Hamiltonian Magnetic Monopoles and an Integral Transform for Kinetic Plasmas

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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

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 relationship 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 differential 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 Explicit Result...... 191 13.4 Fluid Equations in u ...... 191 13.4.1 Relating u and v Moments...... 192 13.4.2 Deﬁning Moments...... 194 13.4.3 Zeroth Moment...... 194 13.4.4 First Moment...... 196 13.4.5 Higher Moments...... 198 13.4.6 Intuition?...... 199

Chapter 14. Nonlocal Information 200 14.1 General Strategy...... 200 14.2 A Simple Example...... 201 14.2.1 G-Transform...... 202 14.2.2 Boundary Conditions...... 205 14.2.3 Solving the Equations of Motion...... 206 14.3 Finite Time and Additional Terms...... 212

Chapter 15. Conclusion 214

Bibliography 218

xii List of Figures

10.1 Contour for the proof of Hilbert transform property2..... 133

11.1 Analytical solution without collisions...... 143

12.1 Comparing the analytical and numerical solutions with collisions 163 12.2 Ratio of collisions to advection...... 166 12.3 Numerical solution with realistic initial conditions...... 174 12.4 Time dependence of each term in G−1-transformed coordinates. 176 12.5 Time dependence of each term in original coordinates...... 177

14.1 Initial Conditions for Analytical Nonlocal Information..... 210 14.2 Analytical Distribution Function at Other Locations...... 211

xiii Chapter 1

Preface

All dissertations take years of research and condense it down into a single document. Some people spend the entire time working towards one goal, so the topic of their dissertation is obvious. A coherent story can be told. That was not the case for me. It is not obvious how to condense my work into a single coherent document.

I focus on the two projects that I spent most of my time on: looking at the theory of magnetic monopoles from a Hamiltonian perspective and simplifying the equations for kinetic plasmas using the G-transform. Many of these results have already been published [53, 54].

Other projects that I have worked on during graduate school are unfor- tunately neglected. I showed that if you take a chaotic map with weak noise, then add a perturbation to deterministic dynamics, you can predict the results using perturbation theory, even though perturbation theory fails for chaotic systems without noise [52]. I have worked on a more accurate matter model for the slurry of neutron chunks swirling around a merging pair of neutron stars than the magnetohydrodynamics currently used in simulations. I have noticed that there seems to be no widely accepted explanation of an obvious

xii feature of clouds, that there is a sharp gradient in opacity, and perhaps other quantities, at their edge. Despite being a theorist (as these pages will attest), I also spent two summers in different labs working on different projects related to the motion of stratified fluids.

There are a few connections between my two main topics: magnetic monopoles and G-transforms. Both approach the physics from a noncanonical Hamiltonian perspective. Both use the kinetic description of plasma given by the Vlasov equation.

But beyond that, the connection fails.

The G-transform is deﬁnitely the more practical of the two. It focuses on a phenomenon that is widely seen (Landau damping) and makes it simpler to calculate and understand. This is the most intuitive way for Landau damping to be taught. There are applications for this technique both in simulations of laboratory plasmas and in analyzing data from spacecraft.

The magnetic monopole part of my dissertation is much more philosophical. Magnetic monopoles have never been observed. Nevertheless, most theorists who work in high energy physics believe that at least a few of them exist. The argument is largely aesthetic. My dissertation argues against that beauty. Although Maxwell’s equations look more symmetric, adding magnetic monopoles makes them lose their Hamiltonian structure. All physical theories we have seen so far are Hamiltonian, so we should be more skeptical of believing in magnetic monopoles.

xiii I am a ﬁrm believer that science should be explainable to the public. Although Einstein’s quote, “If you can’t explain it simply, you don’t understand it well enough,” may not always be true, it is a good goal to work towards. To achieve this goal, I have included two unusual chapters. For each part of my dissertation, I have written a Summary for a General Audience. In them, I show a glimpse of the ideas considered and the arguments made that should be accessible to anyone – not just people with a technical background and certainly not only to plasma physicists.

The dissertation is organized as follows:

Chapter1 is this chapter. It contains an outline of the dissertation.

PartI is about Hamiltonian Magnetic Monopoles. PartII is about the G-transform, An Integral Transform for Kinetic Plasmas.

PartI contains seven chapters. Chapter2 summarizes partI for a general audience. Chapter3 is the introduction. It contains a brief history of magnetic monopoles and a description of the Hamiltonian nature of electromagnetism without magnetic monopoles. Chapter4 adds magnetic monopoles to electromagnetism. It contains the main calculation for this part: a proof that the Jacobi identity for the Poisson bracket is not satisﬁed. Chapter5 considers some consequences of this result. First, by reducing the theory to one electron and one monopole, we see that the problems occur when electrons and monopoles touch. Then, we describe how central the Jacobi identity is to many of the main results of classical mechanics. Chapter6 shows how diﬃculties

xiv that have been previously noticed for quantum theories of magnetic monopoles arise from the classical failure of the Jacobi identity. Quantum theories typically involve some implicit assumption that the locations of the electrons and monopoles never touch. Chapter7 exactly solves a relatively simple example: the motion of an electron in a sphere of monopolium. Chapter8 reiterates the main points and concludes partI.

PartII also contains seven chapters. Chapter9 summarizes partII for a general audience. Chapter 10 is the introduction. It contains a brief overview of the systems where the G-transform might be used and a summary of the properties of the Hilbert transform, an integral transform widely used in signal analysis that the G-transform is built from. Chapter 11 shows the original purpose of the G-transform: to provide an exact solution to the simplest kinetic equation for plasmas, the one-dimensional linearized Vlasov-Poisson system. Chapter 12 considers how the G-transform interacts with a collision operator and contains an analytic solution that shows the interaction between Landau damping and collisions. Chapter 13 considers possible applications of the G- transform to gyrokinetics. These results can be used in simulations or to write non-intuitive ﬂuid equations after G-transforming. Chapter 14 shows how the G-transform can be used to extract nonlocal information from a measurement at a single location. Chapter 15 reiterates the main points and concludes partII.

To make this dissertation numerologically perfect, I should have a third part with seven chapters. Alas, that is not the case.

xv Part I

Hamiltonian Magnetic Monopoles

1 Chapter 2

Summary for a General Audience

2.1 Canonical Hamiltonian Systems

The very ﬁrst problems solved by physics involve balls rolling around on ramps, objects connected by springs, and, of course, planets moving due to their mutual gravitational attraction. These problems, and the problems like them, are the canonical problems of physics.

There is a joke in physics that we only know how to solve two problems: the harmonic oscillator (springs) and the Kepler problem (gravity). Even many of the most impressive modern results in quantum ﬁeld theory and solid state physics come from abstract versions of these problems.

Newton solved almost all of the known problems in physics using his fa- mous three laws. After him, other physicists tried repeatedly to rewrite physics to make it easier to get the same results and to be able to solve new problems. The most successful reformulations of Newton’s laws are called Lagrangian mechanics1 and Hamiltonian mechanics.2 We will focus on Hamiltonian me-

1Joseph-Louis Lagrange (1736-1813) was a Franco-Italian mathematician. Lagrange developed variational calculus and rewrote Newtonian mechanics in this more powerful formalism. He also helped create the metric system in revolutionary France. 2William Rowan Hamilton (1805-1865) was an Irish mathematician. Hamilton developed a single formalism that works for both mechanics and optics. He also invented quaternions,

2 chanics.

The key object in Hamiltonian mechanics is the Hamiltonian. The Hamiltonian is a function of the positions and momenta of all the particles and is usually equal to the total energy.3 This is simpler than Newton’s laws because we work with the Hamiltonian, a scalar, instead of the force, a vector. Any other quantity that we could measure would also depend on the position and momentum.

We can predict how any quantity changes in time using only our knowledge of how it and the Hamiltonian depend on the position and momenta.4 In particular, the way the position changes in time equals how the Hamiltonian depends on momentum and the way the momentum changes in time equals how the Hamiltonian depends on position.5 We have given a name to these relationships: the Poisson bracket. We can also say: The way any quantity an extension of complex numbers and a forerunner of vector calculus. 3For this chapter, equations have been banished to the footnotes. Here’s the notation I will use for the Hamiltonian:

H = H(x, p) . (2.1)

We call the position x and the momentum p. 4Here’s the diﬀerential equation for any quantity A: dA ∂A ∂H ∂H ∂A = − . (2.2) dt ∂x ∂p ∂x ∂p

5These are called Hamilton’s equations of motion: dx ∂H dp ∂H = , = − . (2.3) dt ∂p dt ∂x

3 changes in time is determined by the Poisson bracket of that quantity with the Hamiltonian.6

Any problem that can be written in this way is a canonical Hamiltonian system.

For the simplest examples, the Hamiltonian is a sum of potential energy and kinetic energy. The kinetic energy depends on momentum, but not position. The potential energy depends on position, but not momentum. The way the position changes in time depends on the kinetic energy. The way the momentum changes in time depends on the potential energy, which determines the force.

One key feature of Hamiltonian mechanics is that it make conservation of energy obvious.7 When using Newton’s laws, you have to prove conservation of energy for particular forces. Here, it is built into the structure of the theory. Friction, or other forces that dissipate energy, must be added in post hoc.

6The canonical Poisson bracket is represented by square brackets: dA = [A, H] . (2.4) dt c

7Plug in H for A: dH ∂H ∂H ∂H ∂H = [H,H] = − = 0 . (2.5) dt c ∂x ∂p ∂x ∂p

4 2.2 Extending the Theory

It would be great if we could extend Hamiltonian mechanics beyond the canonical problems (balls, springs, gravity, etc.) considered by Newton. One of the most interesting problems to consider is ﬂuid mechanics.

Fluids are described using a field theory. A field is something that can be measured at many locations and is different at different locations. The spatial variability is key. Weather maps are familiar examples of fields. They might show what the temperature is at every location in the country. Fields can be either scalar (like temperature) or vector. A weather map of wind velocities is a vector field. At every location, there is an arrow that shows the wind speed and direction. This is the image conveyed by the word “field.” In a field of grass, there is a blade of grass at every location, with a particular length and pointing in a particular direction.

How could we think of ﬂuid mechanics in terms similar to the canonical problems in physics?

Think of a fluid as a collection of drops. Treat each drop as an object in Newtonian mechanics. There are an extremely large number of drops in most fluids. We will pretend that there are infinitely many, infinitely small drops. Each drop moves around in the fluid as a result of gravity, pressure, and perhaps some other forces. The gravitational force on the drops is obviously canonical. The pressure is an interaction between drops (which is always repelling). You can think of it as connecting your objects with springs. Fluid

5 mechanics can be formulated as a canonical Hamiltonian system of inﬁnitely many objects, or a Hamiltonian ﬁeld theory.

In order to use a Hamiltonian description of fluid mechanics, our observers have to follow the individual drops in the fluid.8 Your measuring equipment must move with the fluid: think of a boat driven by a storm, or a hot air balloon drifting with the wind, or Kong from Dr. Strangelove riding a ‘drop’ of nuclear ‘fluid’ to its destination.

Often, this is not the most convenient way to arrange our measuring device. Instead, we fix the location of the observer and allow her to watch the fluid as it passes by.9 Ground-based weather stations don’t move with the wind. Videos of whitewater are safer if they’re taken from the shore. An ice-fisherman makes one measurement (Is there a fish?) at one location for many hours in the cold. We want to predict how these observers will see the fluid.

The equations of motion these observers see are different and they’re not obviously Hamiltonian. It is still possible for these observers to measure the fluid’s total energy. We define the Hamiltonian function to be this energy.10 The biggest change will be in the Poisson bracket (what we use to

8This is called the Lagrangian description of the fluid mechanics. It should not be confused with Lagrangian mechanics. They are both named after Joseph-Louis Lagrange. 9This is called the Eulerian description of fluid mechanics. Leonhard Euler (1707-1783) was the greatest mathematician of all time. There is a joke that, in math, everything is named after the second person who discovered it, because otherwise, everything would be named after Euler. 10The Hamiltonian is actually a functional for a field theory, but we won’t worry about that here.

6 determine how things change in time), which can be more complicated than simply describing how two quantities depend on position and momentum. For the ﬁxed observer watching the moving ﬂuid, we can use the new Hamiltonian and the new Poisson bracket to determine the way any quantity changes in time.

This suggests that we can generalize Hamiltonian mechanics.

How can we check if a particular system is Hamiltonian or not? If something does not look like one of the canonical physics problems, it could just mean that you are not using the right observers to describe it.

The answer comes from theorems that connect mathematical properties of the Poisson bracket to properties of the physical system. Two of these properties are antisymmetry and the Jacobi identity. If the Poisson bracket is antisymmetric, then the Hamiltonian (energy) will be conserved.11 If the Poisson bracket also satisﬁes the Jacobi identity equation, then the theorem guarantees that it is possible to pick a group of observers for which the system

11The Hamiltonian also needs to not explicitly depend on time. Antisymmetry means that switching the order adds a negative sign. It can be written as:

[A, B] = −[B,A] . (2.6)

Notice that the subscript c is not there because this is no longer a canonical bracket.

7 looks like one of the canonical problems of physics.12

Hamiltonian mechanics includes any problem which can be written using a Poisson bracket that satisﬁes these properties.

2.3 Electromagnetism

The greatest triumph of nineteenth century physics is the theory of electromagnetism.

James Clerk Maxwell, a Scottish physicist (and poet), was the first to present a complete theory of electromagnetism [80]. He recognized that the key components of the theory are the electric and magnetic fields, each of which has a vector at every location in space. Maxwell claimed that all problems in electromagnetism could be solved using four statements about the geometry of the field lines. Three and a half of these statements were already known; Maxwell completed the set and raised them to prominence.

The statements are: Electric ﬁeld lines start at (diverge from) positive charges and end at (converge to) negative charges.13 Magnetic ﬁeld lines don’t

12This is the hardest property to check:

[[A, B],C] + [[B,C],A] + [[C,A],B] = 0 . (2.7)

The canonical coordinates (the group of observers) only exist locally. If you have some nontrivial topology, you might need to use diﬀerent collections of observers in diﬀerent locations. 13This is Gauss’s Law: div E = Q. (2.8) Since this chapter is for a general audience, we will not distinguish charge from charge density and we will not write all of the constants here.

8 start or end anywhere – they always form closed loops, extend to infinity, or form complicated tangles.14 Curling electric fields are created by changing magnetic fields.15 Curling magnetic fields are created by electric currents and by changing electric fields.16 Precise mathematical statements of these laws are written in terms of vector calculus.

Maxwell used these equations to show that electromagnetic waves should exist and calculated that they should move at the speed of light. He concluded that light is an electromagnetic wave [81] ! This unified the previously distinct fields of electricity, magnetism, and optics. His prediction was confirmed by Hertz’s experiments with radio waves [55].

Is electromagnetism Hamiltonian? There is nothing in the theory that looks like a Newtonian object with a position and momentum. It is obviously not a canonical system.

We should check if there is a Poisson bracket which gives the correct

14This one doesn’t have a widely accepted name:

div B = 0 . (2.9)

15This is Faraday’s Law: dB curl E = − . (2.10) dt

16This is the Amperé-Maxwell Law. Amperédiscovered that curling magnetic fields are created by current. Maxwell argued that they should also be created by changing electric fields: dE curl B = I + . (2.11) dt

9 equations of motion with the correct Hamiltonian. If this Poisson bracket satisﬁes all of the necessary properties, then the theory is Hamiltonian.

Such a Poisson bracket exists. We won’t write it down here because it’s a mess,17 especially once you realize that the charges and currents that create the fields also experience forces as a result of the fields. Direct calculations confirm all of the necessary properties [84].

Before, we claimed that if the Poisson bracket satisﬁes the Jacobi identity, then it is possible to pick a group of observers for which the system looks canonical. What do these observers measure?

The answer was already known before anyone had worked out the Hamiltonian structure for electromagnetism. Maxwell’s equations can be simplified by shifting from the fields (2 vectors) to the potentials (1 scalar, 1 vector).18 The potentials ensure that two of Maxwell’s equations, the ones that don’t include matter (charges and currents), are automatically satisfied. When the motion of a charged particle is written in terms of the potentials, its Hamiltonian is no longer equal to its energy. Its energy must be shifted by the scalar potential and its momentum must be shifted by the vector potential.

Electromagnetism can be written with the same structure as the canonical problems of physics, but it involves using the potentials instead of the ﬁelds.

17We will write it down in (3.4). 18Mathematically, the potentials are also ﬁelds.

10 2.4 Magnetic Monopoles

Physicists love symmetry more than nature.

Maxwell’s equations are incredibly symmetric. They remain unchanged if you shift, reﬂect, or rotate in space or time. They are even symmetric if you do a Lorentz transformation from special relativity, even though special relativity wouldn’t be developed for another 40 years.19

If there are no charges or currents, Maxwell’s equations have another symmetry called the duality transformation. The structure of Maxwell’s equations remains the same if you switch the electric and magnetic ﬁelds.20 The charge messes this up. Although electric charges are ubiquitous, no one has ever seen a magnetically charged object. Permanent magnets always have both a N and a S pole of the same strength and so have zero total magnetic charge. Their magnetic ﬁeld is created by the currents corresponding to electrons or- biting and spinning within each atom.

We could rectify this lack of symmetry by supposing that magnetic charges, or magnetic monopoles, exist and we just haven’t found them yet. A magnetic monopole is a magnet which has a N pole, but no S pole, or vice versa. Maxwell’s equations would then be: Electric ﬁeld lines diverge from electric charges. Magnetic ﬁeld lines diverge from magnetic charges. Curling

19This is why the cosmic speed limit is called the speed of light. Electromagnetism is the only relativistic theory we knew about before relativity. 20Really, it’s the rotation E → B, B → −E.

11 electric fields are created by magnetic currents or changing magnetic fields. Curling magnetic fields are created by electric currents or changing electric fields.21

Is this system of equations Hamiltonian?

Earlier, I said that electromagnetism (without magnetic monopoles) could be written using a Poisson bracket that satisfies the Jacobi identity. There is actually a subtlety here. The Poisson bracket satisfies the Jacobi identity only if magnetic field lines do not diverge from / converge to any location.

When we add magnetic monopoles into the theory, we have to redo a lot of our work. There are new kinds of matter and new kinds of interactions, so we have to extend the Hamiltonian and Poisson bracket accordingly. When we work it all out, we ﬁnd a problem: the Poisson bracket does not satisfy the Jacobi identity, so the system is not Hamiltonian.

We can recover the Jacobi identity only by imposing unphysical constraints. We could have no matter, but this obviously doesn’t match our world. We could insist that everything has the same ratio of electric to magnetic charge, but if our observers measured things diﬀerently, these monopoles

21Here are the new equations:

div E = QE , div B = QM , (2.12) dB dE curl E = −I − , curl B = I + . (2.13) M dt E dt The subscripts E and B refer to electric vs. magnetic charges and currents.

12 would vanish. We could prevent electrically and magnetically charged matter from ever touching. This might seem plausible if every piece of matter were inﬁnitely small, but not for anything with nonzero volume.

Electromagnetism with magnetic monopoles is not Hamiltonian

Why should we care?

Most models of physical systems are Hamiltonian. The ones that are not leave something out of the model. For example, friction is not Hamiltonian because it transfers energy from macroscopic motion to microscopic motion of atoms. The motion of the atoms is not part of the model, so it appears as though the energy disappears. It would be Hamiltonian if we kept track of all the energy.

All ‘fundamental’ physical systems, which don’t interact with motion not described by the model, are Hamiltonian. This is a philosophical statement, but it has survived for hundreds of years, even through the revolutions of relativity and quantum mechanics.

The other dominant modern description of physics is Lagrangian mechanics. Usually, you can switch back and forth between Lagrangian and Hamiltonian mechanics and use whichever is easier for your particular problem. If the Jacobi identity is not satisﬁed, the Lagrangian description of your theory cannot be written.

13 Since the Poisson bracket for magnetic monopoles does not satisfy the Jacobi identity, it is impossible to pick a group of observers for which the system looks like one of the canonical problems of physics. Regardless of how you measure the problem, it will never be like one of the problems we are familiar with solving.

These problems become especially severe if you try to quantize this system. We should be able to describe how a single monopole interacts (mi- croscopically) with a single atom, which requires us to use quantum mechanics. Any process we might use to create monopoles, or any other exotic particles, is quantum. The techniques that we use to transform classical theories into quantum theories all rely on the Hamiltonian structure. You cannot write a quantum system without the Jacobi identity.

No one has observed a magnetic monopole in the lab or in the wild. Nevertheless, many physicists believe that they exist. The justiﬁcation for this belief is mathematical and aesthetic. If magnetic monopoles exist, then Maxwell’s equations become symmetric and certain coincidences can be explained. Despite the symmetry, magnetic monopoles contradict the assumptions of Hamiltonian mechanics. This undermines the beauty of the theory of magnetic monopoles. Magnetic monopoles would have the only ‘fundamental’ theory that is not Hamiltonian. Although this is not, by itself, proof that magnetic monopoles can’t exist, it should make us more skeptical.

14 Chapter 3

Introduction

This part of the dissertation considers the classical gauge-free theory of electromagnetic fields interacting with electrically and magnetically charged matter as a Hamiltonian field theory. In this chapter, we give a brief history of magnetic monopoles and an overview of Hamiltonian structure of the Vlasov- Maxwell theory. In future chapters, we show that adding magnetic monopoles destroys the Hamiltonian structure, discuss some of the implications of this, and consider how this has influenced attempts to create quantum theories of magnetic monopoles.1

We approach the problem of magnetic monopoles from the perspective of plasma physics, although our conclusion is general and independent of the matter model used.

3.1 A Brief History of Magnetic Monopoles

The idea of separate charges for electricity and magnetism existed even before the idea of charge was established. Only later was magnetism under-

1Much of the material in Part I has been previously published in J. M. Heninger and P. J. Morrison. Hamiltonian nature of monopole dynamics. Physics Letters A, 384:126101, 2020.

15 stood as a consequence of electricity.

The earliest analyses of magnetism are from China. These include work on magnetic declination [65], instructions for how to make a compass [66], and the use of compasses in the army [43] and at sea [143]. More history of magnetism and other physics in medieval China can be found in Ref. [95].

In 1269, Petrus Peregrinus was the first person in Europe to perform an experimental study of magnets [31, 124]. He introduced the term ‘pole’ to describe the two ends of a lodestone sphere and showed that splitting a magnet results in two smaller magnets, not separate poles. Magnetism entered the Scientific Revolution when William Gilbert used compasses to argue that the Earth has a global field, almost a century before Newton proposed Earth’s gravitational field [40].

During the 1700s, electricity was considered to be a fluid. Aepinus proposed a similar fluid to describe magnetism [2]. Magnetic poles were places where there was an excess or deficiency of the magnetic fluid. The inverse square law for the force from one end of a long, thin magnet was established by Michell in 1750 [82].

Coulomb revolutionized the theories of electricity and magnetism in the 1780s with his Seven Memoirs on Electricity and Magnetism [29]. He confirmed the inverse square laws for both electric charges and magnetic poles and replaced the one-fluid theories of electricity and magnetism with two-fluid theories. Unlike the electrical fluids, the magnetic fluids were trapped within

16 molecules, so the ﬂuids at most separate to opposite sides of each molecule.

The connection between electrical currents and magnetism was ﬁrst shown by Oersted in 1820 [97]. Amp`ereimmediately used this result to argue that all magnetic phenomena can be described using microscopic electric currents [5]. This view has been dominant ever since.

Nevertheless, some scientists continued to consider magnetic monopoles as hypothetical particles. When Heaviside wrote Maxwell’s equations in their modern (vector calculus) form, he included magnetic charges to emphasize the equations’ symmetry [51]. Curie argued that we should be open to the possibility of magnetic monopoles [26], Poincaréconsidered the motion of an electron in the field of a magnetic monopole [107], and Thomson showed that the fields of a static electrified point and magnetic pole contain angular momentum [133]. More information on the history of magnetism and monopoles before 1930 can be found in Refs. [42, 139].

The quantum theory of magnetic monopoles was developed by Dirac [33, 34]. He showed that, if a single magnetic monopole exists, quantization of angular momentum implies quantization of charge. This remarkable result revived interest in magnetic monopoles. Quantum ﬁeld theories for magnetic monopoles and for dyons (particles with both electric and magnetic charge) were developed by Cabibbo & Ferrari [21] and Schwinger [115, 116, 117].

We will discuss many of the quantum theories that have been developed for magnetic monopoles in more detail in chapter6.

17 Grand unified theories (GUTs) describe the strong, weak, and electromagnetic forces as a single theory whose symmetry is spontaneously broken at lower energies. If the symmetry is not broken in the same direction everywhere, then the fields will be zero at some locations. Around these locations, the fields resemble the fields of a magnetic monopole [130, 106]. Magnetic monopoles are a generic feature of GUTs, including string theories [104]. The existence of magnetic monopoles is often considered to be “one of the safest bets that one can make about physics not yet seen” [105].

If we assume that a GUT exists, then, shortly after the Big Bang, the expanding universe cooled through the critical temperature at which the symmetry is broken. There is no reason to assume that the symmetry would be broken in the same direction at causally disconnected locations. The boundaries between regions with diﬀerently broken symmetry would produce magnetic monopoles and strings [60]. Initial estimates of the number of monopoles produced this way were much too high [108], but the estimates are dramati- cally reduced by inﬂation [46].

The existence of astronomical magnetic fields produces a bound on the number of monopoles. If there were too many monopoles, they could move and screen out large magnetic fields, much like electrically charged matter screens out large electric fields in our universe [99].

Direct observations of magnetic monopoles remain inconclusive. Two early experiments detected candidate events [109, 22], but one was immediately refuted [4] and the other has never been replicated. Extensive searches for

18 monopoles have been done in matter, in cosmic rays, via catalyzing nucleon decays, and at colliders, all with negative results [100].

More information about magnetic monopoles can be found in one of the many relevant review articles [83, 110] or textbooks [9, 118].

3.2 Hamiltonian Form of the Vlasov-Maxwell Equations

In plasmas, the most important dynamics are the collective motions of the particles in collectively generated electromagnetic ﬁelds. These dynamics are best described by the Vlasov-Maxwell model. Many reductions have been developed to convert these equations to more manageable forms like gyrokinetics and magnetohydrodynamics. Since the Maxwell-Vlasov equations are more general than ﬂuid equations, our results are generic for matter models without dissipation.

The relevant dynamical variables are not the individual particles’ degrees of freedom, but rather the distribution function fs(x, v, t) for each species s, which describes the probability density of the particles in phase space, of finding a particle of species s with position between x and x + dx and velocity between v and v+dv. The electric and magnetic fields E(x, t) and B(x, t) are also key components of the theory. The charge and mass of each species are es and ms. If we integrate fs over all positions and velocities, we get the total number of particles of that species, Ns. The temperatures of plasmas are high enough and the densities are low enough that quantum effects are negligible.

19 3.2.1 Equations of Motion

The dynamics of the distribution function are governed by a mean- ﬁeld transport equation. The phase space density is constant along particle trajectories:

df ∂f ∂f e v ∂f s = s + v · s + s E + × B · s = 0 . (3.1) dt ∂t ∂x ms c ∂v

This is combined with Maxwell’s equations for the electric and magnetic ﬁelds, with sources determined by the moments of the distribution function,

Z Z X 3 X 3 ρ = es fs d v , j = es fs v d v . (3.2) s s The resulting Maxwell-Vlasov equations are a closed system of nonlinear partial integro-diﬀerential equations for fs, E, and B.

Two particle interactions can be introduced by adding a collision operator, which scatters particles in velocity, to the right hand side of (3.1). Alternatively, the exact dynamics of a ﬁnite number of interacting particles can be recovered by writing the distribution function as a sum of Dirac delta functions in position and velocity.

3.2.2 Hamiltonian and Poisson Bracket

The Vlasov description of matter coupled with Maxwell’s equations can be written as a Hamiltonian theory using the physical ﬁelds if we allow for a noncanonical Poisson bracket (see Refs. [84, 85, 79, 11, 87] for review). The

20 Hamiltonian functional and noncanonical Poisson bracket for this system are

Z Z X ms 1 H = |v|2f d3x d3v + (|E|2 + |B|2) d3x , (3.3) 2 s 8π s

X Z 1 {F,G} = f (∇F · ∂ G − ∇G · ∂ F ) m s fs v fs fs v fs s s es + 2 fs B · (∂vFfs × ∂vGfs ) msc 4πes 3 3 + fs (GE · ∂vFfs − FE · ∂vGfs ) d x d v ms Z 3 + 4πc (FE · ∇ × GB − GE · ∇ × FB) d x , (3.4)

where subscripts such as Ffs refer to the functional derivative of F with respect to fs.

The Hamiltonian, but not the Poisson bracket, needs to be modiﬁed to make this theory relativistic. Replace the |v|2 in the kinetic energy term with γ|v|2 = c2p1 + |v|2/c2 (e.g. [11]).

3.2.3 Deriving the Equations of Motion from the Hamiltonian and Poisson Bracket

It is straightforward to derive the Vlasov equation (3.1) and the dynamical Maxwell equations by setting

∂f ∂E ∂B s = {f , H} , = {E, H} , = {B, H} . (3.5) ∂t s ∂t ∂t

Going through this calculation also makes it clearer how the functional analysis here works.

21 Before beginning the calculation, determine all of the relevant functional derivatives:

δH 1 δH 1 δH ms 2 HE = = E(x) , HB = = B(x) , Hfs = = |v| , δE(x) 4π δB(x) 4π δfs(x, v) 2 δf δf δE δB δE δB s = s = = = = = 0 , δE δB δfs δfs δB δE 0 0 0 0 δfs0 (x , v ) 0 0 δE(x ) δB(x ) 0 = δss0 δ(x − x ) δ(v − v ) , = = δ(x − x ) I . (3.6) δfs(x, v) δE(x) δB(x) The I at the end of the last expression is a Kronecker delta on the components of the vectors.

When doing the calculation, we will split the bracket into four parts, corresponding to the lines in (3.4). The ﬁrst line is easily written using the single particle canonical Poisson bracket (2.2), so we call it ‘c’:

X 1 Z {F,G} = f [F ,G ] d3x d3v (3.7) c m s fs fs c s s The second line involves a triple product, so we call it ‘t.’ The third line will give the current density in the Amper´e-Maxwell Law, so we call it ‘j.’ The fourth line is for vacuum electric and magnetic ﬁelds, so we call it ‘v.’

Start with the distribution function:

∂fs0 = {f 0 , H} = {f 0 , H} + {f 0 , H} + {f 0 , H} + {f 0 , H} . ∂t s s c s t s j s v

22 Z X 1 δfs0 δH δH δfs0 3 3 {f 0 , H} = f ∇ · ∂ − ∇ · ∂ d x d v s c m s δf v δf δf v δf s s s s s s m The second term is zero because ∇ s |v|2 = 0 . 2 Z X 1 0 0 3 3 = δ 0 δ(v − v ) f ∇δ(x − x ) · m v d x d v m ss s s s s The v integral and sum are trivial. Integrate x by parts. Z 0 3 = −v · δ(x − x ) ∇fs0 d x = −v · ∇fs0 . (3.8)

Z X es δfs0 δH 3 3 {f 0 , H} = f B · ∂ × ∂ d x d v s t m2c s v δf v δf s s s s Rewrite the cross product using index notation. Z X es 0 0 3 3 = δ 0 δ(x − x ) f B ∂ δ(v − v ) (m v ) d x d v ss m2c ijk s i vj s k s s The x integral and sum are trivial. Integrate v by parts. Z es0 0 3 0 = − ijk Bi δ(v − v ) ∂vj (fs vk) d v ms0 c es0 0 0 = − ijk Bi (vk ∂vj fs + fs ∂vj vk) ms0 c es0 0 0 = − kij vk Bi ∂vj fs + fs Bi ijkδjk ms0 c es0 = − (v × B) · ∂vfs0 . (3.9) ms0 c

Z X 4πes δH δfs0 δfs0 δH 3 3 {f 0 , H} = f · ∂ − · ∂ d x d v s j m s δE v δf δE v δf s s s s The second term is immediately zero. Z X 4πes 0 1 0 3 3 = δ 0 δ(x − x ) f E · ∂ δ(v − v ) d x d v ss m s 4π v s s The x integral and sum are trivial. Integrate v by parts. Z es0 0 3 es0 = − E · δ(v − v ) ∂vfs0 d v = − E · ∂vfs0 . (3.10) ms0 ms0

23 The fourth term is automatically zero because all of the functional derivatives are with respect to the electric or magnetic ﬁeld. Put these all back together to get (3.1):

∂fs0 es0 es0 = −v · ∇fs0 − (v × B) · ∂vfs0 − E · ∂vfs0 . (3.11) ∂t ms0 c ms0

Next, look at the electric field: ∂E = {E, H} = {E, H} + {E, H} + {E, H} + {E, H} . ∂t c t j v The first and second parts are both automatically zero because all of the functional derivatives are with respect to the distribution function. Z X 4πes δH δE δE δH {E, H} = f · ∂ − · ∂ d3x d3v j m s δE v δf δE v δf s s s s The first term is immediately zero. Z X 4πes = − f (δ(x − x0) ) · (m v) d3x d3v m s I s s s The x integral is trivial. The sum and v integral remain. Z X 3 = −4π es v fs d v . (3.12) s

Z δE δH δH δE {E, H} = 4πc · ∇ × − · ∇ × d3x v δE δB δE δB The second term is immediately zero. Z 1 = 4πc (δ(x − x0) ) · ∇ × B d3x I 4π = c ∇ × B . (3.13)

When we put these together, we get the Amper´e-Maxwell Law: ∂E = −4πj + c ∇ × B . (3.14) ∂t

24 The current density j is given by sum of the ﬁrst velocity moments of the distribution functions for all of the species of particles (3.2).

