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Wave Electrodynamics

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0. Frontmatter

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1. Introduction

WELDBook/IntroBack to Main Page.Fundamental physics should be dead simple. if it doesn't seem to be, then one can't help but wonder if there might besomething simpler going on, that gives rise to the apparent complexities.There is a framework for understanding the fundamental physics of electrodynamics (electrons and electromagneticfields), based entirely on interactive wave dynamics, that appears to be as simple as possible. This framework iscalled Wave Electrodynamics (WELD), which is the interaction between the electromagnetic field as described byMaxwell's equations [1] and electrons as described by Dirac's wave equation for the electron -- the coupledMaxwell-Dirac system.This book describes this framework and why it is arguably the simplest model of fundamental physics that canpotentially account for all of the known phenomena. Clearly, if it were all this simple, the WELD framework wouldbe much more well known than it is, and you wouldn't be encountering these ideas in this relatively obscure medium.There are several seemingly insurmountable problems with this framework, that prevent the vast majority ofphysicists from even considering it. Thus, another important objective of this book is addressing these concerns, tomake it clear that this framework is viable, if not (yet) definitively more accurate than the prevailing more complex(and paradoxical) accepted model: the standard model [2] (wikipedia link) of physics, based on quantumelectrodynamics (QED).WELD is currently very much a work in progress -- there is a working model of the hydrogen atom, using thecoupled Maxwell-Dirac equations for a single electron trapped in a potential well by a fixed positive nuclear charge,which produces what looks like a ground state standing wave oscillation. Simulating excitation and emission fromthe EM field are the next steps, followed by multi-electron atoms (Helium, Lithium), and then exploration of freeelectrons. The purpose of writing this book during the development of the framework is to provide a means ofworking through the conceptual and technical issues (which is greatly facilitated by attempting to explain everythingin as clear a way as possible), and to hopefully recruit some additional researchers into this endeavor.In this introduction, the central philosophical foundations of the approach are presented, which provide a generalmotivation for considering this approach, in the face of the many hurdles it must overcome.Nobody should be convinced of anything on the basis of philosophical reasoning alone. Ultimately, the value of theWELD approach must rest on its ability to make novel, testable predictions that contrast with the standard model.But in advance of that, perhaps this introduction will encourage enough people to continue reading, to get some morepeople working on developing this approach to the point where it can be properly evaluated.Regardless of one's overall feelings about the WELD approach in particular, this book can serve as a very usefulintroduction to many important concepts in physics, because the approach is so fundamentally simple. We developvery simple equations for wave dynamics, and use numerical simulations to bring these waves to life in living colorusing the EmeWave simulator. By developing increasingly complex versions of these wave equations, we end upsimulating all of electromagnetism and a large range of phenomena in electrodynamics, based on Dirac's waveequation for the electron. Along the way, you will learn all about special relativity [3] (wikipedia link), whichemerges naturally out of the wave dynamics. Thus, even if you are learning electrodynamics or quantum mechanicsin a conventional course, this approach can provide a quick and relatively easy way to really understand theunderlying phenomenology, with a minimum of mathematics. Having this solid conceptual foundation can thenmake it much easier to understand all the standard quantum mechanical material.

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OverviewIn the remainder of this chapter, we situate the WELD approach within the broader field, both from a conceptual andhistorical perspective, focusing on three key issues that shape the approach most strongly: a computational modelingapproach, waves vs. particles as fundamental entities, and local vs. nonlocal interactions. Here, we introduce andsummarize the WELD approach to these issues.

