lippolis on noisy chaos; Harry Swinney, Michael Allshouse, Claudia Cenedese, and Jim McElwaine during my two stints with experiments; Behzat Ergun, Michael Waterbury, and Richard Szabo when I ventured into the unfamiliar territory of high energy physics; David Hatch during my vague gestures in the direction of computation; Robert Littlejohn at MSRI; and, of course, my advisor, Phil Morrison. I also want to acknowledge several communities that have supported me: my family, who encourage excellence, regardless of what you choose to pursue; the Church of Jesus Christ of Latter-day Saints, the basis of my re- lationship with God and Christ; the Society of Physics Students at Georgia Tech, who embrace science, regardless of whether it loves you back; my many excellent teachers, both in physics & math and in other fields; and all of my fel- low graduate students, without whom the past 6 years would have been much less interesting - particularly Jason Derr, the main person I bounce ideas off of, whether they are related to science or otherwise. I have been supported by Department of Energy grants to the Insti- tute for Fusion Studies and by teaching at the University of Texas at Austin. Teaching has encouraged me to understand and explain physics more clearly. v Hamiltonian Magnetic Monopoles and An Integral Transform for Kinetic Plasmas Publication No. Jeffrey Michael Heninger, Ph.D. The University of Texas at Austin, 2020 Supervisor: Philip J. Morrison Classical electromagnetism with magnetic monopoles is not a Hamilto- nian field theory because the Jacobi identity for the Poisson bracket fails. The Jacobi identity is recovered only if all of the species have the same ratio of electric to magnetic charge or if an electron and a monopole can never collide. Without the Jacobi identity, there are no local canonical coordinates or La- grangian action principle. To build a quantum theory of magnetic monopoles, we either must explain why the positions of electrons and monopoles can never coincide or we must resort to new quantization techniques. The linearized Vlasov-Poisson system can be exactly solved using the G-transform, an integral transform that removes the electric field term. We apply the G-transform to several applications. By investigating the interaction vi between the G-transform and the Fokker-Planck collision operator, we obtain an explicit solution that shows the interplay between collisions and Landau damping. Gyrokinetics become simpler after G-transforming. For partial dif- ferential equations solvable by the method of characteristics, we can determine the values of the field at many locations by measurements at a single location. The G-transform makes these characteristics much easier to find and use. vii Table of Contents Acknowledgmentsv Abstract vi List of Figures xiii Chapter 1. Preface xii Part I Hamiltonian Magnetic Monopoles1 Chapter 2. Summary for a General Audience2 2.1 Canonical Hamiltonian Systems.................2 2.2 Extending the Theory.......................5 2.3 Electromagnetism.........................8 2.4 Magnetic Monopoles....................... 11 Chapter 3. Introduction 15 3.1 A Brief History of Magnetic Monopoles............. 15 3.2 Hamiltonian Form of the Vlasov-Maxwell Equations...... 19 3.2.1 Equations of Motion.................... 20 3.2.2 Hamiltonian and Poisson Bracket............. 20 3.2.3 Deriving the Equations of Motion from the Hamiltonian and Poisson Bracket.................... 21 3.2.4 Divergence Maxwell Equations.............. 26 3.2.5 Jacobi Identity....................... 29 3.3 Canonical Hamiltonian Electromagnetism............ 30 viii Chapter 4. Vlasov-Maxwell with Magnetic Monopoles 34 4.1 Modifying the Poisson Bracket.................. 34 4.2 Equations of Motion....................... 35 4.3 Jacobi Identity........................... 36 4.3.1 Cyclically Permuting.................... 37 4.3.2 Bracket Theorem...................... 38 4.3.3 Functional Derivatives................... 39 4.3.4 Terms............................ 41 4.3.5 Boundary Term...................... 54 4.4 Conclusion............................. 56 Chapter 5. Consequences 59 5.1 One Electron and One Monopole................. 59 5.1.1 Restricting the Hamiltonian and Poisson Bracket.... 59 5.1.2 Equations of Motion.................... 63 5.1.3 Jacobi Identity....................... 65 5.1.4 No Touching........................ 