Finally, solve for the dynamics of the magnetic ﬁeld:

∂B = {B, H} = {B, H} + {B, H} + {B, H} + {B, H} . ∂t c t j v

The ﬁrst, second, and third parts are all automatically zero because all of the functional derivatives are with respect to either the distribution function or the electric ﬁeld.

Z δB δH δH δB {B, H} = 4πc · ∇ × − · ∇ × d3x v δE δB δE δB The ﬁrst term is immediately zero. Z 1 = −4πc E · ∇ × δ(x − x0) d3x 4π I Rewrite the cross product using index notation. Z 0 3 {B`, H}v = −c ijk Ei ∇jδ(x − x ) δk` d x

Integrate x by parts. Z 0 3 = c ij` δ(x − x )∇jEi d x = −c ji`∇jEi

{B, H}v = −c ∇ × E . (3.15)

When we put this back together, we get Faraday’s Law:

∂B = −c ∇ × E . (3.16) ∂t

This Hamiltonian and Poisson bracket give the Vlasov equation, the Amper´e-Maxwell Law, and Faraday’s Law.

25 3.2.4 Divergence Maxwell Equations

We are still missing two of Maxwell’s equations. These two equations don’t have any time dependence, so they can’t come from the Poisson bracket of something with the Hamiltonian. Since they don’t have time, they are more like constraints than equations of motion.

In the Hamiltonian framework, the remaining two of Maxwell’s equations appear as Casimirs of the bracket (3.4). A Casimir is a function in the null space of the Poisson bracket, i.e. the bracket of any function with the Casimir is zero. Since the time dependence of any quantity is the bracket of that quantity with the Hamiltonian, Casimirs are conserved for any Hamilto- nian.

The Vlasov-Maxwell Poisson bracket has the following two inﬁnite fam- ilies of Casimirs:

Z Z ! X 3 3 CE = hE(x) ∇ · E − 4π es fs d v d x , (3.17) s Z 3 CB = hB(x) ∇ · B d x , (3.18)

where hE(x) and hB(x) are arbitrary functions.

It is simpler to show that CB is a Casimir. Its only functional dependence is on the magnetic ﬁeld:

δC δC B = 0 , B = 0 . (3.19) δfs δE

To determine the last functional derivative, vary the functional, then integrate

26 by parts:

Z Z δC δC = h ∇ · δB d3x = − δB · ∇h d3x ⇒ B = −∇h . (3.20) B B B δB B

Only the very last term in the Poisson bracket contains a functional derivative of the magnetic ﬁeld:

Z δG δC {C ,G} = −4πc · ∇ × B d3x B δE δB Z δG 3 = −4πc ijk ∇j (−∇khB) d x δEi Z δG 3 = 4πc ijk∇j∇khB d x = 0 ∀ G. (3.21) δEi

The electric Casimir is significantly more difficult to work with because it depends on both the electric field and the particle distribution functions. To determine its functional derivatives, vary the Casimir and integrate by parts.

Z Z ! X 3 3 δCE = hE ∇ · δE − 4π es δfs d v d x s Z Z 3 X 3 3 = − δE · ∇hE d x − 4πes hE δfs d x d v , s

δCE δCE δCE = −4πes hE , = −∇hE , = 0 . (3.22) δfs δE δB Go through the four terms of the Poisson bracket to see that this is a Casimir.

Notice that, because hE is independent of velocity, ∂v δCE/δfs = 0. Many of

27 the same steps we used to derive the equations of motion are used again here. Z X 1 δCE δG δG δCE {C ,G} = f ∇ · ∂ − ∇ · ∂ d3x d3v E c m s δf v δf δf v δf s s s s s s X 1 Z δG = f ∇(−4πe h ) · ∂ d3x d3v m s s E v δf s s s Z X 4πes δG = ∂ f · ∇h d3x d3v . (3.23) m δf v s E s s s Z X es δCE δG {C ,G} = f B · ∂ × ∂ d3x d3v = 0 . (3.24) E t m2c s v δf v δf s s s s

Z X 4πes δG δCE δCE δG {C ,G} = f · ∂ − · ∂ d3x d3v E j m s δE v δf δE v δf s s s s Z X 4πes δG = − f (−∇h ) · ∂ d3x d3v m s E v δf s s s Z X 4πes δG = − ∂ f · ∇h d3x d3v . (3.25) m δf v s E s s s

Z δC δG δG δC {C ,G} = 4πc E · ∇ × − · ∇ × E d3x E v δE δB δE δB Z δG 3 = 4πc ijk (−∇ihE)(∇j ) d x δBk Z δG 3 = 4πc ijk∇i∇jhE d x = 0 . (3.26) δBk

The second and fourth terms are both zero. The ﬁrst and third terms cancel. So this is a Casimir:

{CE,G} = 0 ∀ G. (3.27)

These Casimirs are conserved for any Hamiltonian. If they are zero initially, they will remain zero for all time.

28 Since hE and hB are arbitrary functions of x, if the integrals are zero, then the integrands must be zero as well. This is equivalent to saying that the divergence Maxwell equations are satisﬁed.

The Casimirs CE and CB correspond to the remaining two Maxwell’s equations we were unable to ﬁnd by taking the Poisson bracket of something with the Hamiltonian. The divergence equations are initial conditions of the universe, which are subsequently preserved by the dynamical equations.

3.2.5 Jacobi Identity

There is an important subtlety to this formulation of electromagnetism. In order for the theory to be Hamiltonian, the bracket must be a Poisson bracket. Poisson brackets have certain properties: bilinearity, anticommutativity, and the Jacobi identity for any functionals F , G, and H. If these three properties are satisﬁed, then the Poisson bracket deﬁnes a Lie algebra [127].

Bilinearity and anticommutativity are obvious. Each term of (3.4) is linear in F and G, so it satisﬁes:

{aF + bG, H} = a{F,H} + b{G, H} . (3.28)

Each line contains a pair of terms with opposite signs and F and G switched, except for the second line which anticommutes because of the cross product:

{F,G} = −{G, F } . (3.29)

The Jacobi identity is the most diﬃcult to check:

{{F,G},H} + {{G, H},F } + {{H,F },G} = 0 . (3.30)

29 For the Vlasov-Maxwell system, it was shown in [85, 91] that Z X es {F, {G, H}} + cyc = f ∇ · B (∂ F × ∂ G ) · ∂ H d3x d3v . m3c s v fs v fs v fs s s (3.31) We will not show the proof. It follows from the calculation of the Jacobi identity for Vlasov-Maxwell with monopoles shown in section 4.3 by setting all of the magnetic charges to zero. Jacobi identity calculations are long enough that we will only include one in this dissertation.

In order for the Jacobi identity to be satisfied, the domain of functionals must be restricted to solenoidal vector fields, ∇·B = 0, or equivalently defined on closed but not necessarily exact two-forms. Such a set of functionals is closed with respect to the bracket.

The two Casimirs mentioned above are not symmetric. The value of

CE could, in principle, initially be chosen to be anything. If CB 6= 0, i.e. if ∇ · B 6= 0, then the Vlasov-Maxwell system would cease to be a Hamiltonian ﬁeld theory.

This is our ﬁrst indication that the existence of magnetic monopoles is connected to the Hamiltonian nature of classical electromagnetism.

3.3 Canonical Hamiltonian Electromagnetism

Hamiltonian electromagnetism is typically written in terms of the potentials, not the ﬁelds.

The Hamiltonian for a single particle with charge e and mass m moving

30 nonrelativistically in the ﬁelds created by the potentials Φ(x) and A(x) is 1 H = |p − eA|2 + eΦ . (3.32) 2m The motion of the particle is determined using the canonical Poisson bracket. This is easier to do using index notation:

dxi ∂H 1 = [xi,H]c = = (pi − eAi) , (3.33) dt ∂pi m dpi ∂H e = [pi,H]c = − = (pj − eAj) ∇iAj − e∇iΦ . (3.34) dt ∂xi m To see the Lorentz force law, look at the second time derivative of the position: d2x dp dA m i = i − e i dt2 dt dt e ∂A dx = (p − eA ) ∇ A − e∇ Φ − e i + j ∇ A m j j i j i ∂t dt j i dx ∂A = e j (∇ A − ∇ A ) − e ∇ Φ + i . (3.35) dt i j j i i ∂t A vector identity is useful:

vj(∇iAj−∇jAi) = vj∇Àm(δìδmj−δ`jδmi) = vj∇Àm`mnijn = ijnvj(n`m∇Àm) .

Recognize the deﬁnition of the electric and magnetic ﬁelds in terms of the potentials: d2x dx ∂A m = e × (∇ × A) + e (−∇Φ − ) dt2 dt ∂t = e v × B + e E . (3.36)

This formulation of Hamiltonian electromagnetism gives the correct equations of motion for a single particle. More work would have to be done to turn this into a ﬁeld theory like Vlasov-Maxwell.

31 For decades, this was the only way we knew how to write electromagnetism as a Hamiltonian theory. In 1931, Dirac wrote that “if we wish to put the equations of motion [of electromagnetism] in the Hamiltonian form, however, we have to introduce the electromagnetic potentials” [33]. Born and Infeld were the first to show that a formulation in terms of the fields was possible [16, 17], but their theory only worked for fields in vacuum. It used a noncanonical Poisson bracket corresponding to the last line of (3.4). The Poisson bracket we use here was developed in 1980 [84].

We gain additional perspective on electromagnetism by knowing both Hamiltonian formulations. We now understand that, when we switch from the ﬁelds to the potentials, we switch from noncanonical to canonical coordinates. The same sort of transformation occurs in many areas of physics. In ﬂuid mechanics, Eulerian variables are noncanonical and Lagrangian variables are canonical. In rigid body motion, both Euler angles and yaw, pitch, & roll are

2 noncanonical coordinates systems. L , Lz,& ϕ are the canonical coordinates.

In all of these cases, the original variables are more physical or intuitive. The original Hamiltonian is simple. The complicated physics can be seen in the Poisson bracket. After the transformation, the Hamiltonian becomes more complicated. The Poisson bracket becomes the simple, canonical bracket. The conservation of Casimirs becomes a trivial consequence of the variables we use.

We will not use the potentials in this dissertation. Although canonical variables are often easier to work with, all of the relevant calculations can be done using noncanonical variables. Introducing the potentials is intrinsically

32 tied to magnetic monopoles. If the magnetic ﬁeld is not divergenceless, we cannot write B = ∇ × A. People have tried various ways to work around this, but they all involve some subtlety. It is easy to smuggle in some non- physical assumptions when trying to use the vector potential around magnetic monopoles. We can avoid all of these complications by working directly with the ﬁelds.

33 Chapter 4

Vlasov-Maxwell with Magnetic Monopoles

4.1 Modifying the Poisson Bracket

What happens when we add magnetic monopoles?

We must change the Hamiltonian and/or the Poisson bracket so they produce the new equations of motion.

The duality transformation is the simplest way to see what the appropriate form of the new equations is. Maxwell’s equations are symmetric if you rotate the electric and magnetic ﬁelds and charges [59]:

E0 = E cos ξ + B sin ξ , B0 = −E sin ξ + B cos ξ ,

0 0 es = es cos ξ + gs sin ξ , gs = −es sin ξ + gs cos ξ . (4.1)

0 0 0 In particular, if we set ξ = π/2, we get E = B, B = −E, es = gs, and

0 gs = −es. Notice that the charges of all of the species rotate together under the duality transformation.

The Hamiltonian (3.3) remains unchanged by the duality transformation. Add additional terms into the Poisson bracket (3.4) to make it symmetric

34 under the duality transformation:

X Z 1 {F,G} = f (∇F · ∂ G − ∇G · ∂ F ) m s fs v fs fs v fs s s es + 2 fs B · (∂vFfs × ∂vGfs ) msc gs − 2 fs E · (∂vFfs × ∂vGfs ) msc 4πes + fs (GE · ∂vFfs − FE · ∂vGfs ) ms 4πgs 3 3 + fs (GB · ∂vFfs − FB · ∂vGfs ) d x d v ms Z 3 + 4πc (FE · ∇ × GB − GE · ∇ × FB) d x . (4.2)

All of the new physics for magnetic monopoles lies in the Poisson bracket. The Hamiltonian remains the sum of the kinetic energy of the particles and the energy in the ﬁelds.

This bracket, with only one species of particle, was discussed by Mor- rison in 2013 [91]. In this case, Jacobi identity can be satisfied if the duality transformed magnetic field has zero divergence. Morrison did not recognize that having multiple species of particles with different ratios of magnetic to electric charge prevents a single duality transformation from removing the magnetic charge.

4.2 Equations of Motion

We can now use this Poisson bracket to determine the correct equations of motion for the Vlasov-Maxwell system with magnetic monopoles. These

35 calculations are extremely similar to the ones shown in section 3.2, so we will not show them in detail:

∂E X Z = −4π e v f d3v + c ∇ × B , (4.3) ∂t s s s

∂B X Z = −4π g v f d3v − c ∇ × E , (4.4) ∂t s s s

∂fs0 es0 v gs0 v = −v · ∇fs0 − (E + × B) · ∂vfs0 − (B − × E) · ∂vfs0 . (4.5) ∂t ms0 c ms0 c There are once again two Casimirs, corresponding to the two divergence Maxwell’s equations.

Z Z ! X 3 3 CE = hE(x) ∇ · E − 4π es fs d v d x , (4.6) s Z Z ! X 3 3 CB = hB(x) ∇ · B − 4π gs fs d v d x . (4.7) s

4.3 Jacobi Identity

The ﬁrst two properties of the Poisson bracket, bilinearity and antisymmetry, are immediately obvious. Only one calculation requires eﬀort: the Jacobi identity: {{F,G},H} + cyc = 0 . (4.8)

Throughout this dissertation, cyc refers to cyclically permuting F , G, and H.

This is a rather lengthy calculation. If you just want to see the conclusion, feel free to jump to section 4.4.

36 4.3.1 Cyclically Permuting

Proving the Jacobi identity requires a long and tedious calculation. We look for any tricks to speed up the calculation without causing errors.

One of the simplest tricks involves the cyclic permutation. Instead of writing down all of the terms, we do not write down any term that can be obtained by cyclically permuting F , G,& H. If one such term is zero, the others must be zero as well.

Often, we need the other terms in the cyclic permutation for everything to cancel. It would be nice to have a way to see what would cancel with the permutation.

Suppose you have two terms A and B which depend on F , G,& H in similar ways. You want to see whether they will cancel once you add in all of the rest of the terms. You can cyclically permute F → G, G → H, & H → F in one of the terms without changing the Jacobi identity:

A(F, G, H) + B(F, G, H) + cyc = A(F, G, H) + B(F, G, H)

+A(G, H, F ) + B(G, H, F )

+A(H,F,G) + B(H,F,G) (4.9)

= A(F, G, H) + B(G, H, F ) + cyc .

We will use :=: to shown when expressions are equal up to cyclic permutation:

A(F, G, H) + B(F, G, H) :=: A(F, G, H) + B(G, H, F ) . (4.10)

Cyclic permutations will simplify our calculation on multiple occasions.

37 4.3.2 Bracket Theorem

Write a generic bracket using inner product notation:

{F,G} = hFψ| J Gψi . (4.11)

We could either have ψ be a set of canonical coordinates and F and G be functions on phase space, for a finite degree of freedom Hamiltonian system, or we could have ψ be a function and F and G be functionals, for a Hamiltonian field theory. J could be any anti-self-adjoint operator for which the bracket satisfies the Jacobi identity.

The Bracket Theorem [85] greatly reduces the number of terms we need to consider when calculating the Jacobi identity:

Bracket Theorem: To prove the Jacobi identity, one need only consider the explicit dependence of J on ψ when taking the functional derivative δ{F,G}/δψ.

To prove this theorem, we only need the anti-self-adjointness of J , which is equivalent to the antisymmetry of the Poisson bracket,

hFψ| J Gψi = {F,G} = −{G, F } = −hGψ| J Fψi = −h J Fψ| Gψi , (4.12) and the self-adjointness of second functional derivative.

Start calculating the Jacobi identity:

{{F,G},H} = h{F,G}ψ| J Hψi = hFψ| J Gψiψ J Hψ (4.13)

38 In the second term, move the left J to Fψ using anti-self-adjointness. Then rename G → F,H → G, & F → H. Finally, use the self-adjointness of the second derivative to cancel the second term with the ﬁrst:

hFψ| J Gψψ J Hψi = −h J Fψ|Gψψ J Hψi

:=: −h J Hψ|Fψψ J Gψi

= −h J Gψ|Fψψ J Hψi . (4.14)

Only the part of the functional derivative of the Poisson bracket where ψ directly diﬀerentiates J contributes to the Jacobi identity.

4.3.3 Functional Derivatives

It is convenient to split the bracket into the same four parts we used in section 3.2.3. These correspond to the ﬁrst line, the second & third lines, the fourth & ﬁfth lines, and the sixth line of (4.2), respectively.

We will need to know the functional derivatives of each of these terms. Because of the Bracket Theorem, we can drop anything that has a second order functional derivative. These will all automatically cancel. Use ˙= to

39 designate equality modulo second order functional derivatives:

δ 1 1 {F,G}c ˙= (∇Ffs · ∂vGfs − ∇Gfs · ∂vFfs ) = [Ffs ,Gfs ]c , δfs ms ms δ es gs {F,G}t ˙= 2 B · (∂vFfs × ∂vGfs ) − 2 E · (∂vFfs × ∂vGfs ) , δfs msc msc δ 4πes {F,G}j ˙= (GE · ∂vFfs − FE · ∂vGfs ) δfs ms 4πgs + (GB · ∂vFfs − FB · ∂vGfs ) , ms Z δ X gs {F,G} ˙= − f (∂ F × ∂ G ) d3v , δE t m2c s v fs v fs s s Z δ X es {F,G} ˙= f (∂ F × ∂ G ) d3v . (4.15) δB t m2c s v fs v fs s s The rest of the functional derivatives contain exclusively second order functional derivatives, so the Bracket Theorem guarantees that they will cancel.

When you take a functional derivative with respect to a function of x alone, the integral over velocity remains. The resulting quantity is a function of x alone, even if the functional originally depends on both x and v.

40 4.3.4 Terms

When we plug the four terms of the bracket into the Jacobi identity, we get sixteen things to calculate:

{{F,G},H} = {{F,G}c,H}c + {{F,G}c,H}t + {{F,G}c,H}j + {{F,G}c,H}v + ... | {z } | {z } | {z } | {z } 1 2 3 4 {{F,G}t,H}c + {{F,G}t,H}t + {{F,G}t,H}j + {{F,G}t,H}v + ... | {z } | {z } | {z } | {z } 5 6 7 8 {{F,G}j,H}c + {{F,G}j,H}t + {{F,G}j,H}j + {{F,G}j,H}v + ... | {z } | {z } | {z } | {z } 9 10 11 12

{{F,G}v,H}c + {{F,G}v,H}t + {{F,G}v,H}j + {{F,G}v,H}v . | {z } | {z } | {z } | {z } 13 14 15 16 (4.16)

Note that, for the Jacobi identity, we also have to cyclically permute F , G, and H, so (4.16) would be three times as long. Luckily, these do not have to be calculated separately.

The Bracket Theorem alone ensures that 4 and 12-16 are all zero.

The rest of the terms are not as easy. Some terms (1, 6, 11) are zero by themself. Some will cancel another term (7 with 10, 8 with 3 & 9).

In order for terms to cancel, they must have the same combination of functional derivatives. This helps us group terms to make the calculation more straightforward. For example, (Eff) refers to any terms that have one functional derivative with respect to the electric or magnetic ﬁeld and two functional derivatives with respect to the distribution function.

41 Term 1: (fff)

Z X 1 δ{F,G}c δ{F,G}c {{F,G} ,H} = f ∇ · ∂ H − ∇H · ∂ d3x d3v . c c m s δf v fs fs v δf s s s s (4.17)

2 Focus on the integrand I1. A factor of fs/ms will multiply all of the terms, which we will drop. Use index notation:

X 1 Z {{F,G} ,H} ˙= f I d3x d3v , (4.18) c c m2 s 1 s s

I1 = ∇i(∇jFfs ∂vj Gfs − ∇jGfs ∂vj Gfs ) ∂vi Hfs

+∇iHfs ∂vi (∇jFfs ∂vj Gfs − ∇jGfs ∂vj Gfs ) (4.19)

= (∇i∇jFfs )(∂vj Gfs )(∂vi Hfs ) − (∇i∇jGfs )(∂vj Ffs )(∂vi Hfs )

+ (∇iFfs )(∇i∂vj Gfs )(∂vi Hfs ) − (∇iGfs )(∇i∂vj Ffs )(∂vi Hfs )

− (∇iHfs )(∂vi ∇jFfs )(∂vj Gfs ) + (∇iHfs )(∂vi ∇jGfs )(∂vj Ffs )

− (∇iHfs )(∇jFfs )(∂vi ∂vj Gfs ) + (∇iHfs )(∇jGfs )(∂vi ∂vj Ffs ) :=: 0 .

This will all cancel when we add in the cyclic permutations. The first line cancels by letting F → G, G → H, & H → F in the first term. In the second line, let G → F,H → G, & F → H in the first term and F → G, G → H, & H → F in the second term. Then the second line cancels the third line. The fourth line cancels by letting G → F,H → G, & F → H in the first term. The integrand is zero, so this term contributes nothing to the Jacobi identity.

42 Term 6: (fff) Z X es δ{F,G}t {{F,G} ,H} = f B · ∂ × ∂ H t t m2c s v δf v fs s s s gs δ{F,G}t 3 3 − 2 fs E · ∂v × ∂vHfs d x d v . (4.20) msc δfs

Focus on the integrand I6: es es gs I6 = 2 fs B · ∂v 2 B · ∂vFfs × ∂vGfs − 2 E · ∂vFfs × ∂vGfs × ∂vHfs msc msc msc gs es gs − 2 fs E · ∂v 2 B · ∂vFfs × ∂vGfs − 2 E · ∂vFfs × ∂vGfs × ∂vHfs . msc msc msc (4.21)

The electric and magnetic ﬁeld do not depend on velocity, so we can pull them out of the derivatives. The remaining velocity dependence is the same for all four terms. Shift to index notation: 1 I = e2B B −e g (B E +E B )+g2E E ∂ (∂ F ∂ G ) ∂ H . 6 4 2 s i ` s s i ` i ` s i ` ijk `mn vj vm fs vn fs vk fs msc (4.22)

Everything in front of the ’s is symmetric when i ↔ `. Call it Bi`: 1 Bi` := 4 2 (esBi − gsEi)(esB` − gsE`) , (4.23) msc I6 = Bi` ijk`mn (∂vj ∂vm Ffs )(∂vn Gfs )(∂vk Hfs )+(∂vm Ffs )(∂vj ∂vn Gfs )(∂vk Hfs ) . (4.24) Switch n and m in the second term. This gives an extra minus sign because of the antisymmetry of the Levi-Civita symbol. Then cyclically rotate the second term G → F,H → G, & F → H: I6 :=: Bi` ijk`mn (∂vj ∂vm Ffs )(∂vn Gfs ∂vk Hfs − ∂vk Gfs ∂vn Hfs ) . (4.25)

43 This is a (symmetric) second velocity derivative, which we will call Fjm, multiplied by an antisymmetric combination of ﬁrst velocity derivatives, which we will call Gnk:

I6 :=: Bi` ijk`mn Fjm Gnk . (4.26)

A combination of two symmetric rank two tensors, an antisymmetric rank two tensor, and two antisymmetric rank three tensors must be zero. This can seen by renaming i ↔ `, j ↔ m,& k ↔ n, and then reversing the order of the indices:

Bi` ijk`mn Fjm Gnk = B`i `mnijk Fmj Gkn = −Bi` ijk`mn Fjm Gnk . (4.27)

The integrand is zero, so this term contributes nothing to the Jacobi identity.

Term 11: (EEf)

Z X 4πes δ{F,G}j δ{F,G}j {{F,G}j,H}j = fs HE · ∂v − ·∂vHfs (4.28) ms δfs δE s | {z } =0 4πgs δ{F,G}j δ{F,G}j 3 3 + fs HB · ∂v − ·∂vHfs d x d v . ms δfs δB | {z } =0

44 Focus on the integrand I11:

4πes 4πes I11 = fs HE · ∂v (GE · ∂vFfs − FE · ∂vGfs ) ms ms 4πgs + (GB · ∂vFfs − FB · ∂vGfs ) ms 4πgs 4πes + fs HB · ∂v (GE · ∂vFfs − FE · ∂vGfs ) ms ms 4πgs + (GB · ∂vFfs − FB · ∂vGfs ) ms 16π2 = 2 fs (esHE + gsHB) · ∂v (esGE + gsGB) · ∂v Ffs ms −(esHE + gsHB) · ∂v (esFE + gsFB) · ∂v Gfs . (4.29)

Cyclically permute G → F,H → G, & F → H in the second term. The functional derivatives with respect to E and B are functions of x alone (as discussed in section 4.3.3), so the directional velocity derivatives here commute.

16π2 I11 :=: 2 fs (esHE + gsHB) · ∂v (esGE + gsGB) · ∂v Ffs ms −(esGE + gsGB) · ∂v (esHE + gsHB) · ∂v Ffs = 0 . (4.30)

The integrand is zero, so this term contributes nothing to the Jacobi identity.

Term 7: (Eff)

Z X 4πes δ{F,G}t δ{F,G}t {{F,G} ,H} = f H · ∂ − · ∂ H (4.31) t j m s E v δf δE v fs s s s ! 4πgs δ{F,G}t δ{F,G}t 3 3 + fs HB · ∂v − · ∂vHfs d x d v . ms δfs δB

45 Focus on the integrand I7: 4πes es gs I7 = fs HE · ∂v 2 B · (∂vFfs × ∂vGfs ) − 2 E · (∂vFfs × ∂vGfs ) ms msc msc Z X gs0 3 0 − − f 0 (∂ 0 F × ∂ 0 G ) d v · ∂ H 2 s v fs0 v fs0 v fs m 0 c s0 s 4πgs es gs + fs HB · ∂v 2 B · (∂vFfs × ∂vGfs ) − 2 E · (∂vFfs × ∂vGfs ) ms msc msc Z X es0 3 0 − f 0 (∂ 0 F × ∂ 0 G ) d v · ∂ H . (4.32) 2 s v fs0 v fs0 v fs m 0 c s0 s The second and fourth lines cancel when you exchange the order of sum over s0 and the sum over s outside of the integrand. Switch to index notation. The electric and magnetic ﬁelds don’t depend on velocity. Factor as much as possible: 4π I = f e H ∂ e B ∂ F ∂ G − g E ∂ F ∂ G 7 3 s s E` v` s ijk i vj fs vk fs s ijk i vj fs vk fs msc + gsHB` ∂v` es ijk Bi ∂vj Ffs ∂vk Gfs − gs ijk Ei ∂vj Ffs ∂vk Gfs 4π = f (e H + g H )(e B − g E ) ∂ (∂ F ∂ G ) . (4.33) 3 s s E` s B` s i s i ijk v` vj fs vk fs msc Apply the product rule to the last part of this, then switch j and k in the second term:

ijk ∂v` (∂vj Ffs ∂vk Gfs ) = ijk (∂v` ∂vj Ffs ∂vk Gfs + ∂vj Ffs ∂v` ∂vk Gfs )

= ijk (∂v` ∂vj Ffs ∂vk Gfs − ∂vk Ffs ∂v` ∂vj Gfs ) .(4.34)

Plug this back into the integrand: 4π I = f (e H +g H )(e B −g E ) (∂ ∂ F ∂ G −∂ F ∂ ∂ G ) . 7 3 s s E` s B` s i s i ijk v` vj fs vk fs vk fs v` vj fs msc (4.35) This will cancel with Term 10.

46 Term 10: (Eff) Z X es δ{F,G}j {{F,G} ,H} = f B · ∂ × ∂ H j t m2c s v δf v fs s s s gs δ{F,G}j − 2 fs E · ∂v × ∂vHfs . (4.36) msc δfs

Focus on the integrand I10:

es 4πes I10 = 2 fs B · ∂v (GE · ∂vFfs − FE · ∂vGfs ) msc ms 4πgs + (GB · ∂vFfs − FB · ∂vGfs ) × ∂vHfs ms gs 4πes − 2 fs E · ∂v (GE · ∂vFfs − FE · ∂vGfs ) msc ms 4πgs + (GB · ∂vFfs − FB · ∂vGfs ) × ∂vHfs ms 4π = 3 fs (esB − gsE) · ∂v (esGE + gsGB) · ∂vFfs msc −(esFE + gsFB) · ∂vGfs × ∂vHfs . (4.37)

GE, GB, FE,& FB are independent of velocity. Shift to index notation:

4π I = f (e B − g E ) (e G + g G ) ∂ ∂ F ∂ H 10 3 s s i s i ijk s E` s B` vj v` fs vk fs msc −(esFE` + gsFB` ) ∂vj ∂v` Gfs ∂vk Hfs . (4.38)

In the ﬁrst term, cyclically permute F → G, G → H, & H → F . In the second term, cyclically permute G → F,H → G, & F → H:

4π I = f (e B −g E ) (e H +g H )(∂ ∂ G ∂ F −∂ ∂ F ∂ G ) . 10 3 s s i s i ijk s E` s B` vj v` fs vk fs vj v` fs vk fs msc (4.39) This exactly cancels Term 7.

47 Term 8: (Eff)

Z δ{F,G} δ{F,G} {{F,G} ,H} = 4πc t · ∇ × H − H · ∇ × t d3x . t v δE B E δB (4.40) Integrate the second term by parts. The antisymmetry of the cross product cancels the negative sign from integrating by parts: Z Z δ{F,G}t 3 δ{F,G}t 3 − ijkHEi ∇j d x = ijk ∇jHEi d x + B.T. (4.41) δBk δBk Z δ{F,G}t 3 = − kji ∇jHEi d x + B.T. δBk There is also a boundary term, which we denote by B.T. In section 4.3.5, we will discuss why we neglect it. Z X gs {{F,G} ,H} = 4πc f − (∂ F × ∂ G ) · (∇ × H ) t v s m2c v fs v fs B s s es 3 3 − 2 (∂vFfs × ∂vGfs ) · (∇ × HE) d x d v (4.42) msc X 4π Z = − f (∂ F × ∂ G ) · ∇ × (g H + e H ) d3x d3v . m2 s v fs v fs s B s E s s This will cancel when combined with other terms.

Term 3: (Eff) Z X 4πes δ{F,G}c δ{F,G}c {{F,G}c,H}j = fs HE · ∂v − ·∂vHfs ms δfs δE s | {z } ˙=0

4πgs δ{F,G}c δ{F,G}c 3 3 + fs HB · ∂v − ·∂vHfs d x d v ms δfs δB | {z } ˙=0 Z X 4π 3 3 = 2 fs (esHE + gsHB) · ∂v[Ffs ,Gfs ]c d x d v . (4.43) ms

48 We will later refer to this integrand as I3. This will need to be combined with Term 9, then Term 8, to cancel.

Term 9: (Eff)

Z X 1 δ{F,G}j {{F,G} ,H} = f ,H d3x d3v . (4.44) j c m s δf fs s s s c

Focus on the integrand I9: 1 4πes 4πgs I9 = fs (GE · ∂vFfs − FE · ∂vGfs ) + (GB · ∂vFfs − FB · ∂vGfs ) ,Hfs ms ms ms c 4π = 2 fs (esGE + gsGB) · ∂vFfs ,Hfs c − (esFE + gsFB) · ∂vGfs ,Hfs c . (4.45) ms

This is now in a similar form to Term 3. Use the antisymmetry of the canonical Poisson bracket on the ﬁrst term before adding it to Term 3.

Terms 3 & 9: (Eff)

4π I3 + I9 = 2 fs (esHE + gsHB) · ∂v Ffs ,Gfs c − Hfs , (esGE + gsGB) · ∂vFfs c ms − (esFE + gsFB) · ∂vGfs ,Hfs c . (4.46)

It would be convenient if all of the functional derivatives with respect to the ﬁeld were acting on only one of our three arbitrary functionals. In the second term, permute F → G, G → H, & H → F . In the third term, permute G → F,H → G, & F → H. Refer to the integrand after re-permuting and dropping the common factors as I39:

4π I3 + I9+ :=: 2 fs I39 , (4.47) ms

49 I39 := (esHE + gsHB) · ∂v Ffs ,Gfs c − Ffs , (esHE + gsHB) · ∂vGfs c − (esHE + gsHB) · ∂vFfs ,Gfs c . (4.48)

To simplify the notation a bit, we will give a name to the combination of

ﬁeld functional derivatives that appears: A := esHE + gsHB. As we discussed

in section 4.3.3, A will be a function of x alone. Also deﬁne F := Ffs and

G := Gfs :

I39 = Ai ∂vi F , G c − F , Ai∂vi G c − Ai∂vi F , G c = Ai ∂vi (∇jF)(∂vj G) − (∂vj F)(∇jG) − (∇jF) ∂vj (Ai∂vi G) − (∂vj F) ∇j(Ai∂vi G) − (∂vj G)∇j(Ai∂vi F) − (∇jG)∂vj (Ai∂vi F)

= Ai(∂vi ∇jF)(∂vj G)) + Ai(∇jF)(∂vi ∂vj G)

−Ai(∂vi ∂vj F)(∇jG) − Ai(∂vj F)(∂vi ∇jG) | {z }

−(∇jF)Ai(∂vj ∂vi G) + (∂vj F)(∇jAi)(∂vi G) + (∂vj F)Ai(∇j∂vi G) | {z }

−(∂vj G)(∇jAi)(∂vi F) − (∂vj G)Ai(∇j∂vi F) + (∇jG)Ai(∂vj ∂vi F)

= (∇jAi)(∂vj F ∂vi G − ∂vi F ∂vj G) = (∇jAi)(∂vk F)(∂v` G)(δjkδi` − δj`δik)

= (∇jAi)(∂vk F)(∂v` G) njink` = (∇ × A) · (∂vF × ∂vG) . (4.49)

Substitute in for A, F,& G, and plug this back into the integral:

X 4π Z f ∇ × (e H + g H ) · (∂ F × ∂ G ) d3x d3v . (4.50) m2 s s E s B v fs v fs s s This exactly cancels Term 8 (4.42).