The Computational Modeling ApproachThe overarching goal is to simulate fundamental physics on a computer, in just the way that computer models areused in many other domains to understand how simpler mechanisms interacting over time and space can lead tomore complex phenomena. This is perhaps most well known in the case of weather models, where basic physicalmechanisms of heat, pressure, condensation, etc can be simulated to understand the complex phenomena thatemerge. Although computer models have been used in theoretical physics, analytical mathematics remains thedominant methodology. One advantage of computer models is that they enable one to capture and understandcomplex emergent phenomena arising from nonlinear dynamics, that are otherwise difficult to handle withanalytical approaches. The WELD models show that many fundamental paradoxes in the standard model canpotentially be resolved through these emergent dynamics. Even the first step of thinking about how to implement acomputer model raises a number of very important questions, which get at the heart of many fundamental problemsin the standard model.A number of researchers have argued that the most natural framework for computational modeling of physics is thecellular automaton, which forms the basis of the WELD model (Zuse, 1970; Fredkin & Toffoli, 1982; Fredkin,1990). In a cellular automaton, space is divided into a uniform matrix of discrete cells, with state values in each cellthat are updated in discrete time according to local neighborhood interactions. In its most general form, the cellularautomaton approach is identical to the way that differential equations are solved on computers, using discrete gridsand discrete time integration, which is the only way anyone has been able to actually implement the idealizedcontinuous differential equations that appear everywhere in physics. Instead of viewing this as some kind ofinconvenient artifact, we embrace this as a strong constraint on any concrete understanding of how nature actuallyoperates at the most fundamental level: perhaps our inability to devise any truly continuous mechanism to implementdifferential equations is not just a limitation on our imagination, but instead represents a real constraint on nature aswell. The essence of this constraint has to do with the difference between the set of integers vs. truly continuous realnumbers -- real numbers require an infinity at each point in space in time, and are uncountable (see Cardinality of thecontinuum [4] (wikipedia link). It is just not clear how any physical process could manage this kind of uncountableinfinity at each point in space and time.A defining feature of cellular automata, captured in the name, is that they are fully autonomous -- they just crankaway iteratively and produce physical phenomena, without requiring any outside intervention, aside fromestablishing the initial state of the system. In contrast, analytic mathematics is all about figuring out special tricks tosolve specific kinds of problems in the most efficient way possible, to arrive at a fixed answer to a specific question(e.g., what is the radius of an electron orbit in the ground state of a hydrogen atom?). This process obscures many ofthe issues that otherwise arise when trying to solve the general problem that works for all cases, autonomously,which is presumably what we imagine nature to actually be doing at every moment and every location in space.Thus, we argue that attempting to develop a fully autonomous, fully general model of physics is the best way to trulyunderstand how nature works.

This distinction between fully general, autonomous physical models and analytical solution of specific problems using various calculational tools seems to be generally under-appreciated in the field, and there are many cases where calculational tools are mistakenly treated as physical models, leading to considerable confusion. We attempt to clarify this distinction with numerous examples, so that we can easily classify the various theoretical frameworks

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we encounter as either physical models or calculational tools, and thus have appropriate expectations about how theyhelp us understand nature. Each has important and unique strengths and weaknesses, so we need both, even thoughour overall goal is to develop a physical model.