65 5.2 Importance of the Jacobi Identity................ 68 5.2.1 Geometry of Hamiltonian Mechanics........... 69 5.2.2 Canonical Coordinates................... 70 5.2.3 Lagrangian Mechanics................... 72 Chapter 6. Quantizing Magnetic Monopoles 74 6.1 Nonrelativistic Quantum Theories................ 75 6.1.1 Dirac............................ 75 6.1.2 Wentzel........................... 76 6.1.3 Lipkin, Weisberger, and Peshkin............. 77 6.1.4 Bialynicki-Birula...................... 78 6.2 One Electron's and One Monopole's Wavefunction....... 79 6.3 Quantum Field Theories..................... 81 6.3.1 Cabibbo-Ferrari...................... 81 6.3.2 Schwinger 66........................ 82 6.3.3 Schwinger 68........................ 84 ix 6.4 `t Hooft-Polyakov Monopoles................... 86 6.5 New Quantization Techniques.................. 87 Chapter 7. Motion in a Sphere of Monopolium 90 7.1 An Unusual Dynamics Problem................. 90 7.2 Exact Solution........................... 92 7.2.1 Scalar Quantities...................... 92 7.2.2 Rotating Orthogonal Frame................ 93 7.2.3 Converting so(3) ! su(2)................. 96 7.2.4 Weber Differential Equation................ 97 7.2.5 Initial Conditions..................... 101 7.2.6 Solution........................... 103 7.3 Asymptotics............................ 104 7.3.1 Parabolic Cylinder Functions............... 105 7.3.2 Spinors........................... 106 7.3.3 Rotation Matrix...................... 109 7.3.4 Velocity........................... 110 7.3.5 Reversing Time....................... 112 7.4 Non-Hamiltonian Dynamics................... 113 Chapter 8. Conclusion 115 Part II An Integral Transform for Kinetic Plasmas 117 Chapter 9. Summary for a General Audience 118 9.1 What is a Plasma?........................ 118 9.2 Multiple Velocities......................... 123 9.3 Damping without Collisions................... 125 9.4 Unraveling Landau Damping................... 127 Chapter 10. Introduction 129 10.1 Overview.............................. 129 10.2 Properties of the Hilbert Transform............... 132 10.2.1 Plasma Z Function..................... 136 x Chapter 11. One-Dimensional Vlasov-Poisson 137 11.1 Equations of Motion....................... 137 11.2 G-Transform Definition and Inverse............... 139 11.3 Solution of the Linear Vlasov Equation using the G-Transform 140 11.4 Landau Damping......................... 144 11.5 Intuition.............................. 146 11.5.1 Symmetric Form...................... 146 11.5.2 Complex Functions..................... 146 11.5.3 Van Kampen Modes.................... 148 11.5.4 Hamiltonian........................ 149 Chapter 12. Collisions 150 12.1 Fokker-Planck Collision Operator................ 150 12.2 G-Transform and Collisions.................... 151 12.2.1 's for a Maxwellian Equilibrium............. 151 12.2.2 Properties of the Collision Operator........... 152 12.2.3 Commuting the G-Transform and Collision Operator.. 155 12.3 Dropping the Shielding Term................... 157 12.4 Analytically Solving Advection-Diffusion............ 160 12.4.1 Fourier Transform and Method of Characteristics.... 160 12.4.2 Checking the Solution................... 162 12.4.3 Analysis of the Solution.................. 164 12.4.4 Advection-Diffusion Crossover.............. 165 12.5 Comparing Collision Operators.................. 167 12.6 Other Initial Conditions..................... 169 12.6.1 Different Thermal Velocity................ 169 12.6.2 Polynomial Prefactor................... 170 12.6.3 Realistic Initial Conditions................ 171 12.7 Numerically Solving Advection-Diffusion............ 172 12.7.1 Numerical Solution..................... 172 12.7.2 Comparing Advection and Diffusion........... 175 12.8 Introducing the Shielding Term as a Perturbation....... 178 12.9 Other Collision Operators.................... 181 xi Chapter 13. Gyrokinetics and Drift-Kinetics 182 13.1 Equations of Motion....................... 183 13.2 G-Transforming Gyro-/Drift-Kinetics.............. 184 13.3 Hermite Polynomials....................... 187 13.3.1 Definition.......................... 187 13.3.2 Derivatives......................... 188 13.3.3 Recurrence Relation.................... 189 13.3.4 G−1-Transform....................... 189 13.3.5 Plasma Z Function..................... 189 13.3.6