50 Term 2: (fff)

Z X es δ{F,G}c {{F,G} ,H} = f B · ∂ × ∂ H c t m2c s v δf v fs s s s

gs δ{F,G}c 3 3 − 2 fs E · ∂v × ∂vHfs d x d v . (4.51) msc δfs

Focus on the integrand I2:

1 1 I2 = 2 fs (esB − gsE) · ∂v (∇Ffs · ∂vGfs − ∇Gfs · ∂vFfs ) × ∂vHfs msc ms 1 = f (e B − g E ) ∂ (∇ F ∂ G − ∇ G ∂ F ) ∂ H 3 s s i s i ijk vj ` fs v` fs ` fs v` fs vk fs msc 1 = f (e B − g E ) (∂ ∇ F ∂ G ∂ H + ∇ F ∂ ∂ G ∂ H 3 s s i s i ijk vj ` fs v` fs vk fs ` fs vj v` fs vk fs msc

− ∂vj ∇`Gfs ∂v` Ffs ∂vk Hfs − ∇`Gfs ∂vj ∂v` Ffs ∂vk Hfs ) . (4.52)

Cyclically permute F , G, and H so that the second derivative always is on F .

1 I = f (e B − g E ) ∂ ∂ F (∂ G ∇ H − ∂ H ∇ G ) 2 3 s s i s i ijk vj v` fs vk fs ` fs vk fs ` fs msc − ∂vj ∇`Ffs (∂vk Gfs ∂v` Hfs − ∂vk Hfs ∂v` Gfs ) . (4.53)

This will cancel part of Term 5.

Term 5: (fff)

Z X 1 δ{F,G}t δ{F,G}t {{F,G} ,H} = f ∇ · ∂ H −∇H · ∂ d3x d3v . t c m s δf v fs fs v δf s s s s (4.54)

51 Focus on the integrand I5: 1 es gs I5 = fs ∇ 2 B · (∂vFfs × ∂vGfs ) − 2 E · (∂vFfs × ∂vGfs ) · ∂vHfs ms msc msc es gs −∇Hfs · ∂v 2 B · (∂vFfs × ∂vGfs ) − 2 E · (∂vFfs × ∂vGfs ) msc msc 1 = f ∇ (e B − g E ) ∂ F ∂ G ∂ H 3 s ` ijk s i s i vj fs vk fs v` fs msc −∇`Hfs ∂v` ijk(esBi − gsEi) ∂vj Ffs ∂vk Gfs 1 = f ∇ (e B − g E ) ∂ F ∂ G ∂ H (4.55) 3 s ijk ` s i s i vj fs vk fs v` fs msc +(esBi − gsEi) ∇`∂vj Ffs ∂vk Gfs ∂v` Hfs + ∂vj Ffs ∇`∂vk Gfs ∂v` Hfs −(esBi − gsEi) ∇`Hfs ∂v` ∂vj Ffs ∂vk Gfs + ∇`Hfs ∂vj Ffs ∂v` ∂vk Gfs .

Split the integrand into two parts. Call the part which involves derivatives of the ﬁelds I5b and the part which does not I5a. We will ﬁrst address I5a:

1 I = f (e B − g E )∇ ∂ F ∂ G ∂ H + ∂ F ∇ ∂ G ∂ H 5a 3 s ijk s i s i ` vj fs vk fs v` fs vj fs ` vk fs v` fs msc −∇`Hfs ∂v` ∂vj Ffs ∂vk Gfs − ∇`Hfs ∂vj Ffs ∂v` ∂vk Gfs . (4.56)

In the last term on each line, cyclically permute F , G, and H so that the second derivative always is on F . Also rename indices j ↔ k, which results in an extra minus sign from the Levi-Civita symbol.

1 I :=: f (e B − g E )∇ ∂ F (∂ G ∂ H − ∂ H ∂ G ) 5a 3 s ijk s i s i ` vj fs vk fs v` fs vk fs v` fs msc −∂v` ∂vj Ffs (∇`Hfs ∂vk Gfs − ∇`Gfs ∂vk Hfs . (4.57)

This now exactly cancels Term 2 (4.53).

52 Now, focus on I5b:

1 I = f ∇ (e B − g E ) ∂ F ∂ G ∂ H . (4.58) 5b 3 s ijk ` s i s i vj fs vk fs v` fs msc

We will prove a vector identity below which allows us to switch i and `:

1 I + cyc = f ∇ (e B − g E ) ∂ F ∂ G ∂ H 5b 3 s ijk ` s ` s ` vj fs vk fs vi fs msc 1 = 3 fs ∇ · (esB − gsE) ∂vFfs · (∂vGfs × ∂vHfs ) . (4.59) msc

There are no other terms for this to cancel with, so it must remain in the Jacobi identity.

Vector Identity for Term 5b:

Before we conclude this Jacobi identity calculation, there is one more vector

identity that we need to prove. Let B := (esB−gsE), U := ∂vFfs , V := ∂vGfs , and W := ∂vHfs . We would like to show:

ijk (∇`Bi) UjVkW` + cyc = (∇iBi) abc UaVbWc . (4.60)

Begin the calculation:

ijk (∇`Bi)(UjVkW` + UkV`Wj + U`VjWk)

= UaVbWc ∇`Bi ijk(δajδbkδc` + δakδb`δcj + δa`δbjδck) . (4.61)

We would like to move indices between the Kronecker deltas. In order to do that, we need the following identity:

δaiδbj = mabmij + δajδbi . (4.62)

53 In particular, we will switch the roles of k and ` in each of the three terms. Focus on the ﬁrst term:

ijkδajδbkδc` = ijkδaj(mbcmk` + δb`δck)

= δajijk`mkmbc + ijkδajδb`δck

= δaj(δi`δjm − δimδj`)mbc + iacδb`

= abcδi` − ibcδa` − icaδb` . (4.63)

The other two terms can be determined by cyclically permuting a, b, and c:

ijkδakδb`δcj = cabδi` − iabδc` − ibcδa` , (4.64)

ijkδa`δbjδck = bcaδi` − icaδb` − iabδc` . (4.65)

Add these together:

ijk(δajδbkδc` + δakδb`δcj + δa`δbjδck)

= 3abcδi` − 2(iabδc` + icaδb` + ibcδa`)

= 3abcδi` − 2ijk(δajδbkδc` + δakδb`δcj + δa`δbjδck) . (4.66)

Now we can solve this equation for the combination of Levi-Civita symbols and Kronecker deltas that we were originally interested in:

ijk(δajδbkδc` + δakδb`δcj + δa`δbjδck) = abcδi` ,

ijk (∇`Bi)(UjVkW` + UkV`Wj + U`VjWk) = (∇iBi) abcUaVbWc . (4.67)

4.3.5 Boundary Term

In Term 8, we integrated by parts (4.42) and neglected the boundary term. Is this a legitimate assumption?

54 Boundary terms in velocity space can safely be set to zero. In order to have the density of particles at every location be finite, the distribution function must decay as v → ±∞. If we assume that the energy or other moments of the distribution function are finite, we confine ourselves to an Lp space. Even if the particles move relativistically, we can take our boundaries to be the speed of light. The distribution function for any particle with mass must be zero at the speed of light.

There are situations when boundary terms in position are important. A physical plasma might have walls or a free surface. The Poisson bracket should be able to capture this physics. If the dynamics of the free surface is nontrivial, then the boundary terms in the Poisson brackets and Hamiltonian become important.

We will not consider boundary terms in this dissertation. We will simply write down the boundary term for Term 8, then assume it vanishes:

Z δ{F,G}t 2 B.T. = ijk HEi nˆj d x . (4.68) ∂V δBk

The integral is now over the surface of the volume of interest V , with unit normal nˆ directed outward.

55 Evaluate the functional derivative and simplify:

Z Z ! X es B.T. ˙= − H f ∂ F ∂ G d3v nˆ d2x ijk Ei 2 s k`m v` fs vm fs j m c 3 ∂V s s R Z Z X es = − f H ( ) ∂ F ∂ G nˆ d2x d3v 2 s Ei ijk `mk v` fs vm fs j m c 3 s s R ∂V Z Z X es 2 3 = − 2 fs HEi (∂vi Ffs ∂vj Gfs − ∂vi Gfs ∂vj Ffs )n ˆj d x d v m c 3 s s R ∂V Z Z X es ˆ 2 3 :=: − 2 fs (GE · ∂vHfs − HE · ∂vGfs ) ∂vFfs · n d x d v . m c 3 s s R ∂V (4.69)

Even with the cyclic permutations, this term is not, in general zero.

There are several common situations when this term would be zero. If the distribution function and all relevant functionals are periodic in space, then the integral will have opposite contributions from opposite sides of the periodic cells. If V is R3 and there is a ﬁnite total number of particles, then the distribution function decays as x → ∞. If there is a hard wall along which fs = 0, then we can take ∂V to be the wall. In each case, the integral is zero.

4.4 Conclusion

This 19 page calculation deserves its own conclusion.

In this direct calculation of the Jacobi identity, all of the terms were either zero or canceled with other terms, except for part of Term 5. The Jacobi

56 identity reads:

X 1 Z {F, {G, H}}+cyc = ∂ H ·(∂ F × ∂ G ) f (e ∇ · B − g ∇ · E) d3x d3v . m3c v fs v fs v fs s s s s s (4.70) This is not zero in general.

There are a few special cases in which it is zero.

The simplest case is electromagnetism as we observe it, without magnetic monopoles. In this case, gs = 0 for all species and ∇ · B = 0. We already saw this case in (3.31).

For (4.70) to vanish for arbitrary F , G, H, and fs, we must have

es∇ · B = gs∇ · E , ∀s , (4.71) which is only possible if all species have the same ratio of electric to magnetic charge. The case is addressed in Section 6.11 of Jackson [59]. We can use a duality transformation (4.1) with ξ = arctan(gs/es) to completely remove the magnetic charges. These monopoles are trivial; this theory is equivalent to electromagnetism without monopoles. “The only meaningful question is whether all particles have the same ratio of magnetic to electric charge” [59].

Glance back through the calculation for the Jacobi identity and notice how frequently the duality transformed electric and magnetic ﬁelds,

0 E ∝ esE + gsB ,

0 B ∝ esB − gsE , (4.72)

57 make an appearance. They are seen in all of the terms we calculated except for Terms 1 & 8.

Another way that the Jacobi identity could be satisﬁed is if ∇ · E = ∇ · B = 0. This is obviously not true in general.

There is one further trick you could try to use to make the Jacobi identity zero. For each species, at every location, either fs = 0 or es∇ · B − gs∇ · E = 0. We will discuss this situation farther in section 5.1.4, because the meaning becomes clearer in the context of two point particles than when using distribution functions. It suﬃces for us here to say that this is an unrealistic constraint on the distribution functions.

When we add nontrivial magnetic monopoles to the Vlasov-Maxwell system, the Jacobi identity is not satisﬁed, so it is not a Hamiltonian ﬁeld theory.

Relaxing some of the assumptions for a Poisson bracket results in a twisted Poisson bracket. For a twisted Poisson bracket, although the Jacobi identity is not zero, it is equal to an exact three-form. The dynamics of a single particle in fields created by both electrically and magnetically charged matter is twisted Poisson. If you lift this theory to the space of distribution functions, for some field configurations, the bracket (4.2) is not twisted Poisson [69].

58 Chapter 5

Consequences

5.1 One Electron and One Monopole

If there are only a few magnetic monopoles in the universe, does it even make sense to think of their collective motion in a plasma?

If a single magnetic monopole exists, then there is no reason to think that multiple magnetic monopoles couldn’t exist together. It would be a very strange theory that allowed only isolated magnetic monopoles but didn’t allow interacting groups of magnetic monopoles.

We originally calculated the Jacobi identity for the collective motion of many electrically and magnetically charged particles. The calculation should also be valid when there are only a small number of particles.

5.1.1 Restricting the Hamiltonian and Poisson Bracket

Consider the interaction between a single electron with position Xe, velocity Ve, mass me, electric charge e, and magnetic charge 0 and a single monopole with position Xm, velocity Vm, mass mm, electric charge 0, and magnetic charge g.

The Hamiltonian and Poisson bracket for this system follow from re-

59 stricting the distribution function to a sum of delta functions localized on the particles. Set

fs = δ(x − Xs) δ(v − Vs) , s = e, m . (5.1)

We can directly calculate the reduced Hamiltonian:

Z Z X ms 1 H = |v|2 δ(x − X ) δ(v − V ) d3x d3v + (|E|2 + |B|2) d3x 2 s s 8π s=e,m m m 1 Z = e |V |2 + m |V |2 + (|E|2 + |B|2) d3x . (5.2) 2 e 2 m 8π

For the Poisson bracket, we will need to determine how to restrict the functional derivatives to this distribution function.

All of the relevant functionals are linear in the distribution function:

Z Z 3 3 3 3 F = F(x, v) fs(x, v) d x d v + G(x, v) d x d v . (5.3)

This makes it much simpler to restrict to a δ-function distribution,

Z 3 3 F = F(x, v) δ(x − Xs)δ(v − Vs) d x d v fs=δ(x−Xs)δ(v−Vs) Z + G(x, v) d3x d3v Z 3 3 = F(Xs, Vs) + G(x, v) d x d v , (5.4) and to evaluate functional derivatives,

δF = F(x, v) . (5.5) δfs

60 Restricting the relevant functional derivatives to δ function distributions gives:

δF

= F(x, v) = F(Xs, Vs) , δfs fs=δ(x−Xs)δ(v−Vs) x=Xs,v=Vs δF ∂F(X , V ) s s ∇ = ∇F(x, v) = , δfs fs=δ(x−Xs)δ(v−Vs) x=Xs,v=Vs ∂Xs δF ∂F(X , V ) s s ∂v = ∂vF(x, v) = . (5.6) δfs fs=δ(x−Xs)δ(v−Vs) x=Xs,v=Vs ∂Vs

In future equations, to avoid clutter, we will simply write (... ) instead of δ (... ) . fs=δ(x−Xs)δ(v−Vs) Although the original equations were entirely a ﬁeld theory, the restricted equations are a mixture of a particle theory and a ﬁeld theory. Our restriction will replace F , a functional of fs, E,& B, with F, a function of

Xe, Ve, Xm,& Vm and a functional of E & B. The functional derivatives with respect to E & B remain unchanged when we restrict the distribution function.

Evaluate the restricted Poisson bracket term by term:

X 1 Z {F,G}c = δ(x − Xs) δ(v − Vs) δ m s=e,m s δF δG δG δF 3 3 ∇ · ∂v − ∇ · ∂v d x d v δfs δ δfs δ δfs δ δfs δ 1 ∂F ∂G ∂G ∂F = · − · me ∂Xe ∂Ve ∂Xe ∂Ve 1 ∂F ∂G ∂G ∂F + · − · . (5.7) mm ∂Xm ∂Vm ∂Xm ∂Vm

61 X 1 Z δF δG {F,G}t = δ(x − Xs)δ(v − Vs) esB · ∂v × ∂v δ m2c δf δ δf δ s=e,m s s s δF δG 3 3 −gsB · ∂v × ∂v d x d v (5.8) δfs δ δfs δ e ∂F ∂G g ∂F ∂G = B · × − E · × . 2 2 mec Xe ∂Ve ∂Ve mmc Xm ∂Vm ∂Vm

X 4π Z {F,G}j = δ(x − Xs)δ(v − Vs) δ m s=e,m s δG δF δF δG es · ∂v − · ∂v δE δfs δ δE δfs δ δG δF δF δG 3 3 +gs · ∂v − · ∂v d x d v δB δfs δ δB δfs δ 4πe δG ∂F δF ∂G

= · − · me δE Xe ∂Ve δE Xe ∂Ve 4πg δG ∂F δF ∂G

+ · − · . (5.9) mm δB Xm ∂Ve δB Xm ∂Vm

The last term of the Poisson bracket, which gives equations for electromagnetic ﬁeld in vacuum, is unaﬀected by changing the distribution function.

62 The restricted Poisson bracket is therefore:

1 ∂F ∂G ∂G ∂F {F, G} = · − · me ∂Xe ∂Ve ∂Xe ∂Ve 1 ∂F ∂G ∂G ∂F + · − · mm ∂Xm ∂Vm ∂Xm ∂Vm e ∂F ∂G + B · × 2 mec Xe ∂Ve ∂Ve g ∂F ∂G − E · × 2 mmc Xm ∂Vm ∂Vm 4πe δG ∂F δF ∂G

+ · − · me δE Xe ∂Ve δE Xe ∂Ve 4πg δG ∂F δF ∂G

+ · − · mm δB Xm ∂Vm δB Xm ∂Vm Z δF δG δG δF + 4πc · ∇ × − · ∇ × d3x . (5.10) δE δB δE δB

Now that we no longer have to distinguish between restricted and un- restricted functionals, call all of the functions F instead of F.

5.1.2 Equations of Motion

This Hamiltonian and Poisson bracket give the expected equations of motion: the Lorentz force laws and the dynamical Maxwell equations, with

63 currents proportional to Vs δ(x − Xs):

∂X e = {X , H} = V , (5.11) ∂t e e ∂X m = {X , H} = V , (5.12) ∂t m m ∂V e 4πe 1 e = {V , H} = B · ( × m V ) + E · e 2 I e e I ∂t mec Xe me 4π Xe e e = Ve × B + E , (5.13) mec me ∂V g 4πg 1 m = {V , H} = − E · ( × m V ) + B · m 2 I m m I ∂t mmc Xm mm 4π Xm g g = − Vm × E + B , (5.14) mmc mm

∂E Z 1 = {E, H} = 4πc δ(x − x0) · ∇ × B d3x ∂t I 4π 4πe − I δ(x − Xe) · meVe me

= c ∇ × B − 4π e Ve δ(x − Xe) , (5.15) ∂B Z 1 = {B, H} = −4πc δ(x − x0) · ∇ × E d3x ∂t I 4π 4πg − I δ(x − Xm) · mmVm mm

= −c ∇ × E − 4π g Vm δ(x − Xm) . (5.16)

The divergence Maxwell equations, with delta function sources, appear in the Casimirs:

Z 3 CE = hE(x) ∇ · E − 4πe δ(x − Xe) d x , (5.17) Z 3 CB = hB(x) ∇ · B − 4πg δ(x − Xm) d x . (5.18)

64 5.1.3 Jacobi Identity

The Jacobi identity calculation for (5.10) can be done directly, but it follows easily upon substituting (5.1) and the last line of (5.6) into (4.70):

X 1 Z ∂F ∂G ∂H {{F,G},H} + cyc = · × δ(x − X )δ(v − V ) m3c ∂V ∂V ∂V s s s=e,m s s s s 3 3 es∇ · B − gs∇ · E d x d v e ∂F ∂G ∂H = · × ∇ · B 3 mec ∂Ve ∂Ve ∂Ve Xe g ∂F ∂G ∂H − · × ∇ · E (5.19) 3 mmc ∂Vm ∂Vm ∂Vm Xm

The divergence Maxwell’s equations tell us that this is proportional to a delta function in position:

4πeg 1 ∂F ∂G ∂H {{F,G},H} + cyc = δ(Xe − Xm) 3 · × c me ∂Ve ∂Ve ∂Ve 1 ∂F ∂G ∂H − 3 · × . (5.20) mm ∂Vm ∂Vm ∂Vm

The Jacobi identity is not satisﬁed globally. There is a singularity when the positions of the electron and monopole coincide.

The electromagnetic interaction between a single electron and a single magnetic monopole is not, in general, Hamiltonian.

5.1.4 No Touching

The main result of this section is that the Jacobi identity fails when the position of the electron and monopole coincide. If there were some way to

65 keep electrons and monopoles from ever touching, electromagnetism would be a Hamiltonian theory.

At first, this seems reminiscent of the problems which occur with the self-energy of electrically charged point particles. When we look closer, we see that these singularities are very different. That singularity comes from the Hamiltonian, can only be reached with infinite energy, and is removed if the point particles are replaced by continuous charge distributions. This singularity comes from the Jacobi identity, requires no energy to reach, and becomes worse if the point particles are replaced by continuous distributions because the Jacobi identity is violated at more locations.

Causing two electrically charged point particles to collide classically would require inﬁnite energy. There is nothing like this preventing an electron from colliding with a monopole. A stationary monopole produces a radial magnetic ﬁeld. An electron moving directly towards the monopole experiences a force eVe × B/c = 0. The electron passes through the monopole without experiencing any force at all.

What is required for the Jacobi identity to be satisﬁed for continuous distributions of charge?

Look back at (4.70).

Focus on one species of particle. For example, we could focus on electrons. At any location where fe 6= 0, we must have ∇ · B = 0. At any location where fm 6= 0, we must have ∇ · E = 0. We also might have a species of

66 dyon, d, with both electric and magnetic charge. At any location where its distribution function is nonzero, we must have ∇ · (edB − gdE) = 0.

Now look at the divergence Maxwell’s equations, which are proportional to the integral of the distribution functions. In order for ∇ · B = 0, the distribution functions of all species with nonzero magnetic charge must be zero at that location. In order for ∇ · E = 0, the distribution functions of all species with nonzero electric charge must be zero at that location. In order for ∇ · (edB − gdE) = 0, the distribution functions of all species with a diﬀerent ratio of electric to magnetic charge from this dyon must be zero at that location. We summarize this result as the No Touching Theorem:

No Touching Theorem: If some species’s distribution function is nonzero at some location, then, in order to preserve the Jacobi identity, the distribution function of any species with a diﬀerent ratio of electric to magnetic charge must be zero there.

We can divide space into regions where electrically charged matter exists, regions where magnetically charged matter exists, and regions where dyons exist. In order for the Jacobi identity to remain satisﬁed, these regions must never overlap.

We know of no physical mechanism to preserve these non-overlapping regions. Many equations of motion violate this assumption. The most obvious example is the diﬀusion equation, which appears when we add a collision operator to the right hand side of the Vlasov equation to account for two

67 particle interactions. Diﬀusion operators spread particles out across previously abrupt boundaries.

Since we cannot prevent electrons and monopoles from touching, if magnetic monopoles exist, they break the Hamiltonian nature of electromagnetism.

5.2 Importance of the Jacobi Identity

The existence of magnetic monopoles prevents electromagnetism from satisfying the Jacobi identity.

Why should we care?

All ‘fundamental’ physical theories that we have observed have been Hamiltonian. Dissipative dynamics occurs when our system interacts with microscopic motion that we do not include in our theory. If we were to include this microscopic motion, the dynamics should again be Hamiltonian, although impossible to solve.

This does not mean that non-Hamiltonian theories could not exist, but it should make us much more skeptical. Extracting patterns from nature is the essence of science, so we should not abandon one of our best established patterns lightly.

Hamiltonian mechanics pervades our understanding of classical mechanics. The theorems and intuition we take for granted are built upon its foundations.

68 In this section, we highlight some of the standard results of classical mechanics that depend on the Jacobi identity. We do not intend for this to be an exhaustive list, nor do we indicate what would change if we tried to work through similar proofs with the Jacobi identity replaced by (4.70) or (5.20). Instead, the purpose of this section is to provide a sense of how deeply integrated the Jacobi identity is to our understanding of physics. See also [90] for further discussion of the importance of the Hamiltonian formulation of physics.

5.2.1 Geometry of Hamiltonian Mechanics

There is extensive literature on the algebraic and geometric nature of Hamiltonian mechanics (e.g. Refs. [127, 131,8, 123, 28]). The Jacobi identity is central to these results.

Phase space is a smooth manifold. The manifold has a differential 2- form ω, which defines an interior product, ω(u, v). The 2-from must be closed, i.e. its exterior derivative must be zero. If there are no Casimirs, then ω is nondegenerate and the manifold is a symplectic manifold. One consequence of this, and the inherent antisymmetry of 2-forms, is that the symplectic manifold must have an even number of dimensions. The Poisson bracket is defined using the 2-form:

{f, g} = ω(Xf ,Xg) , (5.21) where Xf and Xg are the vector ﬁelds associated with the derivatives of f and g. The Jacobi identity is equivalent to dω = 0.

69 If there are Casimirs, then the phase space has the more general structure of a Poisson manifold. The phase space has a foliation of symplectic sub-manifolds. These leaves must all have an even number of dimensions. The number of dimensions can vary if the number of Casimirs is different in different locations in phase space. The Poisson bracket is a Lie algebra structure for the vector space of smooth functions on the manifold. The dynamics can be understood as Lie dragging by the Hamiltonian vector field.

Some geometric properties do not depend on the Jacobi identity. Con- servation of the Hamiltonian depends only on the bracket’s antisymmetry. Similarly, anything whose bracket with the Hamiltonian is zero, including Casimirs, is still conserved.

5.2.2 Canonical Coordinates

The existence of canonical coordinates for a noncanonical Hamiltonian system is guaranteed by Darboux’s theorem [27]. Any symplectic manifold can be deformed locally into the linear symplectic space of the same dimension. A coordinate transformation can convert an arbitrarily complicated Poisson bracket (which satisfies the Jacobi identity) into the canonical Poisson bracket. If there are Casimir invariants, then this transformation has to be done on one of the leaves, defined by C = const., which foliate the phase space [74]. If the phase space or the leaves are topologically nontrivial, then the canonical coordinates can only be defined locally in coordinate patches. As we saw in section 3.3, for electromagnetism, this transformation involves introducing the

70 potentials A and Φ.

Since the Jacobi identity is not satisfied for our bracket, we need the inverse of this statement. If we apply an arbitrary coordinate transformation to a bracket that satisfies the Jacobi identity, the new bracket will also satisfy the Jacobi identity [36]. Inverting a coordinate transformation is still a coordinate transformation. If the Jacobi identity is not satisfied, then no coordinate transformation can turn it into a canonical bracket.

Canonical coordinates do not exist, even locally.

Motion under a Hamiltonian vector ﬁeld is a canonical coordinate transformation. The Hamilton-Jacobi equation is a nonlinear partial diﬀerential equation which determines this coordinate transformation. In order to even write the Hamilton-Jacobi equation, we must use canonical coordinates. With- out the Jacobi identity, there are no canonical coordinates to write the Hamil- tonian in and there is no guarantee that the dynamics will be a canonical transformation.

Hamiltonian perturbation theory is a powerful tool to determine the dynamics of systems which are almost integrable. The nearby integrable system must be solved ﬁrst for its action-angle variables. The perturbation theory proceeds by a series of canonical transformations which make the remaining corrections to the dynamics successively smaller. This procedure does not apply if the dynamics is not a canonical transformation. In particular, the KAM theorem [62,7, 93], which tells us when the motion remains on invariant

71 tori after the perturbation and when it becomes chaotic, no longer applies. Similarly, Greene’s method [44] for determining when the last invariant torus breaks (much farther from integrability) is no longer valid.

5.2.3 Lagrangian Mechanics

Most physical theories begin as a Lagrangian action principle.

Action principles are the modern manifestation of Aristotle’s statement: “Nature does nothing in vain” [6]. Action principles are a particularly simple and beautiful way of understanding physics – and have been extremely successful at modeling the world.

If you have a Lagrangian action principle and the Legendre transform exists, then you can transform it into a Hamiltonian system with a Poisson bracket that satisﬁes the Jacobi identity. Even if your Lagrangian is not convex in v, so the Legendre transform does not exist, Dirac’s constraint theory allows you to construct the Hamiltonian formulation [35].

Contrapose this. If your system has a Poisson bracket that doesn’t satisfy the Jacobi identity, then no Lagrangian exists.

No action principle exists for magnetic monopoles. Action principles for the Vlasov-Maxwell equations involve the potentials A and Φ [142], so they cannot be generalized to include magnetic monopoles.

The theorems and intuition of Lagrangian mechanics also fail if the dynamics is determined by a bracket which does not satisfy the Jacobi identity.

72 In particular, Noether’s theorem fails [96]. Although symmetries can still be used to suggest invariant quantities, the proof, which provides a method of calculating the conserved current, no longer applies [1].

73 Chapter 6

Quantizing Magnetic Monopoles

Standard methods of quantization fail without the Jacobi identity. Typ- ically, we replace dynamical variables with operators whose commutation relation algebra matches the algebra of the Poisson bracket. However, commutators automatically satisfy the Jacobi identity, so it is impossible to match this algebra. Transforming to canonical coordinates ﬁrst, then quantizing, is impossible since canonical coordinates do not exist. We cannot build a wave equation from the Hamilton-Jacobi equation because it also requires canonical coordinates. Even path integral quantization is impossible because there is no Lagrangian action principle [41, 112].

Nevertheless, people have written many quantum theories of magnetic monopoles. It has been known for decades that these theories often have problems [73]. In this chapter, we go through some examples of quantum theories of magnetic monopoles from our gauge-free Hamiltonian perspective. These theories subtly smuggle in the assumption that electrons and monopoles can never touch. This assumption needs to be justiﬁed. If it were true, it would restore the Hamiltonian nature of electromagnetism and remove the impediment to quantization, as we discussed in section 5.1.4.

74 6.1 Nonrelativistic Quantum Theories 6.1.1 Dirac

Dirac wrote the ﬁrst quantum theory of the behavior of an electron in the presence of a magnetic monopole in 1931 [33]. He expanded the theory to involve an arbitrary number of electrically and magnetically charged particles in 1948 [34].

Dirac’s calculation proceeds by locally transforming to canonical coordinates - by locally replacing fields with potentials. This comes at a cost. The theory has a semi-infinite ‘string’ extending from the location of each monopole, along which the electrons’ wavefunctions are zero. Along this string, the electromagnetic vector potential is undefined. The phase of the electron is no longer single valued along a loop encircling the Dirac string. In order for observables to be single valued, the phase shift around the loop must be an integer multiple of 2π. Since the (quantized) phase shift is proportional to the product of the electric and magnetic charge, both charges must be quantized.

The direction of the string is arbitrary. Changing it corresponds to a gauge transformation for the ﬁelds and a global phase shift for the wave function. To avoid the string entirely, Wu and Yang, in 1976, deﬁned the vector potential in multiple coordinate patches around the monopole [140].

Dirac claims that this is not a physical constraint since the locations of the strings are arbitrary. Although the directions of the strings are arbitrary, the locations of the ends of the strings are not arbitrary because these are the locations of the monopoles. By assuming that the electrons’ wavefunctions

75 must be zero at the endpoints of the strings, Dirac implicitly assumes that electrons’ and monopoles’ positions could never coincide.

Moreover, Dirac produced a theory of how quantum electrons would behave in the presence of point magnetic monopoles. Each monopole, as the end of a string whose position is always well deﬁned, does not satisfy the Heisenberg uncertainty principle. This is not a fully quantum theory.

6.1.2 Wentzel

Wentzel, in 1966, considered the wave-mechanical description of the monopole motions [137]. This requires a parallel bundle of Dirac’s strings originating from each volume element of the wavepacket.

Wentzel modiﬁes the relationship between the magnetic ﬁeld and the magnetic potential to account for the string:

H = ∇ × A − H(f) , (6.1) where H(f) is the ﬁctitious magnetic ﬁeld localized on the string.

However, the equations of motion remain:

e π = p − A , c 1 π˙ = j × (∇ × A) . (6.2) c

This is only equivalent to the Lorentz force law if j = 0 whenever H(f) = 0, i.e. away from the strings. Wentzel acknowledges this problem: “Then, without a

76 new interaction term, the [theory] gives the complete Lorentz force acting on the electrons provided the electrons stay away from the magnetic ﬂuxlines.”

This model either assumes the electrons never touch the magnetic monopoles or it doesn’t give the Lorentz force law for the electrons and so shouldn’t be called electromagnetism.

6.1.3 Lipkin, Weisberger, and Peshkin

Lipkin, Weisberger, and Peshkin, in 1969, are the ﬁrst to mention the Jacobi identity in relation to magnetic monopoles [73].

They consider the appropriate angular momentum operators for an electron in the presence of a point magnetic monopole. The Jacobi identity for the commutator of angular momentum operators is proportional to ∇ · B, which, for them, is a δ-function at the origin.

Lipkin et al. then dismiss this concern: “The contradiction could be removed at the outset by restricting the domain of x to exclude the origin. . . . However, it turns out that for ﬁnite energy, all radial wave functions vanish at the origin. . . . Therefore, the Jacobi identity is satisﬁed as an operator equation on the Hilbert space formed by those radial wave functions.” The Hilbert space is restricted to only include wavefunctions which are zero at the location of the magnetic monopole. These radial wavefunctions are not a complete basis. They only cover the part of the Hilbert space for which the electron never touches the monopole.

Moreover, this restriction doesn’t work if the monopole is allowed to

77 move or if the monopole is not localized to a single point. A quantum description of the monopole’s motion would have both of these problems.

6.1.4 Bialynicki-Birula

Bialynicki-Birula introduced magnetic monopoles to the hydrodynamic formulation of quantum mechanics in 1971 [10].

Since this formulation involves ﬂuid-like variables and ﬁelds, it removes the ambiguity associated with the phase of the wave function and vector potential.