Waves vs. ParticlesOne of the first decisions in implementing a computer model is how information will be represented, which thenshapes everything else. In fundamental physics, there are two qualitatively different kinds of entities: force fields(e.g., the electromagnetic force) and particles (e.g., electrons). For classical electromagnetic fields, the most naturalrepresentation is a space-filling field model: you represent each point in space, with some number of state values ateach point that are then updated according to interactions that propagate throughout the field. This is essentially thecellular automaton model. Particles are more difficult. In many ways the most natural particle representation is just abig list of particles, with each entry in the list containing the relevant state values (position, momentum, charge, etc).These state values are then updated as a result of interactions between the particles. It is also possible to representdiscrete point-like particles in the field representation, by having state values that indicate the presence or absence ofa particle at that particular location in space, along with perhaps other values needed to encode other properties ofthe particle (momentum, charge etc). But most of space is empty of particles, so the particle-list is generally moreefficient.The field representation is intuitively appealing because we already know that nature has just the right amount of 3Dspace to hold an all-pervasive physical field. In contrast, it is unclear how or where nature would store a list ofparticles, and how interactions between these particles would be mediated in any kind of general, autonomousmanner. Is the list sorted by location? Does every particle interact with every other one on the list (which isexponentially expensive), or is there some kind of neighborhood localization? What happens when particles arecreated and destroyed, as happens all the time? What kind of fully general autonomous process can manage theinsertion and deletion of particles on a list in just the right way at just the right time?There are several problems for the field representation of particles as well. The discretization of space and timemakes it difficult to achieve smooth, isotropic (same in every direction) motion -- just like the diagonal lines on acomputer monitor when you look up close, discretization causes "jaggies." Also, the issues of particle creation anddestruction are potentially challenging here -- how can one convert energy in a field into creation of a particle at onepoint, and vice-versa for destruction? And how does one implement the interactions between forces and particles in adiscrete system? Having spent a fair amount of time programming such models, they always ended up being verycomplex and "hacky" -- it just doesn't seem like nature could operate anything like this.Given all these issues, quantum mechanics (QM) seems to provide a critical advance for our physical model:particles can be represented as waves propagating in a field-like medium. Each particle is associated with a coherentperturbation of this wave field that evolves deterministically over space and time. Thus, it would seem that we canget rid all the complexities of discrete point particles, and instead retain a fully field-based representation of bothforces and particles-as-waves. This simple, elegant solution to all of the above problems forms the conceptual core ofthe WELD approach. Although this wave model may be generally appealing as an idea all by itself, thecomputational modeling perspective elevates it to an absolutely essential principle -- it solves too many problems insuch an elegant way, without any viable alternatives in sight, to be tossed aside without anything less than the utmosteffort to make it work.Unfortunately, it looks like it will indeed take a lot of effort to make this idea work. Although QM has a wave-based description of particles in the form of the Schrödinger wave equation, this wave description is vexingly co-dependent upon a complementary particle-based description, resulting in the paradoxical wave-particle duality that apparently pervades the quantum realm. The uneasy co-existence of wave-like and particle-like behavior represents the central conundrum of QM. The standard interpretation of QM holds that the wave-like aspect of particles is entirely non-physical and instead merely represents the probability of finding a discrete particle in a

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given state in a measurement process, which is thought to engender a sudden collapse of the wave function down tothe discrete outcome actually observed. The wave thus encodes our knowledge about the system -- it is a purelyepistemological entity, enabling a quantum probability calculus that stands as one of the great calculational tools inphysics -- it can be used to accurately calculate the results of many different kinds of experiments. Thus, despite thefact that all the mathematics of QM is based on wave mechanics, all the semantics and experimental data suggeststhat discrete particles are primary, and the wave is this strange non-physical halo that nobody quite knows how tothink about. A quote from E. T. Jaynes is particularly apropos here:

"But our present QM formalism is not purely epistemological; it is a peculiar mixture describing in partrealities of Nature, in part incomplete human information about Nature — all scrambled up by Heisenberg andBohr into an omelette that nobody has seen how to unscramble. Yet we think that the unscrambling is aprerequisite for any further advance in basic physical theory. For, if we cannot separate the subjective andobjective aspects of the formalism, we cannot know what we are talking about; it is just that simple." (Jaynes,1990).

Thus, to make a physically real wave model work, we need to unscramble this omelette that nobody has yet beenable to unscramble. Fortunately, Jaynes and several other pioneers did a lot of unscrambling, by establishing thesemiclassical model of electrodynamics, which features a classical electromagnetic field evolving according toMaxwell's differential equations, interacting with an atomic system that has quantum mechanical properties. Thiscontrasts with the standard QM model of electrodynamics (QED), which treats the electromagnetic field in terms ofdiscrete photon particles, instead of the classical differential equations. Thus, by beating back one purported particle(the photon) and retaining the continuous electromagnetic field, we gain a toe-hold into the fully field-based wavemodel. The confusion surrounding the existence or non-existence of the photon is analogous to a common magictrick, where the magician draws attention to one obvious system (e.g., the rabbit that seems to disappear), when infact there is a less obvious system that is actually responsible (e.g., an extra pocket in the magician's hat, that holdsthe rabbit hidden from view). In this case, the rabbit is the electromagnetic field and its purported particle-likeproperties, which instead can be attributed entirely to the properties of the atomic system with which the EM fieldinteracts (the hidden pocket in the hat).Devising a fully wave-based model of the electron is more difficult than vaporizing photons. Unlike photons,electrons have a lot of strong particle-like properties, such as an entirely consistent and conserved amount of electriccharge, and the apparent propensity to remain localized over time, whereas most wave equations tend to spread outand diffuse over time. A purely wave-based electron would seem to be too likely to splatter and leave drops ofcharge lying around all over the place. A key property of the WELD model of the electron is thus emergentlocalization -- a nonlinear emergent dynamic whereby the electron waves remain tightly localized over time, andresist splattering and fractional loss of charge. Thus, these waves remain tightly localized wave packets, asoriginally envisaged by Schrödinger when he discovered his wave equation in 1928 (he soon abandoned theparticles-as-wave packets idea after realizing that the wave equation spread out too much over time). This kind ofemergent phenomenon is extremely difficult to characterize analytically, but can be observed in computersimulations quite readily.Another key piece of the unscrambling puzzle comes from the pilot wave model of de Broglie (who initially proposed the wave-like property of massive particles like electrons), and Bohm. They were able to establish a fully consistent interpretation of QM where the wave serves to guide the trajectory of particles over time. For any given specific event, the particle has a specific trajectory through space, and the measurement process simply reveals the particle where it ends up, thus avoiding many of the mysteries and conceptual problems associated with the notion of wave function collapse in the standard QM model. This framework unscrambles two key contributions of the wave function: (a) the wave specifies the initial uncertainty in the location and momentum of the particles at the start of an experiment, and the way that this uncertainty inevitably compounds and thus spreads out over time; and (b) the gradient (local slope) of the wave nudges the particles as they move along their trajectory, producing the wave

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interference effects that are the unique signature of the wave-like nature of electrons and other particles. The firstcontribution is clearly purely epistemological -- it reflects our fundamental inability to know and control the states oftiny elemental particles. The second contribution is clearly physical -- it absolutely requires a physical wavepropagating through space and influencing the particle trajectory. Thus, the pilot wave model unscrambles thesephysical and epistemological components of the overall Schrödinger wave function. However, the pilot wave modelitself retains the notion of discrete particles, which cause further difficulties for this model, and we reject onprinciple in the WELD approach. Nevertheless, we can retain the conceptual unscrambling progress from the pilotwave model, while integrating the wave-packet model of particles.In summary, the particle waves in WELD are not synonymous with the Schrödinger waves of standard QM, whichare purely linear and, consistent with the standard theory, are best thought of as calculational tools for computingprobabilities for experimental outcomes. Instead, the WELD physical model is based on nonlinear interactionsbetween the electromagnetic field and the physical electron wave equation described by the Dirac wave function. Toachieve emergent localization of the wave packet, the gravitational warping of space is required, in response to thelocal energy density of the EM and electron waves -- this warping represents a weak force overall, but it is justsufficient to counteract the wave spread that otherwise occurs. To recover the full picture described by the standardSchrödinger waves, these more localized wave packets must be combined with a healthy dose of purelyepistemological uncertainty about initial conditions, which compounds over time. Overall, the resulting system issimilar in character to the de Broglie-Bohm pilot wave model, replacing discrete particles with localized wavepackets. Interestingly, the oscillations of the Dirac waves drive corresponding EM waves that produce the physicalinterference "nudges" that push the wave packets around.