The hydrodynamic formulation is equivalent to a description involving wavefunctions as long as the magnetic ﬁeld is proportional to the vorticity everywhere except on certain singular lines. These singular vortex lines play a similar role to Dirac’s strings.

A problem with this system can be seen by carefully analyzing the expectation values of observables in this system. The expectation value of an observable O can be written as:

Z Z r Z r 3 1/2 i e 1/2 i e O = d r ρ exp − dl · mv − A Oˆ ρ exp dl · mv − A . ~ r0 c ~ r0 c (6.3) A similar expression for the transition probability between two states is found in (10). These expressions have line integrals in the exponentials. These line integrals “can be evaluated along any curve not passing through the singular vorticity lines.” If one of these line integrals ends on one of the singular

78 vorticity lines and especially if it ends on one of the non-arbitrary endpoints of a singular vorticity line, the line integral is not deﬁned. If ρ is also nonzero at this location, the expectation value of the observable gets a contribution that is undeﬁned. In order to get a numerical value for the expectation value of the observable, ρ must be zero along the singular vorticity line. This is equivalent to Dirac’s condition that the wavefunction must vanish along Dirac strings. It implicitly assumes that the location of the quantum particle can never coincide with the location of a magnetic monopole.

While Bialynicki-Birula’s system has both electrically and magnetically charged particles, the electromagnetic fields must be created by point particles (28-31). This is not a fully quantum system if the fields are created by classical particles. It describes the fluid-like motion of quantum particles in fields created by classical particles. The quantum particles are not considered as sources for the fields.

6.2 One Electron’s and One Monopole’s Wavefunction

A nonrelativistic, quantum description of magnetic monopoles should not treat either the electrons or the monopoles as point particles. They should each have a wavefunction that satisﬁes Heisenberg’s uncertainty principle.

Now that we have some familiarity with nonrelativistic quantum theories of magnetic monopoles, we can prove the quantum mechanical version of section 5.1.4.

79 Quantum No Touching Theorem: Wherever the electron’s wavefunction is nonzero, the monopole’s wavefunction must be zero and vice versa.

Proceed by contradiction.

Suppose that there exists some region of space for which both the electron’s wavefunction is nonzero and the monopole’s wavefunction is nonzero. Construct a closed loop entirely within this region. The phase shift of the electron is nonzero since there is a nonzero magnetic charge contained in the loop. The phase shift must be an integer multiple of 2π so the observables remain single-valued.

In order for this to work, there must be at least one point inside the loop where the phase of the electron is undeﬁned, i.e. where the electron’s wavefunction is zero. So every loop in a region where the monopole’s wavefunction is nonzero must contain at least one location where the electron’s wavefunction is zero.

Which loop we choose is arbitrary. Pick any point in the region where the two wavefunctions are both nonzero. You can construct an arbitrarily small loop around that point, inside of which, there must be at least one point where the electron’s wavefunction is zero.

We can go through the same argument again, but with the electron’s and monopole’s roles switched. Since these are only diﬀerent from each other by a duality transformation, the same argument holds.

Wherever one of the wavefunctions is nonzero, the other wavefunction

80 must be zero. These is an extremely strict condition. It should be justiﬁed physically before we accept theories which have this property.

6.3 Quantum Field Theories 6.3.1 Cabibbo-Ferrari

The ﬁrst quantum ﬁeld theory involving magnetic monopoles was written by Cabibbo and Ferrari in 1962 [21].

This is not a closed electromagnetic theory.

The equations of motion they put forth in (10, 15, 19, 22) are:

2 2 Φ − m Φ = 0 , Ψ − m Ψ = 0 , (6.4)

0 − ie R F dσ 0 − ig R F˜ dσ Φ(x, P ) = Φ(x, P ) e 2 S µν µν , Ψ(x, P ) = Ψ(x, P ) e 2 S µν µν ,

∗ ∗ ˜ ∗ ∗ ∂µFµν = jµ = −ie (Φ (∂µΦ) − (∂µΦ )Φ) , ∂µFµν = gµ = ig (Ψ (∂µΨ) − (∂µΨ )Ψ) .

There are also commutation relations speciﬁed for the operators in (20, 20’, 21, 23).

The equations of motion for the particles is completely independent of the electromagnetic fields. At most, they depend on the fields through an un- observable phase shift between different paths. What we have here are electromagnetic fields created by freeing moving electrons and magnetic monopoles.

Φ is not the usual electron ﬁeld. Instead, it is related to the electron ﬁeld according to (9): Z x Φ(x, P ) = ϕ(x) exp −ie Aµ(ξ) dξµ . (6.5) (P )

81 In order to get back to the usual electron ﬁeld, we have to use the vector potential. All of the problems with the vector potential and magnetic monopoles reappear.

6.3.2 Schwinger 66

1 In 1966, Schwinger wrote a quantum field theory for two spin- 2 fields, one of which, ψ, carries electric charge and the other, χ, carries magnetic charge, coupled to the electromagnetic fields [115].

The sources for Maxwell’s equations are 4-currents determined by the electron and monopole ﬁelds:

jµ = eψγ¯ µψ , ∗jµ = gχγ¯ µχ . (6.6)

Schwinger writes two vector potentials - one for the magnetic ﬁeld (A) and one for the electric ﬁeld (B). He separates a transverse component out of each vector potential. The remaining part of the vector potential causes problems: Z 3 0 0 ∗ µ 0 Ag(x) = d x a(x − x ) j (x ) , Z 3 0 0 µ 0 Be(x) = d x b(x − x ) j (x ) . (6.7)

An interesting constraint on the kernals relating the vector potentials to the currents comes from insisting that the commutator between the energy- density of the two types of charges be local: 1 − ∇ = ∇ × a(x) = −∇ × b(x) . (6.8) 4π|x|

82 Commutators are closely related to classical Poisson brackets, so it is not surprising that the problem arises here. The divergence of the left hand side is a Dirac delta function. The divergence of the right hand side is zero. This equation cannot be solved.

Schwinger settled for solving it everywhere except for a single line going through the origin. This line corresponds to a Dirac string. Changing which line you use corresponds to a gauge transformation of the electric and magnetic potentials and a phase shift for the wavefunctions. Schwinger used these phase shifts to derive a condition for the quantization of electric and magnetic charge that is oﬀ by a factor of two from Dirac’s condition.

Although the transformations discussed by Schwinger can change the orientation of the line, all of these lines still go through the origin. There is always a singularity when the argument of a (or b) is zero.

Look back at the definition of Ag. Pick some value of x. Whenever x = x0, a(x − x0) is undefined. The only way for this not to be a problem for the integral is if ∗jµ(x0) is zero here. Otherwise, the integral contains an undefined contribution and so will be undefined. Anywhere ∗j, the magnetic 4- current density, is nonzero, Ag is undefined. Similarly, anywhere j, the electric

4-current density, is nonzero, Be is undeﬁned.

The energy density of this system contains the terms:

¯ − eψγ · Agψ − gχγ¯ · Beχ . (6.9)

In order for the energy density to be well deﬁned, ψ must be zero wherever

83 ∗ Ag is undefined. Ag is undefined wherever j is nonzero, i.e. wherever χ is nonzero. Similarly, χ must be zero wherever Be is undefined, i.e. wherever j is nonzero, i.e. wherever ψ is nonzero. Wherever one of the wavefunctions is nonzero, the other wavefunction must be zero.

6.3.3 Schwinger 68

Schwinger presented a similar quantum ﬁeld theory with magnetic monopoles in 1968 [116].

The theory depends heavily on something that Schwinger refers to as f ν(x − x0). It appears in integrals throughout the paper. The conditions that f ν has to satisfy are:

ν 0 ν 0 ν 0 0 f (x − x ) = −f (x − x) , ∂νf (x − x ) = δ(x − x ) . (6.10)

The problems that appear with a in Schwinger’s 1966 theory [115] appear here with f ν. The function is singular or undeﬁned whenever x = x0. Since f ν always appears in integrals, we have to be more careful.

The ﬁelds are related to the current densities according to (44):

µν µ ∗ µν ∗ µ ∂νF (x) = J (x) , ∂ν F (x) = J (x) ,

∗ µν 1 µνλκ where F = 2 Fλκ . (6.11)

The connection between the ﬁelds and the vector potentials involves f ν, which is singular along the string (47-48). The mixed partials of the vector

84 potentials do not commute along this singularity.

∗ Z h 4 0 0 ∗ 0 0 ∗ 0 i ∂µAν(x) − ∂νAµ(x) = Fµν(x) − d x fµ(x − x ) Jν(x ) − fν(x − x ) Jµ(x ) ,

∗ Z ∗ h 4 0 0 0 0 0 i ∂µBν(x) − ∂νBµ(x) = Fµν(x) − d x fµ(x − x )Jν(x ) − fν(x − x )Jµ(x ) .

Although the string along which f ν has a singularity points in an arbi-

0 trary direction, it must go through the origin. fν(x−x ) is undeﬁned whenever

0 ∗ x = x . At any x where the electric [magnetic] current Jν [ Jν] is nonzero, the

∗ relationship between Fµν [ Fµν] and Aν [Bν] will be undefined. If the fields are determined from the current according to the modified Maxwell’s equations, then the corresponding vector potential at these locations will be undefined.

Anywhere there is a nonzero magnetic current, the usual vector potential is undeﬁned. Anywhere there is a nonzero electric current, the new vector potential is undeﬁned.

The action contains a term of the form (48):

1 Z W = d4xJ µ(x)A (x) + ∗J µ(x)B (x) . (6.12) 2 µ µ

µ This is well deﬁned only if J is zero anywhere that Aµ is undeﬁned and

∗ µ ∗ if J is zero anywhere that Bµ is undefined. Aµ is undefined anywhere Jµ is nonzero and Bµ is undefined anywhere Jµ is nonzero. The only way the action

∗ can be well deﬁned is if Jµ is zero everywhere Jµ is nonzero and vice versa. If there is ever nonzero electric and magnetic current at the same location, then the action is undeﬁned and we no longer have a functioning theory.

85 6.4 ‘t Hooft-Polyakov Monopoles

‘t Hooft and Polyakov, in 1974, independently described magnetic monopoles which form from topologically nontrivial symmetry breaking of a larger gauge theory [130, 106].

Electromagnetism is often considered to be a long wavelength limit of a more fundamental theory with a broken gauge symmetry. When the symmetry group is spontaneously broken in a topologically nontrivial way, its boundary conditions at inﬁnity are not zero. Because of this, the Higgs ﬁeld will be zero at some point, which we can call the origin.

Since ‘t Hooft-Polyakov monopoles have nontrivial ﬁelds at inﬁnity, the boundary terms that we dismissed in section 4.3.5 could be nonzero. It might be possible for these boundary terms to cancel the contribution to the Jacobi identity from touching electrons and magnetic monopoles.

Away from the origin, the electromagnetic fields are shown to be the same as the fields of a magnetic monopole. These (static) fields of the long wavelength theory are of the form (2.8):

a Qa(x, t) = raQ(r) ,Wµ (x, t) = µabrbW (r) , (6.13) so they will also be zero at the origin. Near the origin, the symmetry is unbroken, so you can have nonsingular ﬁelds of the full theory.

This looks like the same sort of situation we’ve seen previously: a theory with electromagnetic ﬁelds that must be zero at the location of the magnetic

86 monopole. However, it comes from a very diﬀerent derivation, which could be done from the Hamiltonian perspective. In future work, we hope to write a classical SU(2) theory using an explicitly gauge invariant noncanonical Poisson bracket and check if it satisﬁes the Jacobi identity. We would then break the gauge symmetry in a topologically nontrivial way to determine which aspects of symmetry breaking are essentially quantum and which are inherited from the classical theory.

6.5 New Quantization Techniques

Classical Poisson brackets are associated with quantum commutation relations. Although we are the ﬁrst to show the failure of the Jacobi identity for a magnetic monopole and electron, the corresponding quantum problem has been previously addressed. Some new quantization techniques are being developed to try to create a quantum theory without the Jacobi identity.

The ﬁrst paper to notice the failure of the Jacobi identity for the quantum commutation relations was in 1969 [73]. We have already discussed it in section 6.1.3. They dismissed the problem by insisting that all wavefunctions be zero at the location of the monopole.

In 1985, multiple groups independently placed the quantum commutators for magnetic monopoles in the context of cohomology theory [58, 45, 141, 18]. These papers all considered the ﬁelds around a point monopole. The failure of the Jacobi identity results in a nonzero three-cocyle around the monopole. Dirac’s quantization condition follows from insisting that the

87 three-cocyle equals 2πN. If this condition did not hold or if there were a more general distribution of magnetic monopoles, then operators would have a non-associative algebra. It is not clear that an acceptable form of quantum mechanics exists without associativity.

Recently, there have been several attempts to create a non-associative quantum mechanics for a theory of electrons in a magnetic ﬁeld generated by an arbitrary distribution of magnetic monopoles. Much of this work is built using techniques developed for a electron in the ﬁeld of a point monopole [37, 122].

One such quantum theory assumes that the commutators form a Mal- cev algebra, a realization of a totally antisymmetric twisted Poisson structure [14, 13]. Moments of the quantum theory can be taken to give semiclassical corrections to the motion of a particle in a magnetic ﬁeld created by monopoles. This theory only works for observables linear in the kinematical momentum. The assumption that the algebra is twisted Poisson is not true if there are many interacting particles [69].

Another theory holds for electrons in a uniform background of magnetic monopoles [129]. The electrons are described using a Wigner function on phase space. Non-associativity is introduced by modifying the star product between Wigner functions. This modiﬁed star product had previously been developed in string theory [94, 68]. This procedure only works for a uniform magnetic charge distribution – for a nonuniform distribution, the needed star product is a formal power series in ~.

88 You can make any equations of motion Hamiltonian by doubling the size of the phase space. This extension is not unique. A corresponding quantum theory on the extended phase space can be written [67]. The reduction back to the original phase space eliminates the parts of the magnetic ﬁeld from monopole sources. Solving problems involving magnetic monopoles on the extended space could give you useful information, even though the reduction doesn’t work.

The geometric structure most relevant for these non-associative quantum theories is the bundle gerbe [20, 128]. The sections of the bundle gerbe form a 2-Hilbert space. We can define magnetic translation operators on this space. It is difficult to represent many operators, including the Hamiltonian, on the 2-Hilbert space. The gerbe can also be mapped to a line bundle over the loop space of the configuration manifold. We can then use standard techniques of geometric quantization on loop space. This involves infinite-dimensional analysis, which makes the computation of observables intractable.

None of these new techniques have created a general quantum theory of interacting electrons and monopoles.

89 Chapter 7

Motion in a Sphere of Monopolium

7.1 An Unusual Dynamics Problem

Most of the problems in dynamics that physicists solve are either Hamil- tonian or dissipative.

The motion of a charged particle in the ﬁeld created by magnetic monopoles is neither. Although energy is exactly conserved, the failure of the Jacobi identity means that the problem is not Hamiltonian. Nevertheless, exact solutions should be possible for some simple geometries.

One of these dynamical problems has been solved, both classically and quantum mechanically: the motion of a single electron in the magnetic field of a point magnetic monopole fixed at the origin [118]. As mentioned in section 5.1.4, if the electron’s motion is entirely radial, it experiences no force and passes straight through the monopole. If the motion is not entirely radial, the electron is confined to a cone. The electron approaches and leaves the monopole from the same asymptotic direction. The angular momentum of the electron and the fields, L = m r × v − eg ˆr, is conserved. In this solution, the positions of the electron and magnetic monopole never coincide for almost all initial conditions, so the Hamiltonian nature of the motion is usually preserved.

90 To see conservative dynamics without the Jacobi identity, the charged particle should go through matter which has a nonzero magnetic charge density. We will call this matter “monopolium.”

Consider the motion of an electron inside of a spherically symmetric distribution of monopolium. The magnetic ﬁeld is radial:

4π B = ρ f(r) r . (7.1) 3 m

We will later focus on a particular example: a uniform distribution of monopolium. In this case, f(r) = 1. If all of space were full of monopolium, then there would be ambiguity as to where the origin is. The orbit of the monopole could change when we reassign the origin. We will assume that this is an extremely large, but ﬁnite, sphere of uniform density, so the origin remains unambiguous. We will see that the motion of the electron can be exactly written in terms of parabolic cylinder functions.

This calculation was done with Robert Littlejohn [75].

The force on the electron is the Lorentz force law:

q 4πρ q F = v × B = m f(r) v × r = −m k(r) r × v , (7.2) c 3c where we deﬁne k(r) := 4πρmq f(r)/2mc. The charge and mass of the electron are q and m, respectively, and ρm is a reference magnetic charge density of the monopolium.

91 Write these as ﬁrst order equations of motion:

r˙ = v ,

v˙ = −k(r) r × v . (7.3)

Notice that the acceleration is perpendicular to both the position and velocity.

7.2 Exact Solution 7.2.1 Scalar Quantities

Consider various dot products between the position, velocity, and acceleration. We can determine the behavior of these scalar quantities using only kinematics and the orthogonality of the force to the position and velocity.

r · r = r2 , (7.4) 1 d r · v = r2 , (7.5) 2 dt d r · a = (r · v) − v2 = 0 , (7.6) dt v · v = v2 , (7.7) 1 d v · a = (v · v) = 0 . (7.8) 2 dt

2 2 The magnitude of the velocity is conserved, v = v0. We can easily solve for how the magnitude of the position changes:

d (r · v) = v2 , dt 0 1 d r · v = v2t + r · v = (r2) . (7.9) 0 0 0 2 dt

92 r2 has a minimum. Without loss of generality, choose t = 0 at this minimum.

This implies that r0 · v0 = 0. The solution is

2 r · v = v0t , 1 d (r2) = v2t , 2 dt 0 2 2 2 2 r = r0 + v0t . (7.10)

There are two special cases where the motion is trivial.

If v0 = 0, then the force is also zero. If the electron begins at rest, it will remain at rest for all time.

If r0 = 0, then the electron moves through the origin. At the origin, the electron’s velocity must be entirely in the radial direction (or zero). As long as the electron’s velocity is in the radial direction, it will experience no force. The electron moves through the monopolium in a straight line at constant velocity.

For the rest of this calculation, we assume that r0 6= 0 and v0 6= 0.

7.2.2 Rotating Orthogonal Frame

There are three mutually orthogonal vectors which all have constant magnitude.

We’ve already found the ﬁrst one: the velocity.

The second one is v × r:

2 2 2 2 2 2 2 2 4 2 2 2 (v × r) = v r − (v · r) = v0(r0 + v0t ) − v0t = v0r0 . (7.11)

93 The third one is v × (v × r):

2 2 2 4 2 (v × (v × r)) = v0(v × r) = v0r0 . (7.12)

These three vectors are obviously orthogonal. If you divide them by their (constant) magnitudes, they are orthonormal. We use them as a basis xˆ0, yˆ0, zˆ0. The primes indicate that this is a rotating basis.

Each of these vectors satisﬁes the same diﬀerential equation. They each rotate with an angular velocity ω = −k(r) r:

d v = −k(r) r × v = ω × v , (7.13) dt d v × r = v˙ × r + v × r˙ = (−k(r) r × v) × r dt = −k(r) r × (v × r) = ω × (v × r) , (7.14) d v × (v × r) = ω × (v × (v × r)) . (7.15) dt

We do not show a direct calculation for the third basis vector. Because it is always orthogonal to the other two and can never pass through zero, it must rotate with them as well.

The three vectors rotate as a rigid body, so we can use the techniques of rigid body rotation.

Write ω in the body frame:

1 v2 1 r = 2 (r · v)v + 2 r − 2 (r · v)v v0 v0 v0 1 0 0 = vt − 2 v × (v × r) = v0txˆ − r0zˆ , (7.16) v0 0 0 ω = −k(r) r = −k(r) v0t xˆ + k(r) r0 zˆ . (7.17)

94 We can write this as a matrix multiplying each of our basis vectors instead of using cross products:

0 v˙i = ijkωjvk =: Ωikvk , (7.18)   0 −k(r) r0 0 0 ω =  k(r) r0 0 k(r) v0t  = −k(r) v0t J x + k(r) r0 J z . (7.19) 0 −k(r) v0t 0 We have also written Ω in terms of the basis of the Lie algebra so(3):

 0 0 0   0 0 1   0 −1 0  J x =  0 0 −1  , J y =  0 0 0  , J z =  1 0 0  . 0 1 0 −1 0 0 0 0 0 (7.20)

Look for an equation of motion for the rotation matrix. Any vector g, attached to the rigid body at time t, equals the rotation matrix at that time applied to the vector’s initial condition. Use this to ﬁnd an equation of motion for the rotation matrix:

g(t) = R(t) g(0) ,

g˙ (t) = R˙ (t) g(0) ,

ω(t) g(t) = R˙ (t) g(0) ,

ω(t) R(t) g(0) = R˙ (t) g(0) ,

R˙ = ω R . (7.21)

In this expression, Ω is expressed in terms of a ﬁxed basis. If instead, we want to express it in terms of the rotating basis, we would have to transform the

95 matrix:

ω0 = R|ωR , (7.22)

ω = Rω0R| ,

R˙ = Rω0 . (7.23)

7.2.3 Converting so(3) → su(2)

This problem becomes simpler if you think of it in terms of spinors instead of a rotation matrix.

The basis of the Lie algebra so(3) satisfy the commutation relations:

[J i, J j] = ijk J k . (7.24)

The Pauli spin matrices satisfy a similar commutation relation:

i i 1 1 i − 2σi, − 2σj = − 4 [σi,σσj] = − 4 2i ijk σk = ijk(− 2σk) . (7.25)

Since the Pauli spin matrices satisfy the same algebra as the basis of so(3), we can solve the problem using Pauli spin matrices and then convert it back into a rotation matrix.

Recall that the Pauli spin matrices are: 0 1 0 −i 1 0 σ = , σ = , σ = . (7.26) x 1 0 y i 0 z 0 −1

Instead of having our dynamical variables be the rotation matrix R, we will use u ∈ su(2) instead: i i r0 v0t u˙ = u(− 2 )(−k(r) v0tσσx + k(r) r0 σz) = − k(r) u . (7.27) 2 −v0t −r0

96 p 2 2 2 We know r(t) = ± r0 + v0t from (7.10), so k(r) is a known function of time for any particular monopolium distribution.

7.2.4 Weber Diﬀerential Equation

The solution will be much simpler if we focus on the special case where the monopole density is uniform, i.e. where f(r) = 1 and k(r) = k =

4πρmq/3mc. The general case is more diﬃcult, but since it is a linear system with time dependent coeﬃcients, it must also be integrable.

Take another time derivative to convert this into a second order diﬀer- ential equation:

i i u¨ = − k u˙ (−v tσσ + r σ ) − k u(−v σ ) 2 0 x 0 z 2 0 x k2 i = − u(−v tσσ + r σ )2 + kv uσ 4 0 x 0 z 2 0 x k2 i = − u(v2t2σ2 + r2σ2 − r v t(σ σ + σ σ )) + kv uσ 4 0 x 0 z 0 0 x z z x 2 0 x k2 i = − u(r2 + v2t2) 1 + kv uσ 4 0 0 2 0 x k2 i = − (r2 + v2t2) u + kv uσ . (7.28) 4 0 0 2 0 x

The matrix u is unitary since it is in su(2). Although general complex 2×2 matrices can have 8 independent variables, the unitary u can only have 3 independent variables. We can parametrize this using two complex numbers, z1 and z2 whose modulus squares add to one: z1 z2 2 2 u = , |z1| + |z2| = 1 . (7.29) −z¯2 z¯1

97 The complex conjugate of z1, for example, isz ¯1.

Do a preliminary calculation for the last term in (7.28): z1 z2 0 1 z2 z1 uσx = = . (7.30) −z¯2 z¯1 1 0 z¯1 −z¯2

Plug the parameterization for the unitary matrix (7.29) into the equations of motion (7.28) to determine equations for z1 and z2:

k2 i z¨ = − (r2 + v2t2) z + kv z , (7.31) 1 4 0 0 1 2 0 2 k2 i z¨ = − (r2 + v2t2) z + kv z , (7.32) 2 4 0 0 2 2 0 1 k2 i −z¯¨ = (r2 + v2t2)z ¯ + kv z¯ , 2 4 0 0 2 2 0 1 k2 i z¯¨ = − (r2 + v2t2)z ¯ − kv z¯ . 1 4 0 0 1 2 0 2

The third and fourth equations are just the complex conjugates of the second and ﬁrst equations. This is good because it ensures that our parametrization of u is still valid for all time.

We can make the two equations of motion independent by shifting to new variables:

a1 = z1 + z2 , a2 = z1 − z2 . (7.33)

k2 i a¨ = − (r2 + v2t2) a + kv a 1 4 0 0 1 2 0 1 1 2 2 1 2 2 2 i = − 4 k r0 − 4 k v0t + 2 kv0 a1 , (7.34) k2 i a¨ = − (r2 + v2t2) a − kv a 2 4 0 0 2 2 0 2 1 2 2 1 2 2 2 i = − 4 k r0 − 4 k v0t − 2 kv0 a2 . (7.35)

98 These look a lot like the Weber diﬀerential equation for the parabolic cylinder functions. We will convert them into explicitly that form [135, 77]:

d2u − ( 1 z2 − 1 − ν)u = 0 . (7.36) dz2 4 2

Shift to a new time variable:

τ 2 τ = eiαωt ⇒ τ 2 = e2iαω2t2 ⇒ t2 = e−2iα , (7.37) ω2 d dτ d d d2 d2 = = eiαω ⇒ = e2iαω2 . (7.38) dt dt dτ dτ dt2 dτ 2

Plug this new time variable into the equations of motion for a1 and a2 (7.35).

d2a 1 1 k2v2 i e2iαω2 1 = − k2r2 − e−2iα 0 τ 2 + kv a , dτ 2 4 0 4 ω2 2 0 1 d2a 1 1 k2v2 i e2iαω2 2 = − k2r2 − e−2iα 0 τ 2 − kv a , dτ 2 4 0 4 ω2 2 0 2 d2a 1 k2r2 1 k2v2 i kv 1 = −e−2iα 0 − e−4iα 0 τ 2 + e−2iα 0 a ,(7.39) dτ 2 4 ω2 4 ω4 2 ω2 1 d2a 1 k2r2 1 k2v2 i kv 2 = −e−2iα 0 − e−4iα 0 τ 2 − e−2iα 0 a .(7.40) dτ 2 4 ω2 4 ω4 2 ω2 2

We would like the coeﬃcient of τ 2 to be 1/4. This does not quite fully determine α and ω:

k2v2 p π − e−4iα 0 = 1 ⇒ ω = kv , α = ± . (7.41) ω4 0 4

99 Plug this partially determined parameter in for the new time variable:

2 2 d a1 ∓iπ/2 kr0 1 2 ∓iπ/2 i 2 = −e + τ + e a1 dτ 4v0 4 2 2 1 2 1 kr0 = τ ± ± i a1 , (7.42) 4 2 4v0 2 2 d a2 ∓iπ/2 kr0 1 2 ∓iπ/2 i 2 = −e + τ − e a2 dτ 4v0 4 2 2 1 2 1 kr0 = τ ∓ ± i a2 . (7.43) 4 2 4v0 We would also like the coeﬃcient of 1/2 to be negative. This selects the lower sign for the ﬁrst equation and the upper sign for the second equation. The new time variable is not the same for the two equations:

2 −iπ/4 iπ/4 kr0 τ1 = e ωt , τ2 = e ωt , p := . (7.44) 4v0

Note that τ2 =τ ¯1.

We have now converted the equations of motion (7.35) into the Weber diﬀerential equation (7.36):

2 d a1 1 2 1 2 = τ1 − − ip a1 , (7.45) dτ1 4 2 2 d a2 1 2 1 2 = τ1 − + ip a2 . (7.46) dτ2 4 2 We can see that ν in the Weber diﬀerential equation is ±ip for our equations.

The solutions are the hyperbolic cylinder functions. Each equation has four solutions, any two of which are linearly independent:

a1(τ1) = Dip(τ1),Dip(−τ1),D−ip−1(iτ1),D−ip−1(−iτ1) , (7.47)

a2(τ2) = D−ip(τ2),D−ip(−τ2),Dip−1(iτ2),Dip−1(−iτ2) . (7.48)

100 We will use the ﬁrst two.

7.2.5 Initial Conditions

We’ve already been using the initial conditions of the magnitudes r0 and v0. They appear in the new time variable and in the constant that appears in the Weber diﬀerential equation (7.41, 7.44).

The rest of the initial conditions can be stated in terms of the rotation matrix. Without loss of generality, choose the rotation matrix at t = 0 to be the identity matrix. At t = 0, the stationary and rotating frames coincide, with v pointing in the xˆ0 direction and r pointing in the −zˆ0 direction.

If the rotation matrix is the (3 × 3) identity matrix at t = 0, then u is the (2 × 2) identity matrix at t = 0. This determines the initial conditions of z1 and z2, which determine the initial conditions of a1 and a2:

z1(0) = 1 , z2(0) = 0 ⇒ a1(0) = 1 , a2(0) = 1 . (7.49)

Since we’ve converted this into a second order equation, we also need initial conditions for the ﬁrst derivatives. These can be found by evaluating the ﬁrst order equation at t = 0: i r0 v0t i u˙ (0) = − k u(0) = − k r0 σz , 2 −v0t −r0 2

i kr √ z˙ (0) = − kr , z˙ (0) = 0 ⇒ a˙ (0) =a ˙ (0) = −i 0 = −i p ω . (7.50) 1 2 0 2 1 2 2

101 We should also convert the t derivative to a τ derivative:

da1 iπ/4 1 iπ/4√ (0) = e a˙ 1(0) = −i e p =: µ1 , (7.51) dτ1 ω da2 −iπ/4 1 −iπ/4√ (0) = e a˙ 2(0) = −i e p =: µ2 . (7.52) dτ2 ω

The solution is a sum of two independent parabolic cylinder functions, with coeﬃcients that will be determined by the initial conditions:

a1 = αDip(τ1) + βDip(−τ1) , (7.53)

a2 = γD−ip(τ2) + δD−ip(−τ2) . (7.54)

The initial conditions for the parabolic cylinder functions involve Γ functions: 1 ν/2 1 (ν−1)/2 Γ( 2 ) 2 0 Γ(− 2 ) 2 Dν(0) = 1−ν ,Dν(0) = ν . (7.55) Γ( 2 ) Γ(− 2 ) Determine the coeﬃcients by setting the initial conditions for a, determined by the initial rotation matrix, to the initial conditions for the parabolic cylinder functions:

1 ip/2 Γ( 2 ) 2 1 = a1(0) = αDip(0) + βDip(0) = (α + β) 1−ip , Γ( 2 ) 1 −ip/2 Γ( 2 ) 2 1 = a2(0) = γD−ip(0) + δD−ip(0) = (γ + δ) 1+ip , Γ( 2 ) da Γ(− 1 ) 2(ip−1)/2 µ = 1 (0) = αD0 (0) − βD0 (0) = (α − β) 2 , 1 dτ ip ip ip 1 Γ(− 2 ) da Γ(− 1 ) 2−(ip+1)/2 µ = 2 (0) = γD0 (0) − δD0 (0) = (γ − δ) 2 , 2 dτ −ip −ip ip 2 Γ( 2 )

102 1 1 µ1 1 1 µ1 α = + 0 , β = − 0 , 2 Dip(0) Dip(0) 2 Dip(0) Dip(0) 1 1 µ2 1 1 µ2 γ = + 0 , δ = − 0 . (7.56) 2 D−ip(0) D−ip(0) 2 D−ip(0) D−ip(0)

7.2.6 Solution

We have integrated the problem. The solution of the motion of an electron in a very large, uniform density sphere of monopolium is

1 Dip(τ1) + Dip(−τ1) µ1 Dip(τ1) − Dip(−τ1) a1 = + 0 , (7.57) 2 Dip(0) 2 Dip(0)

1 D−ip(τ2) + D−ip(−τ2) µ2 D−ip(τ2) − D−ip(−τ2) a2 = + 0 . (7.58) 2 D−ip(0) 2 D−ip(0)

To make sense of this solution, you have to undo all of the steps that we took to get here:

• Plug in the various expressions for the parameters in terms of the initial conditions and magnitude of the magnetic ﬁeld:

1 ν/2 1 (ν−1)/2 Γ( 2 ) 2 0 Γ(− 2 ) 2 Dν(0) = 1−ν ,Dν(0) = ν , ν = ±ip , (7.59) Γ( 2 ) Γ(− 2 )

−iπ/4 iπ/4 iπ/4√ −iπ/4√ τ1 = e ωt , τ2 = e ωt , µ1 = −i e p , µ2 = −i e p , 2 kr0 p p = , ω = kv0 . (7.60) 4v0

• Convert from a to z:

1 1 z1 = 2 (a1 + a2) , z2 = 2 (a1 − a2) . (7.61)

103 • Recombine the z’s into u ∈ su(2):

z z u = 1 2 . (7.62) −z¯2 z¯1

• Convert this back into a rotation matrix:

1 | Rij = 2 Tr(σiu σju) or (7.63)  1 2 2 2 2 i 2 2 2 2  2 (z1 − z2 +z ¯1 − z¯2) 2 (−z1 − z2 +z ¯1 +z ¯2) −z1z2 − z¯1z¯2 i 2 2 2 2 1 2 2 2 2 R =  2 (z1 − z2 − z¯1 +z ¯2) 2 (z1 + z2 +z ¯1 +z ¯2) −i(z1z2 − z¯1z¯2)  . z1z¯2 +z ¯1z2 i(−z1z¯2 +z ¯1z2) z1z¯1 − z2z¯2

• Apply this rotation matrix to the basis elements at t = 0 to determine the basis at every time and write the position in terms of this basis:

0 0 r = v0t xˆ − r0 zˆ = v0t Rxˆ − r0 Rzˆ . (7.64)

This is a giant mess, so we won’t do it in general.

7.3 Asymptotics

What happens when t → ∞?

We suspect that the particle will travel out in a particular direction. We would like to determine what direction that is.

104 7.3.1 Parabolic Cylinder Functions

The parabolic cylinder functions have well known asymptotics that depend on the phase of their argument [77]: 2 ν(ν − 1) 3π D (z) ≈ e−z /4zν 1 − + ... for | arg z| < , (7.65) ν 2z2 4 2 ν(ν − 1) D (z) ≈ e−z /4zν 1 − + ... ν 2z2 √ 2π 2 (ν + 1)(ν + 2) − eνπiez /4z−ν−1 1 + + ... Γ(−ν) 2z2 5π π for > arg z > , (7.66) 4 4 2 ν(ν − 1) D (z) ≈ e−z /4zν 1 − + ... ν 2z2 √ 2π 2 (ν + 1)(ν + 2) − e−νπiez /4z−ν−1 1 + + ... Γ(−ν) 2z2 π 5π for − > arg z > − . (7.67) 4 4

To determine the direction the particle travels as t → ∞, we can drop everything except the leading order term.

Since ν is purely imaginary, z±ν is O(1) and z±ν−1 is O(1/ωt). The one term that remains is the same for any arg(z):

−z2/4 ν Dν(z) ≈ e z . (7.68)

Do some preliminary calculations:

2 2 2 2 2 2 e−z /4 = eiω t /4 , e−z /4 = e−iω t /4 , z=±τ1 z=±τ2

105 ip zν = e−iπ/4ωt = epπ/4(ωt)ip , z=τ1,ν=ip ip zν = e3iπ/4ωt = e−3pπ/4(ωt)ip , z=−τ1,ν=ip −ip zν = eiπ/4ωt = epπ/4(ωt)−ip z=τ2,ν=−ip −ip zν = e−3iπ/4ωt = e−3pπ/4(ωt)−ip . z=−τ2,ν=−ip

The relevant expansions of parabolic cylinder functions are:

iω2t2/4 pπ/4 ip Dip(τ1) ≈ e e (ωt) , (7.69)

iω2t2/4 −3pπ/4 ip Dip(−τ1) ≈ e e (ωt) , (7.70)

−iω2t2/4 pπ/4 −ip D−ip(τ2) ≈ e e (ωt) , (7.71)

−iω2t2/4 −3pπ/4 −ip D−ip(−τ2) ≈ e e (ωt) . (7.72)

Notice that the later two expressions are the complex conjugates of the ﬁrst two.

7.3.2 Spinors

Use these expansions to determine the asymptotics for the spinors, i.e. the rotation matrix in su(2).

Note:

pπ/4 −3pπ/4 −pπ/4 pπ/2 −pπ/2 −pπ/4 pπ e + e = e e + e = 2 e cosh( 2 ) ,

pπ/4 −3pπ/4 −pπ/4 pπ/2 −pπ/2 −pπ/4 pπ e − e = e e − e = 2 e sinh( 2 ) . (7.73)

106 Write the asymptotic solution for a1 and a2:

1 iω2t2/4 ip pπ/4 −3pπ/4 a1 = e (ωt) e + e 2Dip(0) µ1 iω2t2/4 ip pπ/4 −3pπ/4 + 0 e (ωt) e − e 2Dip(0) pπ pπ iω2t2/4 ip −pπ/4 cosh( 2 ) iπ/4√ sinh( 2 ) = e (ωt) e − i e p 0 , (7.74) Dip(0) Dip(0)

1 −iω2t2/4 −ip pπ/4 −3pπ/4 a2 = e (ωt) e + e 2D−ip(0) µ2 −iω2t2/4 −ip pπ/4 −3pπ/4 + 0 e (ωt) e − e 2D−ip(0) pπ pπ −iω2t2/4 −ip −pπ/4 cosh( 2 ) −iπ/4√ sinh( 2 ) = e (ωt) e − i e p 0 .(7.75) D−ip(0) D−ip(0)

These solutions go like e±it2 . They oscillate with increasing frequency for large times.

Convert from a to z:

1 z1 = 2 (a1 + a2) iω2t2/4 ip −iω2t2/4 −ip ! −pπ/4 pπ 1 e (ωt) e (ωt) = e cosh( 2 ) + 2 Dip(0) D−ip(0)

iπ/4 iω2t2/4 ip −iπ/4 −iω2t2/4 −ip ! √ −pπ/4 pπ 1 e e (ωt) e e (ωt) −i p e sinh( 2 ) 0 + 0 2 Dip(0) D−ip(0)

" iω2t2/4 ip # −pπ/4 pπ e (ωt) = e cosh( 2 ) Re Dip(0)

" iπ/4 iω2t2/4 ip # √ −pπ/4 pπ e e (ωt) −i p e sinh( 2 ) Re 0 , (7.76) Dip(0)

107 1 z2 = 2 (a1 − a2) iω2t2/4 ip −iω2t2/4 −ip ! −pπ/4 pπ 1 e (ωt) e (ωt) = e cosh( 2 ) − 2 Dip(0) D−ip(0)

iπ/4 iω2t2/4 ip −iπ/4 −iω2t2/4 −ip ! √ −pπ/4 pπ 1 e e (ωt) e e (ωt) −i p e sinh( 2 ) 0 − 0 2 Dip(0) D−ip(0)

" iω2t2/4 ip # −pπ/4 pπ e (ωt) = i e cosh( 2 ) Im Dip(0)

" iπ/4 iω2t2/4 ip # √ −pπ/4 pπ e e (ωt) + p e sinh( 2 ) Im 0 . (7.77) Dip(0)

This notation is too messy to use to convert from the spinor to the rotation matrix. Introduce four new real variables. R and I refer to whether they contain the real or imaginary parts of the time dependence. The primes refer to whether the term involves the initial condition of the parabolic cylinder function or its derivative:

0 0 z1 = R − iR , z2 = iI + I , (7.78)

" iω2t2/4 ip # −pπ/4 pπ e (ωt) R := e cosh( 2 ) Re , (7.79) Dip(0)

" iπ/4 iω2t2/4 ip # 0 √ −pπ/4 pπ e e (ωt) R := p e sinh( 2 ) Re 0 , (7.80) Dip(0)

" iω2t2/4 ip # −pπ/4 pπ e (ωt) I := e cosh( 2 ) Im , (7.81) Dip(0)

" iπ/4 iω2t2/4 ip # 0 √ −pπ/4 pπ e e (ωt) I := p e sinh( 2 ) Im 0 . (7.82) Dip(0)

108 The spinor and its Hermitian conjugate are:

0 0 z1 z2 R − iR iI + I u = = 0 0 , (7.83) −z¯2 z¯1 iI − I R + iR R + iR0 −iI − I0 u† = . −iI + I0 R − iR0

7.3.3 Rotation Matrix

There are multiple ways of calculating the rotation matrix (7.64). All of them involve a lot of messy algebra. We will use the explicit expression for each component:

 1 2 2 2 2 i 2 2 2 2  2 (z1 − z2 +z ¯1 − z¯2) 2 (−z1 − z2 +z ¯1 +z ¯2) −z1z2 − z¯1z¯2 i 2 2 2 2 1 2 2 2 2 R =  2 (z1 − z2 − z¯1 +z ¯2) 2 (z1 + z2 +z ¯1 +z ¯2) −i(z1z2 − z¯1z¯2)  . z1z¯2 +z ¯1z2 i(−z1z¯2 +z ¯1z2) z1z¯1 − z2z¯2

The components will appear simpler if we introduce the new complex variables:

iω2t2/4 ip −pπ/4 pπ e (ωt) T := R + iI = e cosh( 2 ) , Dip(0) iπ/4 iω2t2/4 ip 0 0 0 √ −pπ/4 pπ e e (ωt) T := R + iI = p e sinh( 2 ) 0 . (7.84) Dip(0)

The time dependence vanishes if we consider |T |2, |T 0|2, or TT¯ 0. The time dependence does not vanish if we take any other product of two of these variables.

109 Evaluate each term individually:

1 2 2 2 2 2 2 2 2 02 02 2 0 2 2 (z1 − z2 +z ¯1 − z¯2) = Re[z1] − Re[z2] = (R + I ) − (R + I ) = |T | − |T | , i 2 2 2 2 2 2 0 0 0 2 (−z1 − z2 +z ¯1 +z ¯2) = Im[z1] + Im[z2] = −2RR + 2II = −2 Re[TT ] ,

0 0 0 −z1z2 − z¯1z¯2 = −2 Re[z1z2] = −2RI − 2R I = −2 Im[TT ] ,

i 2 2 2 2 2 2 0 0 ¯ 0 2 (z1 − z2 − z¯1 +z ¯2) = − Im[z1] + Im[z2] = 2RR + 2II = 2 Re[TT ] ,

1 2 2 2 2 2 2 2 2 02 02 2 02 2 (z1 + z2 +z ¯1 +z ¯2) = Re[z1] + Re[z2] = (R − I ) − (R − I ) = Re[T − T ] ,

0 0 2 02 −i(z1z2 − z¯1z¯2) = 2 Im[z1z2] = 2RI − 2R I = Im[T − T ] ,

0 0 ¯ 0 z1z¯2 +z ¯1z2 = 2 Re[z1z¯2] = 2RI − 2R I = 2 Im[TT ] ,

0 0 2 02 i(−z1z¯2 +z ¯1z2) = 2 Im[z1z¯2] = −2RI − 2R I = − Im[T + T ] ,

2 2 2 2 02 02 2 02 z1z¯1 − z2z¯2 = |z1| − |z2| = (R − I ) + (R − I ) = Re[T + T ] .

Put these together into the rotation matrix:

 1 − 2|T 0|2 −2 Re[TT 0] −2 Im[TT 0]  R =  2 Re[TT¯ 0] Re[T 2 − T 02] Im[T 2 − T 02]  . (7.85) 2 Im[TT¯ 0] − Im[T 2 + T 02] Re[T 2 + T 02]

7.3.4 Velocity

Since the velocity is one of the three vectors which form the rotating basis, its time dependence is simple:

v(t) = Rv(0) = v0Rxˆ . (7.86)

Only the ﬁrst column is needed to determine the velocity. This is also the column for which the time dependence cancels. The other vectors in the

110 basis oscillate with an increasing frequency. As t → ∞, the velocity approaches a single value:

−pπ/2 2 pπ vx 0 2 p e sinh ( 2 ) = 1 − 2|T | = 1 − 2 0 2 , (7.87) v0 |Dip(0)| iπ/4 vy ¯ 0 √ −pπ/2 e = 2 Re[TT ] = p e sinh(pπ) Re 0 , (7.88) v0 D−ip(0)Dip(0) iπ/4 vz ¯ 0 √ −pπ/2 e = 2 Im[TT ] = p e sinh(pπ) Im 0 . (7.89) v0 D−ip(0)Dip(0)

Recall the initial conditions of the parabolic cylinder functions:

1 ν/2 1 (ν−1)/2 Γ( 2 ) 2 0 Γ(− 2 ) 2 Dν(0) = 1−ν ,Dν(0) = ν , ν = ±ip . (7.90) Γ( 2 ) Γ(− 2 )

We will use some Γ function identities [77]:

1 √ 1 √ Γ( 2 ) = π , Γ(− 2 ) = −2 π , (7.91) π Γ(ix)Γ(−ix) = csch(πx) , Γ( 1 + ix)Γ( 1 − ix) = π sech(πx) . x 2 2

We can now explicitly write out the initial conditions of the parabolic cylinder function: 1 Γ( ip ) Γ(− ip ) = 2 2 (7.92) 0 0 1 −ip/2−1/2 1 ip/2−1/2 D−ip(0)Dip(0) Γ(− 2 ) 2 Γ(− 2 ) 2 1 π 1 = √ csch( pπ ) = csch( pπ ) , (−2 π 2−1/2)2 p/2 2 p 2

1 Γ( 1 + ip ) Γ(− ip ) = 2 2 2 (7.93) 0 1 −ip/2 1 ip/2−1/2 D−ip(0)Dip(0) Γ( 2 ) 2 Γ(− 2 ) 2 1 = √ √ Γ( 1 + ip )Γ(− ip ) π(−2 π)2−1/2 2 2 2 1 = −√ Γ( 1 + ip )Γ(− ip ) . 2 π 2 2 2

111 We can simplify the x component of the velocity more:

vx −pπ/2 2 pπ 1 pπ −pπ/2 pπ = 1 − 2 p e sinh ( 2 ) csch( 2 ) = 1 − e 2 sinh( 2 ) v0 p = 1 − e−pπ/2(epπ/2 − e−pπ/2) = e−pπ . (7.94)

√ 2 Recall p = kr0/4v0 and ω = kv0.

The three components of the asymptotic velocity are:

−πkr2/4v v = v e 0 0 , (7.95) x 0 √ r kv 2 2 h 2 2 i 0 0 −πkr /8v0 πkr0 iπ/4 1 ikr0 ikr0 vy = − √ e 0 sinh( ) Re e Γ( + ) Γ(− ) , π 8 4v0 2 8v0 8v0 √ r kv 2 2 h 2 2 i 0 0 −πkr /8v0 πkr0 iπ/4 1 ikr0 ikr0 vz = − √ e 0 sinh( ) Im e Γ( + ) Γ(− ) . π 8 4v0 2 8v0 8v0

The asymptotic velocity calculated here has the same magnitude as the initial velocity. This is guaranteed because the magnitude of the velocity is conserved, or we can check it directly.

7.3.5 Reversing Time

What happens if we let t → −∞ instead of t → +∞?

In the general solution, we have to replace τ1 → −τ1 and τ2 → −τ2.

This is not diﬃcult because our solutions are written in terms of both τ1,2 and

−τ1,2 in an almost symmetric expression:

1 Dip(τ1) + Dip(−τ1) µ1 Dip(τ1) − Dip(−τ1) a1 = + 0 , (7.96) 2 Dip(0) 2 Dip(0)

1 D−ip(τ2) + D−ip(−τ2) µ2 D−ip(τ2) − D−ip(−τ2) a2 = + 0 . (7.97) 2 D−ip(0) 2 D−ip(0)

112 The only eﬀect of negating time is changing the sign of the second

0 0 term. This is equivalent to replacing D±ip(0) → −D±ip(0). The rest of the calculation proceeds in the same way. The net result is T 0 → −T 0.

This means that the negative asymptotic rotation matrix is

 1 − 2|T 0|2 2 Re[TT 0] 2 Im[TT 0]  R =  −2 Re[TT¯ 0] Re[T 2 − T 02] Im[T 2 − T 02]  . (7.98) −2 Im[TT¯ 0] − Im[T 2 + T 02] Re[T 2 + T 02]

The second and third components of the asymptotic velocity vector change sign. We write the velocity as t → −∞ in terms of p to emphasize that the asymptotic directions only depend on one parameter:

−πp vx = v0 e , (7.99) √ p −πp/2 iπ/4 1 ip ip vy = √ e sinh(πp) Re e Γ( + ) Γ(− ) , π 2 2 2 2 √ p −πp/2 iπ/4 1 ip ip vz = √ e sinh(πp) Im e Γ( + ) Γ(− ) . π 2 2 2 2

7.4 Non-Hamiltonian Dynamics

The motion of an electron in a sphere of monopolium is exactly solvable and has simple asymptotics for both t → ∞ and t → −∞.

This solution is very unlike typical problems solved in classical mechanics. The Jacobi identity is satisﬁed nowhere. This system is completely non-Hamiltonian.

One obvious consequence is that Noether’s theorem no longer applies. This system is rotationally symmetric, so Noether’s theorem would suggest

113 that the angular momentum would be conserved. Instead, only the magnitude of the angular momentum is conserved (7.11). The angular momentum vector rotates, along with the velocity, as the electron moves.

Kinetic energy is also conserved. Only the magnetic force is involved. However, the force law (7.13) has secular time dependence from ω ∝ r. The result of this secular time dependence is that the velocity oscillates around its asymptotic value with decreasing amplitude and increasing frequency as the electron moves away from the origin.

Even in this simple, extremely symmetric problem, the weirdness from the failure of the Jacobi identity is apparent.

114 Chapter 8

Conclusion

The existence of magnetic monopoles disrupts the Hamiltonian nature of classical electromagnetism. When the locations of an electron and a monopole coincide, the Jacobi identity for the Poisson bracket is violated. This result is most obviously seen in plasma physics, where the huge number of particles interacting collectively makes collisions between species almost guaranteed.

Without the Jacobi identity, dynamics becomes almost unrecognizable. There is no Lagrangian, no Noether’s theorem, no canonical coordinates, no Hamilton-Jacobi equation, no symplectic geometry, and no KAM theorem. Extremely symmetric problems can still be solved analytically, but the solutions are not like the canonical solutions of classical mechanics.

There are two ways to recover the Jacobi identity, but neither is satisfac- tory. All species could have the same ratio of magnetic to electric charge. This is a duality transformation away from the universe we currently observe without monopoles. Alternatively, we could insist that electrons’ and monopoles’ positions never coincide. Why can’t monopoles collide with ordinary matter? How would this inﬂuence our attempts to detect them?

115 Traditional methods of quantization fail without the Jacobi identity, canonical coordinates, or a Lagrangian action principle. How should we quantize the interactions between arbitrary collections of electrically and magnetically charged particles?

Problems with the quantum theory of electromagnetism with magnetic monopoles have been known for decades [73]. These problems are not inher- ently quantum. The quantum theory merely inherits the problems caused by the failure of the Jacobi identity for the classical Poisson bracket.

Since there is no experimental evidence, the argument for magnetic monopoles is aesthetic. The failure of the Jacobi identity taints this beauty. We should remain skeptical of any theory of magnetic monopoles that does not address the failure of the Jacobi identity.

116 Part II

An Integral Transform for Kinetic Plasmas

117 Chapter 9

Summary for a General Audience

9.1 What is a Plasma?

The most visible example of a plasma is the sun. The core of the sun is hot enough that fusion reactions happen frequently, supplying the power for the sun to continue burning for billions of years. The surface of the sun is much cooler, but still hotter than anything we regularly experience on earth. Like the earth, the sun has a magnetic field. Unlike the earth, the sun’s magnetic field reverses directions every 11 years. Since the surface of the sun is strongly affected by the magnetic field, the number of sunspots and the amount of power the sun emits has an 11 year cycle. A small amount of matter continually boils off the surface of the sun and spreads out through the solar system as the solar wind. ‘Empty’ space is actually full of an extremely low density plasma. When this plasma interacts with the earth’s magnetic field and the plasma in the highest layers of the earth’s atmosphere, it forms aurora. Plasmas can also be seen in lightning and other electrical arcs.

Controlled fusion is the primary goal of plasma physics in laboratories.1

1For a good discussion of fusion research, accessible to a popular audience, read The Future of Fusion Science by Jason Parisi and Justin Ball [98].

118 If we could confine hydrogen in conditions like the core of the sun, we could burn hydrogen into helium to create tremendous amounts of usable energy with minimal fuel and waste. Bottling the core of the sun is difficult, but we hope to achieve it in less than 20 years.2 The experiment that would achieve this goal is called ITER. It is an large international collaboration, based in France, which plans on making its first plasma in 2025 and its first fusion plasma in 2035.3 Other contenders for fusion include SPARC and NIF. SPARC is a privately funded experiment being built in Massachusetts. NIF is at a US weapons lab and is taking a very different approach towards fusion power. Many other plasma experiments, located across the world, have helped us on our path to fusion.

Plasmas are often described as the ‘fourth state of matter.’ I am not particularly fond of this description because ‘state of matter’ has an ambigu- ous meaning. The best definition is that two states of matter are separated by a phase transition (e.g. boiling or freezing). But this definition fails in several ways. There are phase transitions within a state of matter. For example, many materials have a phase transition between a glass (a solid with its atoms arranged randomly) and a crystal (a solid with its atoms arranged in a repeating pattern). At sufficiently high pressures and temperatures, you can transform a gas into a liquid without going through a phase transition.

2There is always a lot of uncertainty in doing anything which has never been done before. There is a saying that “fusion is 30 years away and always will be.” We have reduced that to 20 years away (and always will be?). 3ITER’s ﬁrst plasma will consist entirely of deuterium. In 2035, ITER will switch to a mixture of deuterium and tritium that should produce signiﬁcant fusion power.

119 Some substances have a combination of properties of diﬀerent states of matter. Many foods are somewhere between solid & liquid (jello and mayonnaise come to mind in particular) and ﬁre is somewhere between gas & plasma. Creating a plasma is the next phase transition which occurs as you continue heating up a gas, but phase transitions shouldn’t automatically be called new states of matter.

In a plasma, electrons and ions are not bound together in atoms. Both the electrons and the ions can move freely, so all of the behaviors of ﬂuid mechanics4 are present in plasmas. Because electrons and ions are charged, the electric and especially the magnetic forces are also important. Plasmas contain all of the complexities of both ﬂuid mechanics and electromagnetism. They are one of the most complicated systems that we can currently hope to understand and solve.

There is one other condition which needs to be satisﬁed for something to be considered a plasma. First consider a counterexample. In a metal, electrons can move freely – it is a conductor. When you melt the metal, the ions can also move freely. Even though both the electrons and ions can move freely, we do not consider liquid metals to be plasmas.

In a plasma, particles (electrons or ions) primarily interact with electric and magnetic ﬁelds created by the collective motion of many particles. Particles do not primarily interact with their neighbors.5

4Fluids include both liquids and gases. 5Sometimes, people refer to ‘weakly coupled plasmas’ which satisfy this condition and

120 To understand this last criterion, it is helpful to think of an analogy. Think about walking into a room with 100 people in a bunch of small group conversations. Now think about walking into a room with 100 people singing. In both cases, there is a group of people who are interacting via sound. In the ﬁrst case, people are primarily interacting with their neighbors. Although there might be more information being conveyed locally, from the outside, all you hear is noise.6 In the second case, people are interacting primarily via sounds created by many people. To the outside observer, this situation has much more interesting collective dynamics.

How do we get particles to interact collectively to form a plasma? Atoms do not have choir directors.

How strongly the individual particles interact is obviously relevant, but we rarely have much control over this. All electrons have exactly the same charge and most plasmas contain mostly hydrogen. Hydrogen ions contain one proton, so each one has the same magnitude and opposite sign charge as an electron. Making a plasma from heavy elements is harder.

In a liquid metal, the density of particles is high. As an electron or ion moves, it is continually running into other particles. Instead of moving along a clear trajectory, it bounces randomly between collisions. When all of the particles are moving randomly, no collective ﬁelds form. The result is noise.

‘strongly coupled plasmas’ which do not. I prefer to call the ﬁrst ‘plasma’ and the second ‘warm dense matter.’ 6This is noise in both the technical and nontechnical senses.

121 High density substances are rarely plasmas.

When a plasma’s temperature is large, its particles move faster. Be- cause they have higher momentum, they are less aﬀected by collisions.7 Most plasmas have temperatures of about 10,000 K8 or much higher. High temperature also explains why the electrons are not bound to the atoms.

The plasma parameter measures the number of particles, on average, that any one particle is interacting with now. It increases with increasing temperature or decreasing density.9 Stronger interactions with your neighbors reduce the plasma parameter; weaker interactions with your neighbors means that the cumulative inﬂuence from the group is more important. For something to be a plasma, its plasma parameter must be large.

In a plasma, electrons and ions move freely and interact via electric and magnetic ﬁelds. Plasmas have high temperatures and low densities, so there are few collisions between particles. With only a few collisions, the dynamics is dominated by collective ﬁelds and motions, not by collisional noise.

7Head on collisions have a similar effect, even at higher speeds, but glancing collisions have much less effect. The net result is that there are fewer significant collisions. Also notice that, unlike cars, electrons and ions cannot be damaged by a collision unless nuclear reactions are involved. 8Or 10, 000◦C or 10, 000◦F. When the number is this large and approximate, the temperature scales are all about the same. 9The plasma parameter is typically called Λ. Its equation is

T 3/2 Λ = , (9.1) (4πn)1/2 e3 where T is the temperature, n is the density of particles, and e is the charge of the electron.

122 9.2 Multiple Velocities

When describing a fluid, the most important thing is the velocity field. At every location, there is a single velocity, like the arrows for wind on a weather map. Individual molecules may have different velocities, but they quickly revert back to close to the average velocity at this location because there are frequent collisions.10 No group of particles with a consistently different velocity can form. The group would quickly get disrupted by collisions with the other molecules in the fluid.11

The frequent collisions between particles explain why fluids can’t flow through each other. If you direct two streams of water at each other, they will splash off of each other. While mixing does occur, it is much slower.

In a plasma, collisions are much less signiﬁcant. If you direct two streams of plasma at each other, they will pass through each other. You might suspect that this is not a very stable situation, and you would be correct, but it does happen. Plasmas do not splash oﬀ of each other.

In the extremely low density plasmas that ﬁll space, particles at the

10The temperature of the fluid measures how close individual molecules are to the average velocity. At higher temperatures, there is more variation in the molecules’ velocity. Which atoms are faster and which are slower than average changes continually because of the collisions, so there are no systematic differences. 11We said above that collisions disrupt interesting collective dynamics that can occur in plasmas. Even though fluids do have many collisions, they also have interesting collective dynamics in their velocity fields. Although collisions disrupt deviations from the mean velocity, Galilean relativity forbids them from disrupting the mean velocity itself. If the collisions involved interactions that unambiguously specified which observers were still, e.g. if the fluid were flowing through porous rock or if Aristotelian mechanics were correct, then fluids would have fewer interesting collective behaviors.

123 same location might have come from many diﬀerent sources located light years apart, without interacting with anything else along their path. It should come as no surprise that the particles in these plasmas do not have a normal variation around an average velocity.

There are some plasma phenomena that can be explained using a ﬂuid description, which has only a single velocity at each location (or two velocities, one for electrons and one for ions). Magnetohydrodynamics has been the most successful ﬂuid theory of plasmas.

However, ﬂuid theories cannot fully capture the behaviors of plasmas. Instead, you should describe a plasma using the probability of particles having each velocity at each location. This is called the distribution function.12

In summary, ﬂuids can be described with a single velocity at every location because there are frequent collisions that disrupt any group of particles that deviates too much from normal. Plasmas have far fewer collisions, so particles with diﬀerent velocities can move past each other without directly interacting. To describe a plasma, you need to account for particles moving with independent velocities at the same location.

12The distribution function is referred to as f(x, v, t). It depends on position, velocity, and time. The distribution function changes according to the Vlasov-Maxwell equations (3.1).

124 9.3 Damping without Collisions

When a wave travels through a fluid, it gradually damps, and decreases in strength/amplitude. A sound wave gets weaker the farther it goes from its source – and not just because it spreads out over a larger area. A sound trapped in a box will echo around for a while, but not forever. The collisions between molecules in the fluid disrupt the coherent motion of the sound wave and convert it into random motion. The energy of the sound wave is converted into heat. The changes in density and pressure of the sound wave are stronger than the effects of collisions, so the waves do not damp quickly, but they do damp.

In plasmas, we see something similar. Electrostatic waves, which consist of changes in charge density and the electric ﬁeld, damp as they move through a plasma.13 This phenomenon is called ‘Landau damping.’

This is surprising because plasmas have very few collisions. Without the randomizing eﬀect of collisions, the dynamics of a plasma is reversible. When motion is reversible, it could happen either forward or in reverse. Playing a video of it backward would not look weird. Hamiltonian systems are all reversible.

How can something be reversible and damp?

The solution lies in what we described in the last section. In a plasma,

13Other types of waves exist in a plasma and can damp by a similar mechanism, but we will focus on small electrostatic oscillations in this dissertation.

125 groups of particles at the same location, but with diﬀerent velocities, move independently from one another.

In an electrostatic wave, there are more electrons (or ions) in some locations and fewer electrons in other locations. This charge separation creates the electric ﬁeld.

At one of the locations where there are more electrons, some of the electrons are moving faster than others. These electrons will not stay with the slower electrons that they started out with. Instead, they will overtake the slow electrons from a location which originally had fewer electrons. The original distinction between regions with too many or too few electrons gets smeared out because different electrons move at different speeds. The electric field, which is created by the charge separation, damps.

None of the energy or information is randomized by this process. The initial coherent structure gets converted to an increasingly complicated pattern of how many electrons are moving at each velocity in every location.

The damping is reversible. If we allow a wave in the plasma to damp, then input something that has the opposite effect on the moving electrons, then the electrons would gather back together in their bunches and the electric field would reappear, only to damp again. A simple way to achieve this is to input a similar wave going in the opposite direction. This ‘plasma echo’ was the first unambiguous evidence that Landau damping is correct [78].14

14Showing that a wave damps in a plasma is not conclusive because there are multiple

126 This simple description should actually be more complicated because the electric field influences the motion of the electrons. As the fast electrons travel between regions with more or fewer electrons, they are accelerated and decelerated by the electric field. If more electrons are accelerated than decelerated, they take energy from the field and the wave smears out. If more electrons are decelerated than accelerated, the wave can grow by taking energy from the electrons. When we have two streams of plasma going through each, it is unstable because a wave can take energy from the particles.

9.4 Unraveling Landau Damping

In this part of the dissertation, we present a previously developed mathematical technique, which we called the G-transform, which removes the electric ﬁeld from Landau damping. The simple description above is then correct. We can exactly solve for the behavior of the electrons. We then have to invert the G-transform to make predictions in the physical setting where the electric ﬁeld exists.

The original work in this part of the dissertation is using the G-transform to unravel Landau damping in more complicated situations where it is important.

Although plasmas do not have very many collisions, they do have a few. There are thus two damping mechanisms: collisions and Landau damping. We things that might disrupt coherent structures. Showing that a wave damps and then returns when you add another wave, at the predicted location, is much more conclusive.

127 can use the G-transform to get an explicit solution for this situation, like we did when we assumed that there were no collisions. This solution shows the interplay between the two types of damping.

Gyrokinetics is a simplified version of the Vlasov-Maxwell equations when there is a strong magnetic field. For most plasma experiments, it is simple enough to be solved on a supercomputer and complex enough to describe all of the relevant physics. Since gyrokinetics has Landau damping in the direction of the magnetic field, its behavior can be simplified with the G-transform.

Often, especially in space plasmas, we make measurements in one or a few locations and want to know what the plasma is doing at many locations. Since whatever structure we see will Landau damp, we need to be able to unravel this behavior in order to project our measurements from the observer’s location to determine what the plasma is doing elsewhere.

Landau damping is a phenomenon which is found in many guises throughout plasma physics. The G-transform discussed in this dissertation provides an exact solution for the motion of electrons.

128 Chapter 10

Introduction

10.1 Overview

One of the intriguing features of the Vlasov-Poisson system is the abil- ity of an electrostatic wave to damp, even in the absence of any dissipation mechanism [70]. This can occur because the electrostatic wave couples to a continuum of modes describing all of the possible ways that the distribution function can vary in velocity space.1

The linearized Vlasov-Poisson system can be solved using the Laplace transform [70, 71], normal modes [134], a Green’s function [132, 23], or numerically by a variety of codes [25, 50, 102, 63]. Another method for describing Landau damping, which employs an integral transform in velocity space based on the Hilbert transform, was introduced in Refs. [92, 86, 88]. Applying this ‘G-transform’ to the linearized Vlasov-Poisson equation completely removes the electric ﬁeld term. The resulting advection equation is trivially solved, and then the inverse transform is applied to return the solution to the original coordinates.

1Much of the material in Part II has been previously published in J. M. Heninger and P. J. Morrison. An integral transform technique for kinetic systems with collisions. Physics of Plasmas, 25:082118, 2018.

129 Versions of this integral transform exist for a variety of models [89], and there is one that can be used eﬀectively on any collisionless kinetic model with one velocity dimension.

A simple and important way to extend the linearized Vlasov-Poisson system is to add collisions. The Landau-Boltzmann collision operator for Coulomb collisions is usually unmanageable, so simpler collision operators are often used instead. A good collision operator, such as the Fokker-Planck operator [111, 38, 24, 71, 32], acts as a diffusion in velocity space, conserves particle number, and has a Maxwellian equilibrium. Typically, the collision frequency is assumed to be small. Even if the collision frequency is small, collisions shouldn’t be completely neglected. Landau damping produces fine structures in velocity space for which the diffusion eventually becomes important.

The picture of Landau damping is similar to Kolmogorov turbulence: energy which is input on large velocity scales cascades down to smaller velocity scales until it reaches a dissipation velocity scale where the collisions are important. There is also an intermediate velocity scale where collisions are important for particles with small parallel velocities [138]. Since the damping rate typically depends more on the creation of fine structures in velocity space than on the details of the collision operator, replacing the complete collision operator with a simplified version is reasonable. Landau damping causes the effective dissipation rate to be much higher than the collision frequency. The importance of the fine structures in determining the dissipation rate can be seen by noticing that if the initial conditions have fine structures in velocity

130 space, then the damping rate can be much higher than if the initial conditions diﬀer from a Maxwellian in a smoother way [102].

Gyrokinetics and drift-kinetics are a natural place to look for applications for the G-transform because they only have dynamics in one velocity dimension. In many applications, the plasma has a strong magnetic field. A particle’s motion perpendicular to this field mostly follows small circles around the field lines. This motion does not need to be resolved. Instead, the three-dimensional Vlasov-Poisson equations are integrated over the perpendicular velocity dimensions. Drift velocities which depend on the fields can be added to preserve the important aspects of the perpendicular motion. The resulting system has three spatial dimensions, one velocity dimension and time [39, 19, 64].

The G-transform has been used to analyze data from laboratory experiments [113, 121, 120, 119]. Doppler-resolved laser induced fluorescence measures the dependence of the distribution function on the velocity parallel to the laser beam. G-transforming the data at a single point yields the amplitudes of the van Kampen modes excited by an externally created wave. The spatial dependence of the electric field can be determined from the van Kampen mode amplitudes using a measurement of the velocity distribution function at a single point. The G-transform could also be used on satellite data to measure Landau damping and nonlinear energy transfer between fields and particles in the heliosphere [56, 57].

131 10.2 Properties of the Hilbert Transform

The G-transform is built using the Hilbert transform, so we will review a few properties of the Hilbert transform before proceeding.

The Hilbert Transform is deﬁned by

1Z g(u) H[g](v) := − du , (10.1) π R u − v where R− is the Cauchy principle part of the integral. The integrand is singular when u = v, so we integrate u from −∞ to v − and from v + to +∞, then let → 0.

Below is a list of some basic Hilbert transform properties. The proofs we show are well-known (see e.g. Ref. [61]):

1. H[...] is a linear operator. Proof: Obvious.

2. If g(v) has a Hilbert transform, then there is a function g, analytic in the upper half complex v-plane, that limits to the real v-axis as

g(v) = gR(v) + i gI (v) , (10.2)

where the real and imaginary parts of g are related by

gR = H[gI ] , gI = −H[gR] . (10.3)

Proof: Suppose that the function g(u) is analytic in the upper half plane and decays faster than 1/u as u → ∞ in the upper half plane.

132 Figure 10.1: Contour for the proof of Hilbert transform property2.

Consider the integral I g(u) du (10.4) u − v around the contour shown in ﬁgure 10.1. Since the function g is analytic in this region, the contour integral is zero. The contour can be split into three parts. The integral over the semicircle over the upper half plane is zero because of the decay condition we impose on g. The rest of the integral can be split into the principal value of the integral plus the integral over the semicircle next to the pole. The integral over the semicircle next to the pole is half the integral around the pole, with a minus sign because we are going around it clockwise. The residue is simple to see when the integral is written in this form:

I g(u) Z g(u) 0 = du = − du − iπg(v) , u − v R u − v 1 Z g(u) g(v) = − du . (10.5) iπ R u − v

133 Separate this into its real and imaginary parts: Z 1 gR(u) + igI (u) gR(v) + igI (v) = − du iπ R u − v 1Z g (u) 1Z g (u) = − I du − i − R du π R u − v π R u − v = H[gI ](v) − iH[gR](v) . (10.6)

The real part of this equation is gR = H[gI ]. The imaginary part of this

equation is gI = −H[gR].

3. The inverse of the Hilbert transform is negative itself:

H[H[g]] = −g . (10.7)

Proof: Immediately follows from Hilbert transform property2:

gR = H[gI ] = −H[H[gR]] . (10.8)

4. For two functions g1 and g2, the following convolution identity holds:

H[g1H[g2] + g2H[g1]] = H[g1]H[g2] − g1g2 . (10.9)

Proof: This follows from Hilbert transform property2, applied to the product of the two analytic functions. Construct two analytic functions,

g1 = H[g1] + i g1 , g2 = H[g2] + i g2 , (10.10)

then multiply them:

g1g2 = (H[g1] + i g1)(H[g2] + i g2)

= (H[g1]H[g2] − g1g2) + i(g2H[g1] + g1H[g2]) . (10.11)

This property follows from setting H[(g1g2)I ] = (g1g2)R.

134 5. The Hilbert transform commutes with diﬀerentiation, i.e.

∂g ∂ H = H[g] . (10.12) ∂u ∂v

Proof: Integrate by parts, then switch the u-derivative for a v-derivative:

∂g 1Z ∂g du 1Z ∂ 1 H = − = − − g(u) du ∂u π R ∂u u − v π R ∂u u − v 1Z ∂ 1 = − − g(u) − du π R ∂v u − v ∂ 1Z du ∂ = − g(u) = H[g] . (10.13) ∂v π R u − v ∂v

6. The Hilbert transform has the following multiplication by u identity:

1Z H[ug] = v H[g] + − g(u) du . (10.14) π R Proof:

1Z u 1Z u − v + v H[ug] = − g(u) du = − g(u) du π R u − v π R u − v 1Z v 1Z g(u) 1Z = − g(u) + 1 du = − v du + − g(u) du π R u − v π R u − v π R 1Z = v H[g] + − g(u) du . (10.15) π R

7. The adjoint of the Hilbert transform is negative itself: Z Z − f H[g] dv = −− g H[f] dv . (10.16) R R Proof: Switch the order of integration and switch the names u ↔ v:

Z Z 1Z g(u) Z 1Z f(v) − f H[g] dv = − f(v) − du dv = −− g(u) − dv du R R π R u − v R π R v − u Z 1Z f(u) Z = −− g(v) − du dv = −− g H[f] dv . (10.17) R π R u − v R

135 8. The Hilbert transform reverses the parity of a function: h i Π H[f] = −Π[f] . (10.18)

Proof: Assume g has a well deﬁned parity, g(−u) = ±g(u). If it doesn’t, any function can be split into the sum of a symmetric part plus an antisymmetric part. We would like to show H[g](−v) = ∓H[g](v). 1Z +∞ g(u) H[g](−v) = − du Let u0 = −u . π −∞ u + v Z −∞ 0 Z +∞ 0 1 g(−u ) 0 1 ±g(u ) 0 = − 0 (−du ) = − − 0 du π +∞ v − u π −∞ u − v Z +∞ 0 1 g(u ) 0 = ∓ − 0 du = ∓H[g](v) . (10.19) π −∞ u − v

10.2.1 Plasma Z Function

The Hilbert transform of a Gaussian is the real part of the plasma Z function: u 1 2 2 1 Z 2 2 dv Z := πH √ e−v /vt = √ − e−v /vt . (10.20) vt π π R v − u From Hilbert transform Properties5 and6 and from our knowledge of the derivative of a Gaussian, we can determine that the derivative of the plasma Z function satisﬁes: ∂Z ∂ 1 2 2 √ ∂ 2 2 = π H √ e−v /vt = π H e−v /vt ∂u ∂u π ∂v √ √ 2v 2 2 2 π h 2 2 i 1 Z 2 2 −v /vt −v /vt −v /vt = π H − 2 e = − 2 uH e + e dv vt vt π R 2 u 1 2 2 2 u u = − πH √ e−v /vt + 1 = − 1 + Z . (10.21) vt vt π vt vt vt

136 Chapter 11

One-Dimensional Vlasov-Poisson

11.1 Equations of Motion

The Vlasov equation describes the evolution of the distribution function in phase space, f = f(x, v, t) , (11.1) which describes the probability density that a particle is at location x with velocity v at time t. We will focus on a system with only one velocity and one space dimension for now.

If there are no collisions between particles, then the distribution function obeys Liouville’s theorem in the 1+1+1 dimensional phase space:

df ∂f ∂f e ∂φ ∂f = + v − = 0 . (11.2) dt ∂t ∂x m ∂x ∂v

The relevant force pushing the particles comes from an electric ﬁeld, expressed

∂φ here using the electric potential E = − ∂x . The charge of the particles is e.

The electric ﬁeld has to be determined self-consistently using Poisson’s equation: ∂2φ Z − 2 = 4πe f dv − n0 . (11.3) ∂x R

137 There is also a ﬁxed background n0 (of ions for example) which is independent of position.

This is a nonlinear equation. The last term is nonlinear because φ depends on f through the Poisson equation. We will ﬁrst solve the linearized equation.

The Vlasov-Poisson equation has an equilibrium when f is independent of position. The background density was chosen so there is no electric ﬁeld. This is easy to achieve experimentally - for example, if all of the electrons and ions come from the ionization of an originally neutral gas. Since

∂f ∂v only appears multiplying the electric ﬁeld and the electric ﬁeld is zero at equilibrium, the distribution function can be any function of velocity. We also R factor out the background number density of the particles so f0 dv = 1. The equilibrium is

f(x, v, t) = n0f0(v) . (11.4)

Requiring that the equilibrium be stable puts the additional constraint that f0 monotonically decrease with the magnitude of v.

Consider a perturbation of this equilibrium. Fourier transform the perturbation in position. Write

ikx f(x, v, t) = n0f0(v) + n0fk(v, t) e . (11.5)

Since the equilibrium electric field is zero, the electric field is first order. Also Fourier transform the first order electric potential:

ikx φ = φk(t) e . (11.6)

138 Plug this into the Vlasov-Poisson system, and drop the nonlinear terms:

∂δf e ∂f k + ikvf − ikφ 0 = 0 , (11.7) ∂t k m k ∂v Z 2 k φk = 4πen0 fk dv . (11.8) R It is now easy to merge the Vlasov-Poisson system into a single integro- diﬀerential equation:

2 Z ∂fk ωP 0 ∂f0 + ikv fk − i fk dv = 0 , (11.9) ∂t k R ∂v where ωP is the plasma frequency:

4πe2n ω2 := 0 . (11.10) P m

We will solve this equation using an integral transform based on the Hilbert transform.

◦ Refer to the initial conditions as f k(v) . Throughout this dissertation, initial conditions are indicated by a circle over the variable.

11.2 G-Transform Deﬁnition and Inverse

The G-transform is deﬁned by

f(v) = G[g](v) := R(v) g(v) + I (v) H[g](v) , (11.11) where ω2 ∂f (v) (v) := −π P 0 and (v) := 1 + H[ ](v). (11.12) I k2 ∂v R I

139 In order for an integral transform to be useful, it has to have an inverse. The inverse of the G-transform is (u) (u) g(u) = G−1[f](u) = R f(u) − I H[f](u) , (11.13) |(u)|2 |(u)|2

2 2 2 where || := R + I .

Check that g(u) = G−1[G[g]](u): 1 G−1[ g + H[g]] = ( ( g + H[g]) − H[ g + H[g]]) . (11.14) R I ||2 R R I I R I Focus on the second term. Use Hilbert transform Properties1 and4:

H[Rg + I H[g]] = H[(1 + H[I ])g + I H[g]] = H[g] + H[H[I ]g + I H[g]]

= H[g] + H[I ]H[g] − I g = RH[g] − I g . (11.15)

Plug this back into (11.14): 1 G−1[ g + H[g]] = ( ( g + H[g]) − ( H[g] − g)) R I ||2 R R I I R I 1 = (2 g + H[g] − H[g] + 2g) = g . (11.16) ||2 R R I I R I This is the correct inverse.

11.3 Solution of the Linear Vlasov Equation using the G-Transform

The G-transform provides an exact solution to the one-dimensional linear Vlasov equation [92, 86, 88]. The transform takes you to new coordinates that naturally unravel the phase mixing. In these new coordinates, the electric ﬁeld term vanishes, so the equation is trivial.

140 Applying the inverse G-transform in velocity space to (11.9) gives:

2 Z ∂ −1 −1 ωP 0 −1 df0 G [fk] + ik G [vfk] − i fk dv G = 0 . (11.17) ∂t k R dv We have used the fact that the G-transform and its inverse are both linear and do not eﬀect time.

−1 Deﬁning gk := G [fk], the ﬁrst term is simple, while the other two terms will take a little bit of work.

Use Hilbert transform property6 on the second term. The correction term is the integral of a nonsingular quantity, so we can remove the principal value from the integral:

G−1[vf (v)] = R uf (u) − I H[vf (v)] k ||2 k ||2 k Z R I 1 = 2 ufk(u) − 2 uH[fk(v)] + − fk(v) dv || || π R Z R I 1 I = u 2 fk(u) − 2 H[fk(v)] − 2 fk(v) dv || || π || R Z −1 I = u G [fk] − 2 fk(v) dv . (11.18) π|| R

The third term can be dealt with by recognizing that df0/dv is only a constant away from I and that H[I ] = R − 1:

2 2 −1 df0 −1 k k −1 G = G − 2 I = − 2 G [I ] dv πωP πωP 2 k R I = − 2 2 I − 2 (R − 1) πωP || || 2 k I = − 2 2 . (11.19) πωP ||

141 When we plug these results back into (11.17), we observe the remarkable cancellation:

Z 2 Z 2 ∂gk I ωP 0 k I = −ik ugk − 2 fk dv + i fk dv − 2 2 ∂t π|| R k R πωP || = −ikugk . (11.20)

The solution of (11.20) is trivial,

◦ −ikut gk(u, t) = gk(u) e . (11.21)

The one-dimensional linear Vlasov equation can be solved by ﬁrst in- ◦ verse G-transforming the initial conditions to get gk(u), using the solution (11.21), and then G-transforming back into the original coordinates,

◦ −1 −ikut fk(v, t) = G G [fk(v)] e . (11.22)

This solution is equivalent to van Kampen’s solution, which in turn is equivalent to Landau’s [134].

As a particular example, pick the equilibrium to be Gaussian in v. Choose initial conditions which involve a spatial sinusoidal variation of the density. The perturbation’s velocity dependence is initially the same Gaussian as the equilibrium. We plot the analytical solution, both forward and backward in time, in figure 11.1. In this plot, ωP = 10, k = 2π, vt = 1. Larger ωP makes the linear electric field term larger, so the G-transform has a more significant effect.

142 Figure 11.1: The analytical solution (11.22) for Gaussian equilibrium and initial conditions which are Gaussian in v and sinusoidal in x. All times are measured in units of 1/kvt. The initial conditions are at the top. The left column shows negative times: t = −1, −2, −4. The right column shows positive times: t = 1, 2, 4. 143 11.4 Landau Damping

The solution (11.21) must include Landau damping. If the equilibrium distribution function is monotonically decreasing, then the spatial dependence of a perturbation (and the electric ﬁeld) decays. How does Landau damping appear in this solution?

The distribution function itself doesn’t damp – it could not because this is a Hamiltonian system. Instead, what damps are the density and electrical ﬁeld perturbations, which are proportional to the integral of the distribution function.

The relationship between the integral of a quantity and the integral of its G-transform is simple. Hilbert transform property7 gives Z Z Z f dv = G[g] dv = (Rg + I H[g]) dv R ZR R Z = (g + H[I ]g + I H[g]) dv = g dv . (11.23) R R The density, or the velocity integral of anything else, is the same in both the original velocity coordinates and in the G−1-transformed velocity coordinates.

The density is

Z ◦ Z ◦ −1 −ikut −1 −ikvt nk(t) = G G [fk](u) e dv = G [fk](v) e dv . (11.24) R R Look for the damping in the long time limit.

The Riemann-Lebesgue lemma [125] determines that the following long time limit for any suﬃciently smooth function F (ζ) vanishes: Z lim F (ζ) e−iζt dζ = 0 . (11.25) t→∞ R

144 The integral in (11.24) is of the form speciﬁed in the Riemann-Lebesgue Lemma, if we assume that the initial conditions are suﬃciently smooth and ||2 is never zero along the real axis. We can conclude,

lim nk(t) = 0 . (11.26) t→∞

The density perturbation of the plasma (and thus the electric ﬁeld) decays to zero as time → ∞.

This can be seen in figure 11.1. The density and electric field are proportional to the integral over velocity of the distribution function – an integral along a vertical line in these plots. After a long time (relative to 1/kvt), there has been significant phase mixing, so integrals over velocity contain ap- proximately equal amounts of positive and negative perturbations from the equilibrium. The distribution function does not damp. Instead, it becomes more uniform in space while getting increasingly fine structure in velocity.

The Riemann-Lebesgue Lemma also tells us how the density will decay in terms of how smooth the integrand is. If the function F (ζ) in (11.25) is Ck, then the integral with decay at least as fast as t−k. If F (ζ) is analytic for real ζ, then the integral will decay at least exponentially fast. The decay rate for this exponential is the distance from the real axis to the nearest pole of F (ζ). In our case (11.24), the poles of F (ζ) are the places where ||2 = 0 because the inverse G-transform involves dividing by ||2. Landau’s solution displays this decay rate as the dominant term in his inverse Laplace transform.

145 11.5 Intuition

At this point, the G-transform feels a bit like magic.

At ﬁrst glance, it looks arbitrary, but when you apply it to an integro- diﬀerential equation, it removes everything but advection.

Where did this integral transform come from? Why would we think that this could simplify the linearized Vlasov-Poisson equations?

11.5.1 Symmetric Form

Plugging in the deﬁnitions of I , R, and the Hilbert transform allows us to write the G-transform in a much more symmetric way:

G[g](v) = (1 + H[I ](v))g(v) + I (v)H[g](v)

= g(v) + H[I ](v)g(v) + +I (v)H[g](v) Z 2 1 ωP ∂f0(u) du = 1 + − −π 2 g(v) π R k ∂u u − v 2 Z ωP ∂f0(v) 1 du + −π 2 − g(u) k ∂v π R u − v 2 Z ωP ∂f0(u) ∂f0(v) du = g(v) − 2 − g(v) + g(u) . (11.27) k R ∂u ∂v u − v Although this doesn’t explain where the G-transform comes from, revealing the symmetry makes it feel less arbitrary.

11.5.2 Complex Functions

Hilbert transform property2 showed us that Hilbert transforms connect the real and imaginary parts of analytic functions. The G-transform and

146 its inverse look much simpler when written in terms of these entire analytic functions.

If the equilibrium f0 is analytic along the real axis, then e := R + iI is analytic in the upper half plane.

If we assume that the solution fk is analytic along the real line, we can use the Hilbert transform to construct another analytic function in the upper half plane:

f := fk − i H[fk] . (11.28)

In terms of these two functions, the G-transform is simply:

G[fk] = Re e f = Re (R + iI )(fk − i H[fk]) = Re (Rfk + I H[fk]) + i(I fk − RH[fk])

= Rfk + I H[fk] . (11.29)

Similarly, the inverse G-transform is 1 G−1[f ] = Re e? f k |e|2 1 = Re ( − i )(f − i H[f ]) |e|2 R I k k 1 = Re ( f − H[f ]) − i( f + H[f ]) |e|2 R k I k I k R k 1 = ( f − H[f ]) . (11.30) ||2 R k I k

If we were to work entirely in terms of the complete analytic functions, the G-transform would simply be multiplying by e. The inverse of the G-

147 transform is immediately obvious: multiply by e?, then divide by |e|2. The complications arise because we want to solve for fk = Re[f], not f.

11.5.3 Van Kampen Modes

A general integral operator is typically written as the integral of some kernal acting on the original function. Write the G-transform in this form:

G[g](v) = R(v) g(v) + I (v) H[g](v) Z 1Z g(u) = δ(u − v) R(v) g(u) du + I (v) − du R π R u − v Z 1 1 = − R(v) δ(u − v) + I (v) g(u) du . (11.31) R π u − v The kernal of this operator is a van Kampen mode.

Van Kampen was the ﬁrst to solve the linearized Vlasov-Poisson equation as an eigenvalue problem [134]. He found that, for stable equilibrium distribution functions, there is a continuum of modes:

1 1 K(u, v) = (v) δ(u − v) + (v) . (11.32) R π I u − v

These modes are singular, so they are never seen experimentally. Instead, you see the initial value problem. An initial wave mode with nonzero electric ﬁeld and density perturbation damps as all of its energy converts to the continuum of modes associated with the velocity dependence of the distribution function and never returns.

The G-transform is a projection onto van Kampen modes. Instead of taking the inverse G-transform of the equations of motion (11.17), we could

148 instead write: Z fk(v) = G[gk](v) = − K(u, v) gk(u) du , (11.33) R and substitute this into the Vlasov-Poisson equations. We would then have to move the (forward) G-transform outside of the equations of motion to get the simple equation for gk(u).

11.5.4 Hamiltonian

The G-transform was originally introduced to convert the linearized Vlasov-Poisson system into a canonical Hamiltonian system, then to action- angle variables [92, 90]. The action-angle variables (Jα, θα) are deﬁned in terms of the transformed distribution function. When I < 0 , s s 16| | 16| | g (u) = eiθα I J , g (u) = e−iθα I J , (11.34) k kV ||2 α −k kV ||2 α and when I > 0 , s s 16| | 16| | g (u) = e−iθα I J , g (u) = eiθα I J . (11.35) k kV ||2 α −k kV ||2 α

In terms of these variables, the Poisson bracket is canonical,

X Z ∞ δF δG δG δF {F,G} = du − , (11.36) δθα δJα δθα δJα α(k) −∞ and the Hamiltonian is a functional of the Jα alone. The dynamics is trivial.

149 Chapter 12

Collisions

12.1 Fokker-Planck Collision Operator

Real plasmas have collisions between particles.

The complete description of the plasma would involve the motion of every individual particle. However, a N + N + 1 dimensional phase space is untenable. Instead, we make approximations of the collision operator which can be added to the 1 + 1 + 1 dimensional system of (11.9).

We now add a collision operator to the right hand side of the Vlasov- Poisson dynamics (11.9): 2 Z ∂fk ωP 0 df0 + ikvfk − i fkdv = C[f] . (12.1) ∂t k R dv Since (12.1) is no longer Hamiltonian, an attractor might exist. Any good collision operator C[f] will have asymptotic stability to a Maxwellian distribution. A common choice for describing collisions is the Fokker-Planck operator [111, 38, 24, 71, 32]: v2 ∂2f ∂f ∂ v2 ∂f C[f] := ν t k + v k + f = ν t k + vf . (12.2) 2 ∂v2 ∂v k ∂v 2 ∂v k This operator gives zero when it acts on the Maxwellian

1 −v2/v2 f0(v) = √ e t . (12.3) πvt

150 If the advection and electric ﬁeld terms are set to zero, then any other initial function of velocity will decay to this Maxwellian.

If ν is small, we might be tempted to treat the collision operator as a perturbation on the original problem. This is not easy because we are faced with a singular perturbation. The small parameter multiplies the highest derivative of f with respect to v. If f has structure on extremely small velocity scales, then the highest derivative of f can become O(1/ν), making a conventional perturbation theory illegitimate. For the Vlasov-Poisson system, we are guaranteed that f will eventually get ﬁne structure in v, since it behaves as e−ikvt for large t.

This motivates our use of the G-transform to attack this problem.

12.2 G-Transform and Collisions 12.2.1 ’s for a Maxwellian Equilibrium

If there are no collisions, there is a continuum of possible equilibria. Adding the Fokker-Planck collision operator selects (12.3) as the only equilibrium. We no longer work with general I (v) and R(v). Instead we can be

151 content with a special case of the Maxwellian, where

2 2 ωP ∂f0 ωP 2v 1 2 2 √ −v /vt I = −π 2 = −π 2 − 2 e k ∂v k vt πvt 2 √ 2ω v 2 2 P −v /vt = π 2 2 e , (12.4) k vt vt 2 2 ωP ∂f0 ωP ∂ 1 2 2 √ −v /vt R = 1 + H[I ] = 1 + H −π 2 = 1 − π 2 H e k ∂v k ∂u πvt 2 2 ωP ∂ v ωP 2 v v = 1 − 2 Z = 1 − 2 − 1 + Z k vt ∂v vt k vt vt vt vt 2 2 2ωP 2ωP v v = 1 + 2 2 + 2 2 Z . (12.5) k vt k vt vt vt

Z is the real part of the plasma Z function (10.20).

12.2.2 Properties of the Collision Operator

We ﬁrst state some properties of the collision operator. When the collision operator interacts with the G-transform, we will need to know how the collision operator acts on I , R, the Hilbert transform, and products.

1. C[...] is a linear operator. Proof: Obvious.

2. The collision operator acting on I is simple:

C[I ] = −νI . (12.6)

152 2 2 Proof: Recall I ∝ df0/dv ∝ v exp[−v /vt ] and consider its derivatives:

2 ∂ 2 2 v 2 2 −v /vt −v /vt ve = 1 − 2 2 e , ∂v vt 2 2 ∂ 2 2 v v v 2 2 −v /vt −v /vt 2 ve = −4 2 − 2 2 1 − 2 2 e ∂v vt vt vt 3 v v 2 2 −v /vt = −6 2 + 4 4 e . (12.7) vt vt

Apply the collision operator to this:

2 2 2 2 vt ∂ 2 2 ∂ 2 2 2 2 C[ve−v /vt ] = ν (ve−v /vt ) + v (ve−v /vt ) + ve−v /vt 2 ∂v2 ∂v 2 3 v v v 2 2 t −v /vt = ν −6 2 + 4 4 e 2 vt vt 2 ! v 2 2 2 2 −v /vt −v /vt +v 1 − 2 2 e + ve (12.8) vt 3 3 v v 2 2 2 2 −v /vt −v /vt = ν −3v + 2 2 + v − 2 2 + v e = −νve . vt vt

−v2/v2 Since I ∝ v e t , we can also conclude that the collision operator

acting on I is just −ν times itself. 3. The product rule for the collision operator is

∂A ∂B C[AB] = B C[A] + A C[B] − νAB + νv2 . (12.9) t ∂v ∂v

153 Proof: Use the Leibniz rule several times and rearrange:

v2 ∂2 ∂ C[AB] = ν t (AB) + v (AB) + AB 2 ∂v2 ∂v

v2 ∂2A ∂A ∂B ∂2B = ν t B + 2 + A 2 ∂v2 ∂v ∂v ∂v2 ! ∂A ∂B +v B + A + AB ∂v ∂v v2 ∂2A ∂A ∂A ∂B = ν t B + v B + AB + ν v2 2 ∂v2 ∂v t ∂v ∂v v2 ∂2B ∂B +ν t A + vA + AB − ν AB 2 ∂v2 ∂v ∂A ∂B = B C[A] + A C[B] − νAB + ν v2 . (12.10) t ∂v ∂v

4. The collision operator commutes with the Hilbert transform:

C[H[A]] = H[C[A]] . (12.11)

Proof: Use Hilbert transform properties5 and6 and note that everything decays at inﬁnity:

v2 ∂2 ∂ C[H[A]] = ν t H[A] + νv H[A] + νH[A] 2 ∂v2 ∂v v2 ∂2A ∂A = ν t H + νvH + νH[A] 2 ∂u2 ∂u 2 2 Z vt ∂ A ∂A ν ∂A = νH 2 + νH u − − du + νH[A] 2 ∂u ∂u π R ∂u v2 ∂2A ∂A = νH t + u + A = H[C[A]] . (12.12) 2 ∂u2 ∂u

5. The collision operator’s eﬀect on R is

C[R] = ν (2 − R) . (12.13)

154 Proof: Use R’s deﬁnition (11.12) and collision operator property2:

C[R] = C 1 + H[I ] = ν + H C[I ]

= ν − νH[I ] = ν (2 − R) . (12.14)

12.2.3 Commuting the G-Transform and Collision Operator

We now apply the inverse G-transform in velocity space to (12.1). Be- cause we have already seen the cancellation that occurs with the Vlasov part of this equation, we need only examine its aﬀect on the collision operator:

∂g k + ikug = G−1 [C[G[g ]]] . (12.15) ∂t k k

We will have to determine the commutation relations between the G-transform and the collision operator.

In order to evaluate C[G[gk]], we ﬁrst use linearity (collision operator property1). Next we use the product rule (collision operator property3) on each term, and then apply the collision operator on I using collision operator property2 and on R using collision operator property5,

C[G[gk]] = C[Rgk] + C[RH[gk]] , ∂ ∂g C[ g ] = C[g ] + g C[ ] − ν g + νv2 R k R k R k k R R k t ∂v ∂v ∂ ∂g = C[g ] + 2ν(1 − )g + νv2 R k , R k R k t ∂v ∂v ∂ ∂H[g ] C[ H[g ]] = C[H[g ]] + H[g ] C[ ] − ν H[g ] + νv2 I k I k I k k I I k t ∂v ∂v ∂ ∂g = H[C[g ]] − 2ν H[g ] + νv2 I H k . (12.16) I k I k t ∂v ∂u

155 We can also commute the collision operator with the Hilbert transform using collision operator property4, and note that a derivative can move inside of a Hilbert transform using Hilbert transform property5. We obtain

∂ ∂g C[G[g ]] = C[g ] + 2ν(1 − )g + νv2 R k k R k R k t ∂v ∂v ∂ ∂g + H[C[g ]] − 2ν H[g ] + νv2 I H k I k I k t ∂v ∂u = RC[gk] + I H[C[gk]] + 2νgk − 2ν(Rgk + I H[gk]) ∂ ∂g ∂ ∂g +νv2 R k + I H k t ∂v ∂v ∂v ∂u

= G[C[gk]] + 2ν(gk − G[gk]) ∂ ∂g ∂ ∂g +νv2 R k + I H k . (12.17) t ∂v ∂v ∂v ∂u

Using this result in (12.15), we can determine how the collision operator interacts with the G-transform:

∂g k + ikug = G−1 [C[G[g ]]] (12.18) ∂t k k −1 = G G[C[gk]] + 2ν(gk − G[gk]) ∂ ∂g ∂ ∂g +νv2 R k + I H k t ∂v ∂v ∂v ∂u −1 = C[gk] + 2νG [gk] − νgk ∂ ∂g ∂ ∂g +2νv2 G−1 R k + I H k . t ∂v ∂v ∂v ∂u

When dealing with the G−1 of the last term, we use Hilbert transform prop-

156 ∂R ∂I erty4 to simplify the H[...H[...]] term. Note that ∂v = H[ ∂v ].

∂ ∂g ∂ ∂g G−1 R k + I H k ∂v ∂v ∂v ∂u

1 ∂ ∂g ∂ ∂g = R k + I H k ||2 R ∂v ∂v ∂v ∂u ! ∂ ∂g ∂ ∂g + H H I k + I H k (12.19) I ∂v0 ∂u ∂u ∂v0 ! 1 ∂ ∂g ∂ ∂g ∂ ∂g ∂ ∂g = R k + I H k + H I H k − I k ||2 R ∂v ∂v R ∂v ∂u I ∂u ∂u I ∂v ∂v 1 ∂ ∂ ∂g ∂ ∂ ∂g = R − I k + I + R H k . ||2 R ∂v I ∂v ∂v R ∂v I ∂v ∂u

Finally, we obtain the G-transformed one-dimensional Vlasov equation with collisions:

∂g k + ikug = C[g ] + 2ν(G−1[g ] − g ) (12.20) ∂t k k k k νv2 ∂ ∂ ∂g ∂ ∂ ∂g + t R − I k + I + R H k . ||2 R ∂u I ∂u ∂u R ∂u I ∂u ∂v

The left hand side of this equation is simply advection: the electric ﬁeld term has vanished. The right hand side of this equation has the collision operator in terms of u, but it also has other terms. The rest of the right hand side describes how the electric ﬁeld and the collisions interact. We will call all of them the shielding term, S[gk].

12.3 Dropping the Shielding Term

What have we gained by doing the G−1-transformation?

157 At first, it doesn’t look like we have gained very much. If the collision operator is not there, the G-transform transforms an integro-differential equation into a differential equation. The equations of motion are dramat- ically simpler since they are local in k and v. However, the shielding term that arises from the G-transform of the collision operator is also nonlocal. We have replaced an integro-differential equation with another integro-differential equation. And, the new one looks more complicated.

However, if we knew that these terms are small, it would be a reasonable approximation to drop all of the complicated terms on the right hand side and add them in later as a perturbation.

In most physical situations when the Vlasov equation is relevant, the collision frequency ν is assumed to be smaller than the other frequencies in the system: ∂g νg k , kug . (12.21) k ∂t k All of the terms on the right hand side are proportional to ν, so they all appear to be smaller than the terms on the left hand side of (12.21). This suggests that it might be possible to treat the collision term as a perturbation in the original problem and not worry about G-transforming the collision operator.

Having a small collision frequency is not enough to make sure that the right hand side remains small. The collision operator has a term proportional

2 2 to ∂ gk/∂u , which is the highest derivative with respect to u in the problem. Although the left hand side may be originally dominant, the dynamics create

158 small-scale structure in velocity space, with terms proportional to exp[ikvt]. These small scales in velocity space make the collision operator signiﬁcant, even for small collision frequencies.

Note, unlike the Fourier transform, the Hilbert transform preserves scale. If g(u) = H[f(v)](u) and f is rapidly varying, say f(v/λ) for λ << 1, then we also have g(u/λ). Thus rapid variation is also preserved by the G- transform: a solution with a rapid scale of variation exp[ikut] in u will have a scale of variation exp[ikvt] in v, and vice versa.

The highest derivative in the shielding term is ∂gk/∂u. When a function has extremely small scales, its higher order derivatives are larger than its lower order derivatives. Even when the right hand side of this equation is important, the collision operator still dominates the shielding term.

This same argument could also be used for the last two terms of the collision operator (12.2). They both are also multiplied by the small parameter ν and neither has a second derivative. If we are justiﬁed in dropping the shielding terms, then we are also justiﬁed in dropping the latter two terms of the collision operator.

We thus have two reasonable approximations that we could make: drop the shielding terms and keep the entire collision operator or drop everything except the second derivative. We call the second derivative by itself D[gk] for diﬀusion and everything else can be lumped together into an expanded set of

0 shielding terms S [gk].

159 If we keep only the second derivative, the resulting solution is easier to solve and analyze analytically. However, it’s long-time limit is strange. The full collision operator relaxes to a Maxwellian with width vt. The collision operator D results in a Gaussian whose width increases without bound. This is not too big of a concern for us since our perturbations will decay on a much shorter time than the collision time 1/ν. A comparison of the analytical solutions for both of these reasonable approximations will be considered in section 12.5, after we show and analyze the simpler solution. The two solutions agree anywhere they are signiﬁcantly diﬀerent from zero.

The shielding term is a good term to initially ignore, and then add back in as a perturbation.

12.4 Analytically Solving Advection-Diﬀusion

The arguments of section 12.3 imply that we can proceed with the partial diﬀerential (but not integral) equation:

∂g v2 ∂2g k + ikug = ν t k , (12.22) ∂t k 2 ∂u2 which can be exactly solved.

12.4.1 Fourier Transform and Method of Characteristics

First Fourier transform in velocity:

∂gˆ ∂gˆ 1 − k + νv2η2gˆ = 0 , (12.23) ∂t ∂η 2 t

160 (n) whereg ˆ := F[gk ]. This reduces the equation to being ﬁrst order in time and velocity instead of being ﬁrst order in time and second order in velocity.

We can then use the method of characteristics to solve this problem with the Fourier transformed initial conditions:

◦ t(r, 0) = 0 , η(r, 0) = r , gˆ(r, 0) = gˆ(r) , (12.24) dt dη dgˆ 1 = 1 , = −k , = − νv2η2gˆ . (12.25) ds ds ds 2 t

The ﬁrst two are trivial to solve:

t = s , η = r − ks . (12.26)

The third can be solved using an integrating factor:

d 1 (logg ˆ) = − νv2(−ks + r)2 , ds 2 t s 1 Z s 1 1 2 2 0 2 02 0 2 2 2 2 3 logg ˆ = − νvt (r − 2rks + k s ) ds = − νvt (r s − rks + k s ) , 0 2 0 2 3 ◦ − 1 νv2r2s+ 1 νv2rks2− 1 νv2k2s3 gˆ(r, s) = gˆ(r) e 2 t 2 t 6 t . (12.27)

Now substitute back in the solutions for s(t) and r(η, t):

◦ − 1 νv2(η+kt)2t+ 1 νv2(η+kt)kt2− 1 νv2k2t3 gˆ(t, η) = gˆ(η + kt) e 2 t 2 t 6 t ◦ − 1 νv2t(η2+ktη+ 1 k2t2) = gˆ(η + kt) e 2 t 3 . (12.28)

We could immediately Fourier transform back in velocity to get (12.38) or we could ﬁrst plug in some initial conditions, then Fourier transform back in velocity.

161 One simple initial condition is Gaussian in u:

◦ 1 2 2 ◦ 1 2 2 −u /vt −η vt /4 gk(u) = √ e ⇒ gˆ(η) = √ e . (12.29) πvt 2π Plug this in:

1 − 1 v2( 1 η2+ktη+ 1 k2t2) − 1 νv2t(η2+ktη+ 1 k2t2) gˆ(t, η) = √ e 2 t 2 2 e 2 t 3 , (12.30) 2π 1 v2 1 + νt 2 1 2 + νt gˆ(t, η) = √ exp − t (1 + 2νt) η + kt − k2v2νt3 . 2π 4 1 + 2νt 12 t 1 + 2νt Then inverse Fourier transform to get the solution.

Given the equation (12.22) and an initial condition which is Gaussian in velocity and has a wave number k in position, the solution to (12.22), is

2 1 1 u 1 + νt 1 2 2 3 1 + 2 νt g(u, t) = √ √ exp − 2 −ikut − vt k νt . πvt 1 + 2νt vt (1 + 2νt) 1 + 2νt 6 1 + 2νt (12.31)

12.4.2 Checking the Solution

It is easy to evaluate the terms in (12.22) on the solution:

2 ∂gk ν u iku 2 2 = 2 2 2 − (1 + 2νt + 2ν t ) ∂t (1 + 2νt) vt ν 1 − k2v2t2(1 + νt)2 − (1 + 2νt) g , (12.32) 2 t

A[gk] = iku gk , (12.33) 2 ν u 1 2 2 2 2 D[gk] = 2 2 2 − vt k t (1 + νt) (1 + 2νt) vt 2 −(1 + 2νt) + 2ikut(1 + νt) gk , (12.34) and see that the equations of motion are satisﬁed. An existence and uniqueness theorem guarantees that this is the only solution.

162 Figure 12.1: The solution of linearized Vlasov-Poisson with collisions for Gaus- sian equilibrium and initial conditions which are Gaussian in u and sinusoidal in x. The analytical solution (12.31) (left) and numerical solution (right) look almost identical. The solution is plotted at times 0, 0.5, 1.0, 1.5 tD. More details about the numerical solution are discussed in the text in section 12.7.1. 163 12.4.3 Analysis of the Solution

The exponential of (12.31) has three terms. The first term is the result of the diffusion part of the equation. If we ignore advection here, the remaining heat equation will cause a Gaussian to spread out in time. The variance in velocity space increases, but the amplitude of the perturbation does not change. The time scale for this process is 1/ν. The perturbation will already have decayed away before this increased variance becomes significant.

The second term is primarily the result of the advection part of the equation. It is a velocity dependent phase shift of the initial conditions. This doesn’t change the amplitude of the perturbation to the distribution function. As we saw in section 11.4, it does result in Landau damping for the perturbation to the density and electric ﬁeld.

The rate at which this occurs is modulated by an extra factor resulting from the collision operator. This factor is unity at t = 0 and decays to 1/2 as t → ∞. This change occurs with a time scale 1/ν, so the perturbation will already have damped away before its effect becomes significant. However, as we will see in section 12.6, this effect can occur much sooner for other initial conditions.

The third term is the result of the interaction between the two terms. It is also where the damping of the perturbation to the distribution function comes from. Neither term individually damps the distribution function, but their interaction does.

164 This term is modulated by a factor which varies from 1 at t = 0 to 1/4 as t → ∞. The time scale over which this happens is 1/ν, so it will be close to 1 until the perturbation has almost entirely decayed.

If we deﬁne the decay time, tD, for this equation to be the time for the perturbation to the distribution function to decay to 1/e of its original value, and then approximate the modulating factor to be 1, we ﬁnd

1/3 1 2 2 3 6 − 1 = − k vt νtD ⇒ tD = 2 2 . (12.35) 6 k vt ν

Since this goes as ν−1/3, this time is much shorter than the time scale associated with the collisions by themselves.

Shortly after the ﬁrst experiment demonstrating the existence of the plasma echo [78], Su and Oberman investigated the eﬀects of adding the Fokker-Planck collision operator [126]. They were able to show that the perturbation should decay as e−νt3 , which implies a decay time proportional to ν−1/3. Our derivation assumes that the shielding term is small instead of doing singular perturbation theory for small ν, and we solve to arbitrary order in that parameter in Section 12.8. Similar problems and results continue to be of interest in increasingly complicated physical situations [23].

12.4.4 Advection-Diﬀusion Crossover

We have already plugged our solution into each of the terms of the equations of motion (12.22). This can tell us when each of the terms dominates – when advection dominates the damping and when diﬀusion dominates the

165 damping.

Nondimensionalizing the ratio of the diﬀusion term to the advection term using ε = ν/kvt , τ = kvtt , and ξ = u/vt gives

D[gk] −i ε 2 1 2 2 = 2 2ξ − τ (1 + ετ) − (1 + 2ετ) + 2iξτ(1 + ετ) . (12.36) A[gk] ξ(1 + 2ετ) 2

In Fig. 12.2, we plot the magnitude of the ratio of (12.36) for various nondimensionalized times and velocities to see when each term dominates and when the two terms are comparable. For this plot, we ﬁx ε = 0.1.

Figure 12.2: The magnitude of the ratio of the collision term to the advection term, plotted as a function of the non-dimensional time τ and velocity ξ.

The advection term dominates for small times and large velocities. The diﬀusion term dominates for large times and small velocities. For suﬃciently

166 large velocities, diﬀusion starts becoming important again, but the distribution function is so small at these velocities that this is not signiﬁcant.

12.5 Comparing Collision Operators

After G-transforming the Fokker-Planck collision operator, we get the same collision operator plus some corrections which we call the shielding terms. The shielding terms are expected to be small because they are multiplied by the small parameter ν and do not contain the highest order velocity derivative. Two of the terms in the collision operator are also the same order of magnitude as the shielding operator. We have a choice between dropping everything but the second velocity derivative term or just dropping the shielding term. Both equations can be solved analytically for general initial conditions.

If we drop everything except the second velocity derivative, the resulting equation, ∂g v2 ∂2g k + iku g = ν t k , (12.37) ∂t k 2 ∂u2 ◦ with initial conditions gk(u), has solution:

Z 0 2 1 0 ◦ 0 h i 0 (u − u ) 1 2 2 3i gk(t, u) = √ √ du gk(u ) exp − k(u+u )t− 2 − k vt νt . πvt 2νt 2 2vt νt 24 (12.38) When we plug in Gaussian initial conditions (12.29), we get the solution ana- lyzed before (12.31).

If we drop the shielding terms, but keep the entire collision operator,

167 the resulting equation,

∂g v2 ∂2g ∂g k + iku g = ν t k + u k + g , (12.39) ∂t k 2 ∂u2 ∂u k

◦ with initial conditions gk(u), has solution: Z 1 0 ◦ 0 gk(t, u) = √ √ du gk(u ) (12.40) −2νt πvt 1 − e −νt 0 −νt 2 2 2 −νt h k 0 1 − e (u − u e ) k vt 1 − e νti exp − i (u + u ) −νt − 2 −2νt + 2 −νt − . ν 1 + e vt (1 − e ) ν 1 + e 2 When we plug in Gaussian initial conditions (12.29), we get:

1 h k u2 k2v2 i √ −νt t −νt νt gk(u, t) = exp − i u(1 − e ) − 2 + 2 (1 − e )(3 − e ) − 2νt . πvt ν vt 4ν (12.41)

We know that the solutions will decay at a rate given by (12.35), which is much shorter than 1/ν. By the time νt gets close to one, the perturbation will have already decayed to close to zero.

To compare these solutions, expand (12.41) for small νt.

1 − e−2νt ≈ 2νt , (12.42) 1 − eνt νt νt ν3t3 = tanh ≈ − . (12.43) 1 + eνt 2 2 24

When we substitute these in and drop anything higher than ﬁrst order in ν, we ﬁnd that (12.41) agrees with (12.38).

Some of the terms we drop also contain velocity. For example, we drop ikuν2t3/24. This term could become large at suﬃciently large velocities:

24 u (12.44) & kν2t3

168 Since ν is a small parameter, this will be large compared to vt until well past the decay time. Since any reasonable initial condition decays at large velocity, the distribution function here will be negligibly small.

The two solutions are almost equal whenever the distribution function is significantly different from zero. Since the shielding terms are the same order of magnitude as the difference between these two collision operators, this analysis gives further indication of the legitimacy of dropping them.

12.6 Other Initial Conditions 12.6.1 Diﬀerent Thermal Velocity

The perturbation to the distribution function could be a very diﬀerent temperature than the equilibrium temperature that appears in the collision operator. Let us choose as an initial condition in u a Gaussian with a diﬀerent initial thermal velocity vI :

1 −u2/v2 gk(u, 0) = √ e I . (12.45) πvI

We will not show the calculation because it follows the same steps as before.

2 2 Deﬁne α := vI /vt . The solution is

2 1 1 u α + νt 1 2 2 3 3α + 2 νt g(u, t) = √ √ exp − 2 −ikut − vt k νt . πvt α + 2νt vt (α + 2νt) α + 2νt 6 α + 2νt (12.46) This has the same asymptotic behavior as before, but the time it takes for the modulation factors to transition from close to unity to close to 1/2 is now α/ν.

169 This transition is observable if this time is shorter than the decay time, i.e.

1/3 α vI ν . tD ⇒ . . (12.47) ν vt kvt

In order to measure the modulation of the terms, you would have to use initial conditions with thermal velocities much smaller than the equilibrium thermal velocity.

12.6.2 Polynomial Prefactor

We could also consider initial conditions in u that are a polynomial times a Gaussian:

◦ 2 2 −u /vI gk(u) = P(u) e . (12.48)

The calculation proceeds as before. The only diﬀerence is that there is now a polynomial multiplying everything.

The Fourier transform of a polynomial times a Gaussian is another polynomial times a Gaussian. Call this new polynomial Pˆ:

◦ −η2v2/4 gˆ(η) = Pˆ(η) e I . (12.49)

The solution in terms of η is

− 1 v2( 1 η2+ktη+ 1 k2t2) − 1 νv2t(η2+ktη+ 1 k2t2) gˆ(t, η) = Pˆ(η + kt) e 2 I 2 2 e 2 t 3 . (12.50)

Once again, we have to take the Fourier transform of some Gaussian multiplied by a polynomial. The solution will be (12.46), multiplied by some polynomial in u and t.

170 An appropriately chosen set of polynomials, such as the Hermite polynomials, will form a complete basis. Any other bounded function that decays quickly enough in velocity can be written as a superposition of these functions. Our analysis of the solution in section 12.4.3 also applies for many initial conditions.

12.6.3 Realistic Initial Conditions

Gaussian initial conditions in velocity are not physically realistic in u space. The initial conditions (12.29) are commonly used in kinetics because they represent a purely spatial perturbation to the distribution function. We have transformed the velocity coordinate, so we should use diﬀerent initial conditions for the advection-diﬀusion equation:

◦ ◦ −1 −1 1 −v2/v2 gk(u) = G fk(v) = G √ e t . (12.51) πvt

We assume that the thermal velocity for the equilibrium is the same as the thermal velocity of the initial conditions. We have already calculated I

(12.4) and R (12.5). The Hilbert transform of these initial conditions is

◦ 1 v2 1 u H[f k] = H √ exp − 2 = Z . (12.52) πvt vt πvt vt

171 Thus the initial condition of (12.51) in G−1-transformed space is

◦ 1 ◦ ◦ g (u) = f − H[f ] k ||2 R k I k 2 2 1 2ωP 2ωP u u 1 2 2 √ −u /vt = 2 1 + 2 2 + 2 2 Z e || k vt k vt vt vt πvt 2 ! √ 2ω u 2 2 1 u P −u /vt − π 2 2 e Z k vt vt πvt vt

2 2 2ω2 √1 −u /vt P πv e (1 + 2 2 ) = t k vt , (12.53) A2 + B2 where

2 2 2 √ 2ω u 2 2 2ω 2ω u P −u /vt P P A := π 2 2 e ,B := 1 + 2 2 + 2 2 Z(u/vt) . (12.54) k vt vt k vt k vt vt

We use (12.53) in ﬁgure 12.3.

12.7 Numerically Solving Advection-Diﬀusion 12.7.1 Numerical Solution

We also numerically solve the advection-diffusion equation (12.22) to compare with our analytical solution. We use a finite difference method in x and v and implicit Euler in time. Our code is written in MATLAB, and is simple enough to be run on a personal laptop. This code was written with David Hatch.

The equilibrium distribution function is Gaussian:

1 −v2/v2 f0(v) = √ e t . (12.55) πvt

172 The units in velocity and position are chosen so vt = 1 and k = 2π. We use 300 cells in x, with periodic boundary conditions, and 300 cells in v that range between ±3 vt , with Dirichlet boundary conditions. In our units of time, the collision frequency is ν = 0.05 and the plasma frequency is ωP = 10. The decay time (12.35) for these parameters is tD = 1.45. The non-dimensional parameters for this run are

kv ω t = 40π , P = 200. (12.56) ν ν

We run the advection-diﬀusion equation for 300 time steps, until t = 1.5 tD, at which point the initial perturbation has almost completely decayed.

We ran this simulation for two initial conditions: Gaussian in u and Gaussian in v. The solution which was initially Gaussian in u is compared with our analytical solution from section 12.4.1 in figure 12.1. The initial conditions which are Gaussian in v,(12.53), are too messy to do analytical calculations. The solution, in both u and v, is shown in figure 12.3. Since the G-transform only depends on velocity, the two solutions must have the same spatial dependence. The only observable difference is that the distribution function has a narrower peak in u than in v.

173 Figure 12.3: The solution of linearized Vlasov-Poisson with collisions for Gaus- sian equilibrium and realistic initial conditions, which are Gaussian in v and sinusoidal in x. The solution is plotted at times 0, 0.5, 1.0, 1.5 tD. The solution is shown both as a function of u (left) and as a function of v (right).

174 To construct the realistic initial conditions (12.53), we used MATLAB’s built in Hilbert transform to numerically apply the G−1-transform to Gaussian initial conditions in the original coordinates. Any other initial conditions for the original coordinates could be similarly numerically G−1-transformed. After ﬁnding the solution, we numerically apply the G-transform to get the solution in the original velocity coordinate.

We emphasize that our finite difference solution of the advection-diffusion equation did not include the shielding term since it should be small. This assumption is checked below.

12.7.2 Comparing Advection and Diﬀusion

We can compare the sizes of various terms in both the original coordinates and in the G−1-transformed coordinates. Throughout this subsection, we use the solution with realistic initial conditions.

In both velocity coordinates, there is a time derivative, an advection term, and a Fokker-Planck collision operator. Even though the terms look the same, the solution is different in the two coordinates, so we will get different results when we evaluate the term on the solution. The original coordinates also have a term from the electric field and the transformed coordinates also have the shielding term.

We take the solution for the advection-diﬀusion equation and evaluate it on all of the terms in both coordinates. The shielding term, evaluated on the solution calculated without it, should be small. We also G-transform the

175 shielding term evaluated on the solution back into the original coordinates, so we can see how signiﬁcant what we neglected is there.

We L1 integrate all of these terms in both position and velocity and plot the resulting magnitudes of each as a function of time in Figure 12.4 and Figure 12.5.

Figure 12.4: The time dependence of the terms of the collisional Vlasov equation, L1 integrated in position and velocity, in the G−1-transformed coordinates. The terms are an advection term, a Fokker-Planck collision operator, and the shielding term. Neglecting the shielding term when ﬁnding the solution is justiﬁed since the shielding term is always at least an order of magnitude smaller than the dominant term.

For small times, the advection term is dominant in both the original and transformed coordinates. In the original coordinates, the electric ﬁeld

176 Figure 12.5: The time dependence of the terms of the collisional Vlasov equation, L1 integrated in position and velocity, in the original coordinates. The terms are an advection term, a Fokker-Planck collision operator, and an electric field term. We also G-transformed the shielding term evaluated on the solution to see how significant the dropped term is. It is always at least an order of magnitude below the dominant term. term is the same order of magnitude as the advection term. One way that we could have reached the (local) advection-diffusion equation is by dropping the electric field terms and not G-transforming. We see here that this plan is illegitimate. Increasing the plasma frequency increases the significance of the electric field at small times.

For large times, the collision term is dominant in both the original and

177 transformed coordinates. This is unsurprising since the advection term creates small scale structures in velocity space. The collision operator has the highest order velocity derivative in the equation, so it becomes more signiﬁcant when there is lots of small scale structure.

At an intermediary time, there is a crossover where collisions replace advection as the dominant term.

The shielding term is always at least an order of magnitude lower than the other terms. This gives us conﬁdence that neglecting it is reasonable.

12.8 Introducing the Shielding Term as a Perturbation

Since we know that the shielding terms and part of the collision operator will be small compared to the other terms, we can stick a small parameter, δ, in front of these terms and write the solution g as a series in this parameter,

(0) (1) 2 (2) gk = gk + δgk + δ gk + ··· . (12.57)

The resulting hierarchy of equations reads

∂g(0) k + ikug(0) = D[g(0)] , (12.58) ∂t k k ∂g(1) k + ikug(1) = D[g(1)] + S0[g(0)] , (12.59) ∂t k k k ∂g(2) k + ikug(2) = D[g(2)] + S0[g(1)] , (12.60) ∂t k k k . . where D and S0 were deﬁned in section 12.3.

178 The zeroth order equation is the one solved above in section 12.4.1. All of the equations of other orders are inhomogeneous versions of this one. The only addition is a known function of velocity, determined by the lower order solutions, added to the right hand side.

The initial conditions of the higher order terms are all taken to be zero, i.e., the initial conditions are entirely included in the zeroth order equation.

(n) Fourier transforming in velocity space, u → η, and lettingg ˆ := F[gk ] ˆ 0 (n−1) and S := F[S [gk ]] gives

∂gˆ ∂gˆ 1 − k + νv2η2gˆ = Sˆ(η) . (12.61) ∂t ∂η 2 t

We solve (12.61) using the method of characteristics, which gives the following set of diﬀerential equations for t(r, s), η(r, s), andg ˆ(r, s):

◦ t(r, 0) = 0 , η(r, 0) = r , gˆ(r, 0) = gˆ(r) , (12.62) dt dη dgˆ 1 = 1 , = −k , = − νv2η2gˆ + Sˆ(η) . (12.63) ds ds ds 2 t

These equations are straightforward to solve. The ﬁrst two are trivial:

t(r, s) = s , η(r, s) = r − ks . (12.64)

Next we plug these into the equation forg ˆ(r, s), and solve it using an integrating factor:

" s # 1 νv2 R η2(s0)ds0 d 2 t 1 νv2 R s η2(s0))ds0 gˆ e 0 = e 2 t 0 Sˆ(η(s)) ds

179 Since we explicitly know what η(r, s) is, we can evaluate the integrals in the exponentials. We cannot evaluate the ﬁnal integral unless we specify explicitly

0 (0) what F[S [gk ]] is. The solution is

Z s 00 − 1 νv2 R s η2(s0)ds0 1 νv2 R s η2(s000)ds000 00 00 gˆ(r, s) = e 2 t 0 e 2 t 0 Sˆ(η(s )) ds 0 h νv2 i = exp − t (k3s3 + 3k2s2r + 3ksr2) (12.65) 6k Z s hνv2 i exp t (k3s003 + 3k2s002r + 3ks00r2) Sˆ(r − ks00) ds00 . 0 6k

We now invert (12.64) to get s(t, η) and r(t, η),

s = t , r = η + kt , (12.66) and substitute these in to getg ˆ(η, t). Note that we do not substitute anything in for s00 since it is the integration variable. From this point onward, we refer to it as s. We obtain:

h νv2 i gˆ(η, t) = exp − t (3η2kt + 3ηk2t2 + k3t3) (12.67) 6k Z t hνv2 i exp t (3η2ks + 3ηk2s2 + k3s3) Sˆ(η + k(t − s)) ds . 0 6k

(1) Finally the inverse Fourier transform of (12.68) in velocity space yields gk (v, t). We will not write this out explicitly since our expressions are long enough already.

The higher order corrections can be done using exactly the same technique.

180 12.9 Other Collision Operators

The strategy for any collision operator is the same. We show that the commutator between the collision operator and the G-transform is small, then neglect it. The dramatic simpliﬁcation of the left hand side only creates a small correction of the right hand side.

If we choose some local collision operator with a ﬁnite number of velocity derivatives, say

∂n ∂ n−1 Q [g ] := c g + O g , (12.68) n k ∂vn k ∂v k then we may consider how this collision operator acts on the G-transform.

∂n ∂ n−1 Q [G[g ]] = c g + H[g ] + O g n k ∂vn R k I k ∂v k ∂ng ∂ng ∂ n−1 = c k + H c k + O g R ∂vn I ∂vn ∂v k ∂ng ∂ n−1 = G c k + O g ∂vn ∂v k ∂ n−1 = G[Q [g ]] + O g (12.69) n k ∂v k

n−1 The commutator between Qn and G is O [∂/∂v] gk. As long as the original collision operator is multiplied by a small collision frequency, this shielding term will always be much smaller than either the advection term or the collision term. It can be neglected. The only problems could arise when gk is close to the kernal of Qn and the velocity is small.

181 Chapter 13

Gyrokinetics and Drift-Kinetics

A three-dimensional highly magnetized plasma is often approximated by gyrokinetic or drift-kinetic equations [39, 19, 64]. Although the Vlasov- Maxwell equations provide a complete picture of the plasma’s behavior, they are intractable to solve – analytically or numerically. There are two main sources of this difficulty: high dimensionality and high frequencies. The equations describe a field theory in 3+3+1 (space + velocity + time) dimensional phase space. Two fast time scales are associated with the time it takes for light to transit the plasma and the gyration of the particles in the magnetic field. The largest force on the particles is from the magnetic field, so the particles mostly move in circles perpendicular to the magnetic field and freely stream along the magnetic field. The combination is helical. We can integrate over the phase of the circular motion (the gyrophase) to obtain gyrokinetics. The radius of the circular motion is determined by the adiabatic invariance of the

2 magnetic moment, µ = mv⊥/2B. Also integrating over the magnetic moment results in drift-kinetics. Any deviations from circular motion perpendicular to the magnetic ﬁeld are captured in additional “drift” terms. These integra- tions reduce the equations to a 3+1+1 dimensional phase space and remove the fastest time scales. Gyrokinetics and drift-kinetics capture enough of the

182 relevant physics to make predictions for laboratory plasmas, while remaining simple enough to run on supercomputers.

As a specific example, we consider small electrostatic perturbations around an equilibrium with slab geometry. The equilibrium has no electric field and the magnetic field is constant and pointed in the z direction. The motion of the ions is driven by equilibrium density and temperature gradients in the x direction. The motion of the electrons is determined by quasineutrality, not by additional equations of motion. To make the drift-kinetic approximations for the ions, we have to assume that the equilibrium fields do not vary on a length scale shorter than the Larmor radius and that the relevant time scales are long compared to the Larmor frequency. The perturbed quantities are allowed to vary on a length scale comparable to the Larmor radius, so the E × B nonlinearity is significant.

The system obtained by the above approximations is the subject of ongoing numerical studies [48, 114].

13.1 Equations of Motion

Our drift-kinetic equations, in three spatial dimensions, one velocity dimension, and time, are ∂f ∂f ∂ϕ 1 + v + vF + ρ v [ϕ, f] = C[f] + χ , (13.1) ∂t ∂z ∂z M 2 i t x,y where Z mvt ZTe 1 −v2/v2 ρi = , ϕ = f dv , FM = √ e t , (13.2) eB Ti πvt

183 2 ρivt ∂ϕ 1 v 1 1 χ = − + 2 − FM , (13.3) 2 ∂y Ln vt 2 LT where Ln and LT are the length scales for the density and temperature gradients, respectively. The nonlinear Poisson bracket term of (13.1) is the perpendicular advective derivative caused by the E × B drift. Recall that B ∝ ρizˆ and E = −∇ϕ :

1 V · ∇ f = ρ v (zˆ × ∇ ϕ) · ∇ f ⊥ ⊥ 2 i t ⊥ ⊥ 1 ∂ϕ ∂f ∂ϕ ∂f = ρ v − 2 i t ∂x ∂y ∂y ∂x 1 = ρ v [ϕ, f] . (13.4) 2 i t x,y

13.2 G-Transforming Gyro-/Drift-Kinetics

The gyrokinetic and drift-kinetic equations can be simpliﬁed using a slightly diﬀerent form of the G−1-transform. Quasineutrality replaces the Poisson equation, so there are no derivatives in the relationship between ϕ and R f dv. We do not have to Fourier transform this equation in position before applying the G−1 transform; the 0s are independent of k.

The only change is the deﬁnition of I . For drift-kinetics, the new I is

ZTe I (v) := π vFM (v) . (13.5) Ti

The rest of the G-transform and its inverse remain the same (11.11-11.13).

The gyrokinetic equations are extremely similar. Instead of evaluating the ﬁelds at the gyrocenter, the ﬁelds are evaluated at a gyroradius away from

184 the gyrocenter. This introduces some additional dependence on the perpendicular spatial directions to the Poisson equation. I will also reﬂect this: −k2 π vF (v) e ⊥ (v) = M , (13.6) I T −k2 i + 1 − e ⊥ I (k ) ZTe 0 ⊥ where I0 is the zeroth order modiﬁed Bessel function. This does depend on k⊥, but it does not require you to Fourier transform in the parallel spatial direction.

We will focus on the drift-kinetic equations because they are simpler and the calculations for both are similar.

Take the G−1-transform of all terms of (13.1) and recognize that the spatial and time derivatives commute with the G−1-transform, as does any function of only space and time, such as ϕ:

∂ −1 ∂ −1 ∂ϕ −1 1 h −1 i −1 −1 G [f]+ G [vf]+ G [vFM ]+ ρivt ϕ , G [f] = G [C[f]]+G [χ] . ∂t ∂z ∂z 2 x,y (13.7) Deﬁne g := G−1[f] and consider each term of (13.7). The ﬁrst term is simple. Use Hilbert transform property6 for the second term: Z −1 R I R I 1 G [vf] = 2 uf − 2 H[vf] = 2 uf − 2 uH[f] + − f dv || || || || π R Z −1 I = u G [f] − 2 f dv . (13.8) π|| R

The third term can be dealt with by recognizing that vFM is only a constant away from I and that H[I ] = R − 1: −1 Ti −1 Ti R I G [vFM ] = G [I ] = 2 I − 2 H[I ] πZTe πZTe || || Ti R I Ti I = 2 I − 2 (R − 1) = 2 . (13.9) πZTe || || πZTe ||

185 The sum of these two terms of (13.7) simplify. Note that the 0s depend only on velocity, so they commute with spatial derivatives,

Z Z ∂ −1 ∂ϕ −1 ∂g I ∂ 0 ZTe ∂ 0 Ti I G [vf] + G [vFM ] = u − 2 f dv + f dv 2 ∂z ∂z ∂z π|| ∂z R Ti ∂z R πZTe || ∂g = u . (13.10) ∂z

The nonlinear Poisson bracket term of (13.1) involves something that depends only on space acting on f, thus the G−1-transform acts directly on f. This term also involves ϕ, which is proportional to the integral of f, and we need to express ϕ in terms of g. We saw in (11.23) that the integral of any quantity is the same in the original velocity coordinates and in the G−1- transformed velocity coordinates. ϕ remains unchanged.

Z −1 1 1 ZTe G ρivt[ϕ, f]x,y = ρivt g du , g . (13.11) 2 2 Ti R x,y

We use the Fokker-Planck collision operator on the right hand side of (13.1). This is convenient because its interaction with the G−1-transform has already been determined in section 12.2. As before, after G−1-transforming, we get the Fokker-Planck collision operator plus the shielding term, and the shielding term is small and can be neglected.

The source term, χ, is some particular function of velocity. After G−1- transforming, it becomes a new particular function of u. Call itχ ¯(u). In particular, (13.3) is a sum of the zeroth and second Hermite polynomials, which are considered in section 13.3.

186 The G−1-transformed drift-kinetic equations are:

Z ∂g ∂g 1 ZTe + u + ρivt g du , g = C[g] + S[g] +χ ¯(u) . (13.12) ∂t ∂z 2 Ti R x,y

We have shown that if the source and collisions are neglected, the parallel electric field can be exactly eliminated. Also, in the case of collisions, the shielding term is small, so we can eliminate the electric field and retain C alone. We can solve a simpler equation in u and then G-transform back to the original velocity coordinates after the calculation of the dynamics is finished.

13.3 Hermite Polynomials

Some gyrokinetic codes use Hermite polynomials multiplied by a Gaus- sian as a basis for velocity space [49, 48]. To use the G-transform to simplify these gyrokinetic codes, we will need to know how to G−1-transform Hermite polynomials.

13.3.1 Deﬁnition

We deﬁne the Hermite polynomials as:

n n (−1) ζ2 d −ζ2 Hn(ζ) := √ e e . (13.13) pn! 2n π dζ

There are multiple normalizations used for the Hermite polynomials. Changing the normalization does not substantially change our results.

187 13.3.2 Derivatives

The derivatives of the Gaussian can be written explicitly using the

Hermite polynomials. Deﬁne the constants out front to be an:

n d 2 q √ 2 e−ζ = (−1)n n! 2n π H (ζ) e−ζ dζ n −ζ2 =: anHn(ζ) e . (13.14)

√ Note that an = − 2n an−1.

There is a simple expression for the derivative of a Hermite polynomial:

n d d 1 ζ2 d −ζ2 Hn(ζ) = e e dζ dζ an dζ n n+1 2ζ 2 d 2 1 2 d 2 = eζ e−ζ + eζ e−ζ an dζ an dζ n n 2ζ 2 d 2 1 2 d 2 = eζ e−ζ + eζ −2ζ e−ζ an dζ an dζ n n n−1 2ζ 2 d 2 2ζ 2 d 2 2n 2 d 2 = eζ e−ζ − eζ e−ζ − eζ e−ζ a dζ a dζ a dζ √n n n = 2n Hn−1(ζ) . (13.15)

Iterate this to get the expression for an arbitrary derivative of a Hermite polynomial. For any integers n, m with m ≤ n, s d m 2m n! H (ζ) = H (ζ) . (13.16) dζ n (n − m)! n−m

For m > n, the derivative is zero.

188 13.3.3 Recurrence Relation

There is an explicit recurrence relation for Hermite polynomials: 2 d 2 dH 2 d 2 a H = eζ a H e−ζ = a n + H eζ e−ζ (13.17) n+1 n+1 dζ n n n dζ n dζ √ = an 2n Hn−1 − 2ζ anHn = −2n an−1Hn−1 − 2ζ anHn .

13.3.4 G−1-Transform

G−1-transform the Hermite polynomials, multiplied by a Gaussian. Use Hilbert transform property5. Recognize the plasma Z function (10.20) when it appears:

n n n −1 1 d −ζ2 1 R d −ζ2 I d −ζ2 G n e = 2 n e − 2 H n e an dζ an || dζ || dζ n n 1 R d −ζ2 I d −ζ2 = 2 n e − 2 n H[e ] an || dζ || dζ n n 1 d 2 d = R e−ζ − √ I Z(ζ) . (13.18) 2 n 2 n an || dζ π|| dζ

13.3.5 Plasma Z Function

The derivative of the plasma Z function is given by (10.21). The expression for an arbitrary derivative of the plasma Z function can be proved by induction:

n−1 n−1−j dnZ X d = a H (ζ) Z(ζ) − 2 a H (ζ) . (13.19) dζn n n j dζ j j=0

Consider the ﬁrst derivative of the plasma Z function. Note that

189 a0H0(ζ) = 1 and a1H1(ζ) = −2ζ : 0 −j dZ X d = a H Z − 2 a H dζ 1 1 j dζ j j=0 d0 = a H Z − 2 a H = −2ζ Z − 2 . (13.20) 1 1 0 dζ0 0

This agrees with (10.21), for ζ = u/vt.

Now assume that (13.19) holds and take its derivative. See if the validity of the nth expression implies the validity of the (n + 1)st expression: n−1 n−1−j ! dn+1Z d X d = a H Z − 2 a H dζn+1 dζ n n j dζ j j=0 n−1 n−j dHn dZ X d = a Z + a H − 2 a H n dζ n n dζ j dζ j j=0 n−1 n−j dHn X d = a Z + a H a H Z − 2 a H − 2 a H n dζ n n 1 1 n n j dζ j j=0 (n+1)−1 (n+1)−1−j X d = a H Z − 2 a H . (13.21) n+1 n+1 j dζ j j=0 This completes the proof by induction of (13.19).

Combine (13.19) with (13.16) to get an expression for the derivatives of the plasma Z function explicitly in terms of Hermite polynomials. Plug in the explicit expression for aj (13.14). Note that the derivatives in the sum will

n−1 only be nonzero if n − 1 − j ≤ j, i.e. j ≥ 2 : n−1 s dnZ X q √ 2n−1−j j! = a H (ζ) Z(ζ) − 2 (−1)j j! 2j π H (ζ) dζn n n (j − (n − 1 − j))! j n−1 j≥ 2 n−1 s X 2n−1 = a H (ζ) Z(ζ) − 2 (−1)j j! H (ζ) . (13.22) n n (2j − n + 1)! j n−1 j≥ 2

190 13.3.6 Explicit Result

We can now get an expression for the G−1-transform of the Hermite polynomials in terms of a Gaussian, the plasma Z function, and Hermite polynomials in the new velocity coordinate:

n 1 d 2 1 2 G−1 e−ζ = R a H (ζ) e−ζ − √ I a H (ζ) Z(ζ) n 2 n n 2 n n an dζ an || π||

n−1 s ! X 2n−1 −2 (−1)j j! H (ζ) (2j − n + 1)! j n−1 j≥ 2 2 = R H (ζ) e−ζ − √ I H (ζ) Z(ζ) (13.23) ||2 n π||2 n s n−1 j n−1 2 I X (−1) j! 2 +√ √ Hj(ζ) . π ||2 np n (2j − n + 1)! n−1 (−1) n! 2 π j≥ 2 Note that the G−1-transform of the Hermite polynomials only involves other Hermite polynomials of lower order and the plasma Z function. After you truncate the Hermite series for your numerical scheme, the G−1-transform will not reintroduce the higher order Hermite polynomials.

13.4 Fluid Equations in u

One technique for solving gyro-/drift-kinetics is to take moments in velocity space. The resulting hierarchy of equations are known as the gyroﬂuid equations.

Alternatively, we can write ﬂuid equations in the G−1-transformed velocity coordinate. Instead of taking the v moments of (13.1), take the u moments of (13.12). We call these the u-ﬂuid equations.

191 13.4.1 Relating u and v Moments

In (11.23), we saw that the zeroth moments in u and v of any function are the same. We will now prove similar results for the ﬁrst and second moments of any function.

For the ﬁrst moment, consider

Z Z u G[g] du = (u R g + u I H[g]) du . (13.24) R R Focusing on the second term on the right hand side of (13.24), apply Hilbert transform property7 to move the Hilbert transform oﬀ of g. Then apply

Hilbert transform property6 to separate u from I . The correction term is zero since I is a total derivative: Z Z u I H[g] du = − H[u I ] g du R R Z 1 Z Z = − u g H[ ] du − g du du0 I π I ZR R R = − u g H[I ] du . (13.25) R This allows us to simplify this moment:

Z Z u G[g] du = (u (1 + H[I ]) g − u g H[I ]) du R ZR = u g du . (13.26) R Now let f = G−1[g] and change the name of the integration variable to v:

Z Z v f dv = v G−1[f] dv . (13.27) R R

192 The ﬁrst moment of any function is invariant under G−1-transforms.

For the second moment, we will prove something similar by considering Z Z 2 2 2 u G[g] du = (u R g + u I H[g]) du . (13.28) R R First focus on the second term of (13.28). Apply Hilbert transform property7 to move the Hilbert transform oﬀ of g, then Hilbert transform property6 twice to separate u from I : Z Z 2 2 u I H[g]du = − g H[u I ] du (13.29) R R Z 1 Z Z = − g u H[u ] du − g du u0 du0 I π I ZR Z R ZR Z Z 2 1 0 1 0 0 = − g u H[I ] du − u g du I du − g du u I du . R π R R π R R

The second term of the above is zero since I is a total derivative. The integral of I in the last term can be explicitly evaluated for any deﬁnition of I :(11.12), (13.5), or (13.6). We focus on (13.5) and ﬁnd Z Z Z 2 1 0 0 1 0 ZTe 0 0 ZTe 0 2 0 ZTe vt u I du = u π u FM du = (u ) FM du = . π R π R Ti Ti R Ti 2 (13.30) Once again, let g = G−1[f] and change the name of the integration variable to v. Recall that the zeroth moment of any function is invariant under G−1, so Z Z Z 2 Z 2 2 2 ZTe vt u G[g] du = u R g du − u H[I ] g du − g du R R R Ti 2 R Z ZT v2 Z = u2 g du − e t g du , (13.31) R Ti 2 R Z Z ZT v2 Z v2 f dv = v2 G−1[f] dv − e t f dv , R R Ti 2 R Z Z v2 v2 G−1[f] dv = v2 f dv + t ϕ (13.32) R R 2

193 On the last line, we used the deﬁnition of the electric potential (13.2).

The transformed second moment is equal to the original second moment plus a correction proportional to the zeroth moment. If we continue this pattern, we ﬁnd that each transformed moment is equal to the original moment plus some corrections proportional to lower order moments with the same parity.

13.4.2 Deﬁning Moments

Define the zeroth, first, and second moments in u space as: Z Z n := g du = f dv , (13.33) ZR RZ n U := u g du = v f dv , (13.34) ZR R Z Z Z P¯ := (u − U)2 g du = u2 g du − 2 U u g du + U 2 g du R R R R Z Z v2 = u2 f dv − n U 2 = v2 f dv + t ϕ − n U 2 . (13.35) R R 2 The density and fluid velocity are the same as in v space, but the pressure contains an additional term proportional to the density:

Z Z v2 P := (v − U)2 f dv = v2 f dv − n U 2 = P¯ − t ϕ . (13.36) R R 2

13.4.3 Zeroth Moment

The zeroth moment of (13.12), in u space, gives

Z ∂g ∂g 1 Z du + u + ρivt [ϕ, g]x,y = du (C[g] + S[g] +χ ˜) . (13.37) R ∂t ∂z 2 R

194 The ﬁrst term is the time derivative of the zeroth moment.

The second term is the z derivative of the ﬁrst moment.

The nonlinear term is zero since it reduces to the perpendicular Poisson bracket of something with itself: Z Z Z 1 1 ZTe ρivt [ϕ, g]x,y du = ρivt g du , g du = 0 . (13.38) R 2 2 Ti R R x,y The right hand side is much easier to evaluate in v space. This is not a problem because the zeroth moment of anything is the same in u and v space: Z Z Z du (C[g] + S[g] +χ ˜) = du G−1C[f] + χ = dv C[f] + χ . (13.39) R R R The collision operator is a total derivative and the distribution function decays as v → ∞, so its integral is zero: Z Z ∂ v2 ∂f C[f] dv = ν t + vf dv = 0 . (13.40) R R ∂v 2 ∂v The source term is an explicit function of velocity: Z Z 2 ρivt ∂ϕ 1 v 1 1 χ(v) dv = − + 2 − FM dv R R 2 ∂y Ln vt 2 LT Z 2 2 ρivt ∂ϕ 1 vt ∂ FM = − FM + 2 dv 2 ∂y R Ln 4LT ∂v ρ v ∂ϕ = − i t . (13.41) 2Ln ∂y

Only zeroth and ﬁrst moments appear in this calculation. Everything will be the same in both u and v space.

The resulting zeroth moment equation is ∂n ∂ ρ v ∂ϕ + (nU) = − i t . (13.42) ∂t ∂z 2Ln ∂y

195 13.4.4 First Moment

We next consider the ﬁrst order moment of (13.12):

Z ∂g ∂g 1 Z du u + u + ρivt [ϕ, g]x,y = du u (C[g] + S[g] +χ ˜) . (13.43) R ∂t ∂z 2 R

The ﬁrst term is the time derivative of the ﬁrst moment.

The second term is the z derivative of the second moment, which can be related to the pressure (13.35):

∂ Z ∂ du u2 g du = P¯ + nU 2 . (13.44) ∂z R ∂z

The nonlinear term is the perpendicular Poisson bracket of the zeroth and ﬁrst moment:

Z Z Z 1 1 ZTe 1 ZTe du u ρivt [ϕ, g]x,y = ρivt g du , u g du = ρivt [n, nU]x,y . R 2 2 Ti R R x,y 2 Ti (13.45)

The source term is most easily dealt with using parity. We note that, typically, the equilibrium distribution function is even. Derivatives and Hilbert transforms reverse parity (Hilbert transform property8). I ∝ ∂f0/∂v (or

I ∝ vFM ) is odd and R = 1 + H[I ] is even. Thus the G-transform and its inverse also preserve parity. Because the source term of (13.3) in the original coordinates is even, the G−1-transformed source termχ ¯(u) is also even. The ﬁrst moment of any even function is zero, so there is no contribution from the source.

196 We can deal with the collision and shielding terms using the invariance of the ﬁrst moment (13.27) twice. After integrating by parts, the collision operator reduces to linear drag:

Z Z u (C[g] + S[g]) du = u G−1[C[G[g]]] du R R Z Z ∂ v2 ∂ = v C[G[g]] dv = ν v t G[g] + v G[g] dv R R ∂v 2 ∂v Z v2 ∂ Z = −ν t G[g] + v G[g] dv = −ν v G[g] dv R 2 ∂v R Z = −ν u g du = −ν n U . (13.46) R Putting it all together, we ﬁnd that the ﬁrst moment of (13.12) is

∂ ∂ ¯ 2 1 ZTe (nU) + (P + nU ) + ρivt [n, nU]x,y = −νnU . (13.47) ∂t ∂z 2 Ti

There is a significant difference between this moment and the corresponding first v moment of (13.1):

Z ∂f ∂f ∂ϕ 1 Z dv v + v + vFM + ρivt [ϕ, f]x,y = dv v (C[f] + χ) . R ∂t ∂z ∂z 2 R (13.48) The first term is the time derivative of the first moment. The second term is the z derivative of the second moment, which can be related to the usual (v) pressure. The nonlinear term is the perpendicular Poisson bracket of the zeroth and first moment. The source term is zero because of parity. The collision operator reduces to linear drag, with fewer steps in the calculation.

There is one additional term, related to the linear electric ﬁeld, which

197 is an explicit function of velocity whose integral can be easily calculated:

Z Z 2 ∂ϕ 2 ∂ϕ 2 vt ∂ϕ v FM dv = v FM dv = . (13.49) R ∂z ∂z R 2 ∂z

So the ﬁrst order moment of (13.1) is

2 ∂ ∂ 2 vt ∂ϕ ρivt ZTe (nU) + (P + nU ) + + [n, nU]x,y = −νnU . (13.50) ∂t ∂z 2 ∂z 2 Ti

If all of the ﬂuid variables are uniform in space and time, (13.50) reduces to Ohm’s Law: E = −∂ϕ/∂z ∝ nU ∝ j. Since we are only looking at the perturbation of a single species, the current is proportional to its (perturbed) density times its ﬂuid velocity.

We search the first u moment equation (13.47) for something corresponding to Ohm’s law in vain. The only way that all of the u fluid quantities can be uniform in space and time is if the current is zero. The G−1-transform removes the electric field, so there is nothing to balance the equilibrium u current against.

13.4.5 Higher Moments

We could continue taking higher moments, with the usual hierarchy where the nth equation is coupled to the (n + 1)th moment through the advection term. To close the ﬂuid equations, we have to make additional assumptions, for example, we could assume that the second moment is some speciﬁed function of the lower moments: P = P (n, U).

198 13.4.6 Intuition?

In both u and v space, we get a hierarchy of coupled fluid equations. The u equations could be used as an alternative to gyrofluid equations. They are simpler because we have eliminated the parallel electric field.

The challenge is that we do not intuitively understand what the u moments mean. We cannot apply the G-transform to get back to the usual gyrofluid variables because we have already integrated over the velocity. This also makes it more difficult to determine an appropriate closure. Some closures are similar. If a barotropic closure is appropriate for the original moments, then a barotropic closure is also appropriate for the u moments because the difference between the two pressures only depends on the potential (13.36), which is proportional to the density. However, the u pressure would not be expected to have a single polytropic index, even if the original pressure does. Other closure ideas may also apply (e.g. those of [101]). Once we have developed some intuition about dynamics in u space from gyrokinetic models, we would be better able to interpret and use these new gyrofluid equations.

199 Chapter 14

Nonlocal Information

14.1 General Strategy

The objects in plasma theories are typically ﬁelds – quantities that are functions of space and time (and sometimes velocity). To fully measure a ﬁeld, you would need measuring devices at every location in the plasma. This is impossible in practice. Laboratory plasmas have measuring devices along their edge or that integrate along a line of sight through the plasma. Space plasmas are measured by spacecraft that follow a single orbit.

We want to get as much information as possible from the measurement of the distribution function at a single location. In particular, information about the distribution function at other locations would be useful. The G- transform can help us accomplish this goal.

Here is a strategy we could use to determine nonlocal information from a measurement at a single location:

1. Convert whatever equations of motion you have into a ﬁrst order PDE. This typically involves applying the G−1-transform.

2. Solve the PDE using the method of characteristics. By following the

200 characteristic curves, you can transfer information from the location you measure to other locations in the plasma.

3. Undo whatever transformations you applied to the equations of motion at locations away from your measurements.

We will apply this procedure to a simple example in one dimension, where the entire calculation can be done analytically. Given a measurement of the distribution function at one location for all velocities and times, we can determine the distribution function everywhere.

For more complicated situations, the same procedure could be followed, but the calculations, both of the G-transform and the characteristic curves, would have to be done numerically.

14.2 A Simple Example

Suppose you have some ions which are well modeled by a one dimensional linear Vlasov equation:

∂f ∂f e ∂ϕ ∂f + v + 0 = 0 . (14.1) ∂t ∂x m ∂x ∂v

The electric ﬁeld is determined by an assumption about the adiabaticity of the electrons. For the simplest case, assume that this involves a velocity integral over the ions’ distribution function, but does not have any spatial dependence: Z ϕ = α f dv . (14.2)

201 One example is the drift-kinetics model in the last chapter (13.2).

These equations of motion are symmetric if you reverse both v and t or if you reverse both x and t, but not if you do any other parity transformations.

14.2.1 G-Transform

Deﬁne the relevant G-transform here:

∂f := −π α 0 , := 1 + H[ ] , I ∂v R I G[g] = g + H[g] ,G−1[f] = R f − I H[f] . (14.3) R I ||2 ||2

In order to evaluate the functions R(v) and I (v), we need to choose an equilibrium distribution function. In section 12.2.1, we chose the equilibrium to be Gaussian because that equilibrium is selected by the collision operator. We do not have a collision operator here, so we are free to choose otherwise.

We will select the Lorentzian1 as our equilibrium:

1 w0 f0(v) = 2 2 . (14.4) π v + w0

The Lorentzian is a singly peaked function with width w0. It is not a physically

R 1 2 realistic distribution function because its energy, 2 v f0 dv, is undeﬁned. We select this distribution function because it is particularly simple to Hilbert transform.

1This function has a long history and many names. The most interesting name for it is the ‘Witch of Agnesi’, based on a mistranslation of a Latin word in the ﬁrst surviving mathematics book written by a woman [3, 136]. We will call it the ‘Lorentzian’, although ‘Cauchy distribution’ is also common. Of course, none of these names are associated with the ﬁrst person to consider this function, an unnamed colleague of Fermat [30].

202 The Lorentzian is a limiting case of a kappa distribution. Kappa distributions, in one dimension, are deﬁned as:

1 Γ(κ + 1) v2 −(κ+1) f0(v) = √ √ 1 1 + 2 . (14.5) π w0 κ Γ(κ + 2 ) κw0 Kappa distributions have been seen since the first measurements of space plasmas. In 1966, less than 10 years after the first artificial satellite reached orbit, data from IMP-B showed that electrons in the magnetosphere and solar wind follow a kappa distribution [12]. Since then, kappa distributions have been seen in almost all space plasmas [76]. Properties of kappa distributions and their Hilbert transform are known [103], but the calculations are more complicated than for the Lorentzian.

We will need the Hilbert transform of this function. It is easiest to calculate the Hilbert transform using complex analysis. The Hilbert transform avoids the singularity at u = v by using the principle value of the integral. Consider the integral along the contour shown in ﬁgure 10.1. This integral can be split into three parts: the principal value integral, the integral over the upper half plane, and the integral around the vanishingly small semicircle. Because the integrand decays as |v| → ∞ as |v|−3, the integral over the upper half plane vanishes. The integral over the semicircle next to the pole is half the integral around the pole, with a minus sign because we are going around it clockwise. Refer to this integral with ‘SC’: Z I Z 1 1 w0 dv w0 1 w0 dv H[f0] = − 2 2 = 2 − 2 2 . π R π v + w0 v − u π SC π v + w0 v − u (14.6)

203 The contour integral contains a single pole at v = iw0: I w0 dv w0 1 = 2πi (14.7) π2 (v − iw )(v + iw )(v − u) π2 (v + iw )(v − u) 0 0 0 v=iw0 2iw0 1 1 u + iw0 = = − 2 2 . π(2iw0) iw0 − u π u + w0 The semicircle integral is minus half the integral around the pole v = u: Z w0 dv w0 1 2 = −πi 2 2 2 (14.8) π SC (v − iw0)(v + iw0)(v − u) π (v + w0) v=u i w0 = − 2 2 . π u + w0 When we combine these two, the imaginary parts cancel: I Z w0 1 w0 dv 1 u + iw0 i w0 1 u 2 − 2 2 = − 2 2 + 2 2 = − 2 2 . π SC π v + w0 v − u π u + w0 π u + w0 π u + w0 (14.9) In light of Hilbert transform property2, we could write a single function corresponding to this equilibrium with real part f0 and imaginary part −H[f0]:

f 1 w0 + iu 0 = 2 2 , (14.10) π u + w0 which is obviously analytic whenever w0 6= 0.

I and R are related to the derivative of f0 and its Hilbert transform.

First, calculate I :

∂ 1 w0 2 α v w0 I = −πα = . (14.11) ∂v π v2 + w2 2 22 0 v + w0

To calculate R, use Hilbert transform property5:

2 2 ∂ 1 v v − w0 R = 1 + H[I ] = 1 − πα − = 1 − α . (14.12) ∂v π v2 + w2 2 22 0 v + w0

204 Also calculate ||2:

4 2 2 2 2 2 2 2 v + 2(w0 − α)v + (w0 + α) || = I + R = . (14.13) 2 22 v + w0

14.2.2 Boundary Conditions

The next piece of information that we need is our measurements at the single location. Since this is a simple example designed to show the technique, we will state an analytic function instead of inputting real data here. We also will not consider the complications which arise if the measuring device is moving. Call the location of this ‘measured’ boundary data x = 0.

The disturbance from equilibrium will be a short pulse which changes the density but not the shape of the distribution function.

The velocity dependence of the measured boundary conditions is a Lorentzian with the same width as the equilibrium. The time dependence will also be a Lorentzian: the density increases initially, then decreases:

1 1 1 w0 f(0, v, t) = 2 2 2 2 . (14.14) π 1 + γ t π v + w0

This particular boundary condition was also chosen to make the Hilbert transforms as easy as possible.

Applying the Hilbert transform to this boundary condition is the same as applying the Hilbert transform to the equilibrium:

1 1 1 u H[f(0, v, t)] = − 2 2 2 2 . (14.15) π 1 + γ t π u + w0

205 Apply the G−1-transform to this boundary condition to determine that the corresponding boundary condition for g is

g(0) = G−1[f(0)] = R f(0) − I H[f(0)] ||2 ||2

(u2 + w2)2 u2 − w2 1 1 w = 0 1 − α 0 0 u4 + 2(w2 − α)u2 + (w2 + α)2 2 22 π2 1 + γ2t2 u2 + w2 0 0 u + w0 0 ! 2 α u w0 1 1 u − 2 2 − 2 2 2 2 2 u + w0 π 1 + γ t u + w0 2 2 w0 u + w0 + α = 2 2 2 4 2 2 2 2 . (14.16) π (1 + γ t ) u + 2(w0 − α)u + (w0 + α)

We now need to solve the equations of motion for g(x, u, t).

14.2.3 Solving the Equations of Motion

When we apply the G−1-transform to the one-dimensional linear Vlasov equation (14.1), it reduces to advection:

∂f ∂f e ∂ϕ ∂f ∂g ∂g G−1 + v + 0 = + u = 0 . (14.17) ∂t ∂x m ∂x ∂v ∂t ∂x

The characteristics for this equation are trivial: u = const, x = ut+x0.

We can use these characteristics to connect the g distribution function at one location to the g distribution function at other locations. Because the distribution function is constant along characteristics, its value at x at time t is equal to its value at x = 0 at time t − x/u:

g(x, u, t) = g(0, u, t − x/u) . (14.18)

206 We know what the boundary conditions are (14.16), so we can immediately write down the solution:

2 2 w0 u + w0 + α g(x, u, t) = 2 2 2 4 2 2 2 2 (14.19) π (1 + γ (t − x/u) ) u + 2(w0 − α)u + (w0 + α) 2 2 2 w0 u u + w0 + α = 2 2 2 2 2 2 2 4 2 2 2 2 . π (1 + γ t )u − 2γ txu + γ x u + 2(w0 − α)u + (w0 + α)

We will once again need to do a Hilbert transform to convert back to the original coordinates:

1Z w u2(u2 + w2 + α) du H[g] = − 0 0 . 2 2 2 2 2 2 2 4 2 2 2 2 π R π (1 + γ t )u − 2γ txu + γ x u + 2(w0 − α)u + (w0 + α) u − v (14.20) The easiest way to do this is using complex analysis. As before, integrate around the upper half plane and on a semicircle around the pole at u = v.

√ √ 2 2 There are now poles at u = α±iw0, u = − α±iw0, u = γx(γt±i)/(1+γ t ), and u = v. Refer to the locations of these time dependent poles as a ± ib:

γ2xt γx a := , b := . (14.21) 1 + γ2t2 1 + γ2t2

In terms of the residues of these poles, the Hilbert transform is

w H[g] = 0 I , where π3(1 + γ2t2) I Z u2(u2 + w2 + α) du I = − 0 2 2 4 2 2 2 2 SC (u − a) + b u + 2(w0 − α)u + (w0 + α) u − v √ √ = 2πi Res u= α + iw0 + 2πi Res u=− α + iw0

+2πi Res (u=a + ib) + πi Res (u=v) . (14.22)

207 We need the residue of four poles:

u2(u2 + w2 + α) Res(v) = 0 , (14.23) (u − a)2 + b2u4 + 2(w2 − α)u2 + (w2 + α)2 0 0 v

u2(u2 + w2 + α) Res(a + ib) = 0 , (u − v)(u − a + ib)u4 + 2(w2 − α)u2 + (w2 + α)2 0 0 a+ib

√ u2(u2 + w2 + α) Res(iw + α) = √ 0 √ , 0 2 2 2 2 (u − v) (u − a) + b (u − α + iw0) (u + α) + w0 √ iw0+ α

√ u2(u2 + w2 + α) Res(iw − α) = √0 √ . 0 2 2 2 2 (u − v) (u − a) + b ) (u − α) + w0 (u + α + iw0) √ iw0− α For compactness, we do not write ‘u =’ in the Residue or when stating where to evaluate these functions.

The ﬁrst residue can be evaluated immediately. Evaluating the other residues is straightforward, but signiﬁcantly more complicated.

Next, we substitute in for a and b (14.21).

The solution in the original coordinates is given by the G transform:

f(x, v, t) = G[g(x, u, t)] = R(v) g(x, v, t) + I (v) H[g(x, u, t)] . (14.24)

We will not show the remainder of the steps in detail. They are tedious, but not diﬃcult.

208 The solution is

w Av4 + Bv2 + Cv f(x, v, t) = 0 , (14.25) 2 2 22 2 2 2 2 2 2 4 π v + w0 (1 + γ t )v − 2γ xtv + γ x w6 where

4 2 2 2 2 A := w1 + 2αw2 + α (1 + γ t ) ,

2 4 4 2 2 B := w0w1 + 2αw4 + α w3 ,

3 2 2 C := 2αγ x t(w0w1 + αw5) ,

2 2 2 2 2 2 w1 := (1 + γ t )w0 + 2γxw0 + γ x ,

2 2 2 2 2 2 2 2 2 2 2 w2 := (1 + γ t ) w0 + 2γx(1 + γ t )w0 + γ x (1 − γ t ) ,

2 2 2 2 2 4 4 2 2 2 2 w3 := (1 + γ t ) w0 + 2γx(1 − γ t )w0 + 2γ x (1 − 3γ t ) ,

4 2 2 2 4 2 2 2 2 3 2 2 2 2 2 w4 := (1 + γ t ) w0 + γx(1 + γ t )(3 − γ t )w0 + 2γ x (2 − γ t )w0

3 3 2 2 4 4 +γ x (3 − γ t )w0 + γ x ,

2 2 w5 := (1 + γ t )w0 + 2γx ,

4 4 2 2 2 2 2 w6 := w1 + 2αw2 + α (1 + γ t ) . (14.26)

All of the new variables introduced here only depend on position and time. Each of the w’s has dimensions of [v]. The dimensions of A, B, and C, are [v]4,[v]6, and [v]7, respectively.

Even for these simple boundary conditions (14.14), the analytical solution is complicated.

209 Figure 14.1: Initial conditions (14.27) corresponding to Lorentzian boundary data. v ranges from ±4w0. x ranges from ±4w0/γ.

If we consider x = 0, we obtain the given boundary conditions (14.14). There is a discontinuity at x = 0, v = 0. Along the v-axis, the limit is zero.

2 2 2 Along the x-axis, the limit is 1/(π w0(1+γ t )). This discontinuity occurs because particles with zero velocity cannot carry information to other locations.

The initial conditions that would result in this measurement can be found by plugging in t = 0:

w v2(v2 + w2)(w + γx)2 + α(v2 + w2 + 2γxw + 2γ2x2) f(x, v, 0) = 0 0 0 0 0 . 2 2 22 2 2 2 2 π v + w0 (v + γ x ) (w0 + γx) + α (14.27) A plot of these initial conditions is shown in ﬁgure 14.1. Plots of f(v, t) at various locations are shown in ﬁgure 14.2.

210 Figure 14.2: The analytical solution (14.25) for Lorentzian boundary data vs. t and v. v ranges from ±4w0. t ranges from ±4/γ. All positions are measured in units of w0/γ. The boundary data x = 0 are at the top. The left column shows negative positions: x = −0.5, −1, −1.5. The right column shows positive positions: x = 0.5, 1, 1.5. 211 14.3 Finite Time and Additional Terms

In this simple example, we could calculate everything analytically. We took perfectly precise data, for all velocities and for arbitrarily far into the past and future, at one location, and used it to calculate similar information at all locations.

For a real system, there will be additional complications. If we have a ﬁnite amount of measured data or if we use approximations in determining our equations of motion, how well will this procedure work?

Look back at how we determined the solution in the G−1 transformed space using the method of characteristics (14.18). Problems can arise when u = 0. We cannot know the center of the distribution function away from the measurement location. Characteristic curves along which particles are not moving or are moving extremely slowly cannot carry information to or from other locations.

Since our example had analytical data for all time, we could wait however long it took for the information to arrive. Actual experiments only last for a ﬁnite time, τ. We will not be able to predict the distribution function anywhere x v . (14.28) . τ The ﬁnite measurement time prevents us from determining information at small velocities and at large distances.

What if there are additional terms in our equations of motion?

212 Not all plasmas can be described by a one dimensional Vlasov equation with quasineutrality.

The general procedure described above should work whenever we can massage the equations into something that can be solved by the method of characteristics.

If the only addition is the Fokker-Planck collision operator (12.2), then an analytical calculation is still possible, using the solution in section 12.4.1. This solution is not exact because it involves dropping the shielding term.

Including nonlinear drift terms from gyrokinetics would be even more diﬃcult. However, if you can solve them using the method of characteristics, the procedure works.

Alternatively, we can drop the complicated terms and estimate where this solution is valid. If τ is the time scale associated with the largest term dropped, then (14.28) again determines where, in phase space, we have suﬃ- cient information to calculate the distribution function.

In future work, we hope to apply this technique to data in a linear plasma device. By solving the equations of motion using data from a single location and then making measurements at multiple locations, we can test where the solution is valid. Once it is veriﬁed in experiment, we could apply this procedure to spacecraft data to determine the behavior of plasma at more than a few trajectories through the solar system.

213 Chapter 15

Conclusion

In partII of this dissertation, we presented the G-transform, an integral transform, and showed how it simpliﬁes the equations of motion of kinetic plasmas with one velocity dimension.

A key beneﬁt of the G-transform is pedagogical.

The one dimensional Vlasov-Poisson system is one of the foundational models of plasma physics. It is the simplest system that captures a phenomenon unique to plasmas: Landau damping. Landau’s solution [70], which is taught in all plasma physics textbooks and courses, does not make the un- derlying physics intuitive. It uses a Laplace transform to make the equations solvable algebraically. The physics of Landau damping is hidden in the contour for the inverse Laplace transform. Most plasma physics instructors supplement Landau’s solution with other heuristic arguments.

The G-transform provides an exact solution to the one dimensional Vlasov-Poisson equations. The connection between Landau damping and the phase mixing of pure advection can be easily seen. The decay occurs as the result of the Riemann-Lebesgue Lemma. As we discussed in section 11.5, the G-transform provides connections to other ideas in plasma physics (van Kam-

214 pen modes) and classical mechanics more generally (action-angle variables).

We encourage anyone teaching plasma physics to explain Landau damping using the G-transform.

We also showed how the G-transform can be used to simplify other kinetic theories with one velocity dimension. The G-transform is designed to eliminate the linear electric field. Slight modifications are needed if the Poisson equation is replaced by a quasineutrality assumption. Adding more physics to the theory involves adding additional terms to the kinetic equation. The strategy is similar for each new term. If the G-transform and the new operator commute or almost commute, then the G-transform removes the linear electric field term without making the other terms more complicated.

We showed that the G-transform almost commutes with the Fokker- Planck collision operator (Section 12.3). The additional term, which we call the ‘shielding term,’ is small for small ν, regardless of whether the distribution function has large velocity gradients. If there is no small scale structure in velocity space, then the collision term and the shielding term are small since they are multiplied by a small parameter, the collision frequency. If there is small scale structure in velocity space, then the collision term is signiﬁcant since it contains the highest order velocity derivative. The shielding term is still unimportant since its velocity derivatives are all lower order than the collision term.

If we drop the shielding term, then the resulting advection-diﬀusion

215 equation can be exactly solved. We wrote an explicit solution for simple initial conditions and used it to determine when advection dominates and when collisions dominate the equation (Figure 12.2). We then numerically solved the advection-diﬀusion equation for more realistic initial conditions and showed that our conclusions are not substantially diﬀerent from the simple initial conditions (Figures 12.4 and 12.5). We then discussed how the shielding term could be included as a perturbation.

Although we focused on the Fokker-Planck collision operator, similar arguments also apply for any other collision operators that are local in v. In future work, we hope to extend this argument to more complicated collision operators, such as the one dimensional linearized Landau-Boltzmann operator [15, 72] and pitch angle scattering [47].

The G-transform could find fruitful applications in gyro-/drift-kinetics, which have one dynamical velocity dimension, and still capture much of the relevant physics for tokamak and space plasmas. The G-transform removes the linear electric field term, leaves the nonlinearity unchanged, and only modifies the collision operator by a small perturbation (13.12).

The G-transform could speed up gyrokinetic codes by removing the need to calculate one of the terms at each time step. The G-transform only needs to be applied to the initial data and the ﬁnal output. Computing the G-transform involves computing a Hilbert transform and storing the real and imaginary parts of the plasma dielectric function. Eﬃcient Hilbert transform algorithms are widely available.

216 Gyrokinetic codes sometimes use Hermite polynomials times a Gaussian as a basis in velocity space. We give an explicit expression for the transformed basis elements (13.24). In this basis, the G-transform removes the coupling between each Hermite polynomial and the zeroth Hermite polynomial. Her- mite polynomials are still coupled to their neighbors through the advection term. The perpendicular nonlinearity, which couples diﬀerent perpendicular wave numbers, remains unchanged.

We compared gyrofluid equations found by taking moments in both the original velocity space and in the transformed u space. There is nothing corresponding to Ohm’s law in the transformed gyrofluid equations: the electric field has been removed, so the only things that can balance the first u moment are spatial and time derivatives of other u moments.

The G-transform could also be helpful in extracting nonlocal information from measurements taken only at one or a few locations. If the equations of motion can be coerced into a form solvable using the method of characteristics, the data from a single location can be projected along these characteristics to determine the plasma’s behavior elsewhere. We analytically solved a one dimensional Vlasov equation with quasineutrality and Lorentzian boundary conditions at x = 0 for the distribution function for all locations (14.25).

The G-transform is a simpler, analytical way to understand Landau damping. As more people understand Landau damping this way, we will ﬁnd many more applications of the G-transform.

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