Local vs. Nonlocal InteractionsOne of the appealing features of the cellular automaton field model is that it can implement long-range physicalinteractions via strictly local interactions between adjacent cells in the matrix. The local nature of these interactionsis important for the generality and autonomy of the physical model: the moment one starts to consider nonlocalinteractions, questions arise as to how one part of space "knows" to interact with another distal region. The only fullygeneral nonlocal interaction would be for every point in space to interact with every other point in space, which issuch a fantastically intractable model that the mind reels to even contemplate it. Thus, another bedrock principle ofthe WELD approach, motivated from the goal of achieving a fully autonomous, fully general computational modelof physics, is that nature supports strictly local interactions at the fundamental level.However, again we face a strong challenge from the standard QM model, which is strongly nonlocal in a couple ofways. First, the process of wave function collapse must occur instantaneously across the entire spatial extent of thewave function, which can in principle encompass vast distances. This is an intrinsically nonlocal process, and mostphysicists believe that the nonlocality of wave function collapse has been definitively established experimentally, ina series of celebrated tests of Bell's inequalities, based on the initial framework of Einstein, Podolsky & Rosen(EPR) in 1935. However, these experiments have a set of fundamental flaws, that are begrudgingly acknowledged,but seemingly not fully appreciated by the broader field. Furthermore, the pilot wave model suggests that thiscollapse process is largely an artifact of epistemological uncertainty, and not an actual physical process -- theunderlying particles are always definitely somewhere, and measurement simply lifts the veil of our ignorance.The pilot wave model however contains the second form of nonlocality, where the wave function for multiple particles must be defined in a high-dimensional configuration space, which becomes exponentially large as the number of interacting particles increases (e.g., requiring values to represent the 3D positions of n interacting particles). This configuration space representation supports fully nonlocal interactions among particles, enabling the pilot wave model to account for the apparent nonlocality observed in the Bell's inequality experiments. One important reason that standard QM models, including the pilot wave model, require this configuration space representation is that they are based fundamentally on the linear Schrödinger wave equation -- linear equations are

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incapable of capturing particle interactions, because the waves simply superpose (additively combine) past eachother, without impacting each other at all.In contrast, the nonlinear interactions present in the WELD approach enable it to capture particle interactions inregular 3D space, potentially avoiding the need for the problematic configuration space representation.Nonlocality is not just an aesthetic consideration -- nonlocal systems are fundamentally computationally intractable.Indeed, this is why the hot new field of quantum computing is generating so much excitement in the field: quantumphysics appears to achieve something that standard computers fundamentally cannot do in an efficient way. Thisinefficiency is of the exponentially explosive sort, such that simulating quantum systems with more than a handful ofparticles requires more memory and time than there are atoms in the universe. This is the same kind of dynamic thatleads to surprising facts, such as the extreme difficulty of folding paper more than 8 times, or the exponentially fastgrowth of populations.It is not just that people are not sufficiently creative or current computers are too limited: there really is no way froma mechanistic, information processing perspective to understand how nature could pull off a fundamentally nonlocalcomputation. If we had rock-solid evidence that nature was indeed nonlocal in the way that standard quantummechanics describes, then we would just have to accept this incredible state of affairs. But the evidence is anythingbut solid, despite the widely-held beliefs to the contrary of most in the physics community. As we discuss in detaillater, there are a number of detailed physical models of the Bell's inequality experiments that produce the observedresults, using purely local interactions (Marshall, Santos, & Selleri, 1983; Marshall & Santos, 1985; Thompson,1996; Adenier & Khrennikov, 2003; Santos, 2005; Aschwanden et al., 2006; Adenier & Khrennikov, 2007).

Opportunity, and Prospects for SuccessThe obvious question for any alternative to the standard model is, given that people have been wrestling with theseproblems for 100 years or so, why do you think you have any chance of solving problems that even a genius likeEinstein couldn't solve? In other words, what basis is there for thinking that this approach will succeed where somany others have failed? Here are a number of relevant points:• Most practicing physicists are no longer trying to solve these problems, and have fully accepted the basic

framework of standard QM. In effect, most people are sufficiently indoctrinated into the current paradigm by thestandard physics education, that they don't even consider alternative perspectives. So there aren't that many peopleactually working on these kinds of alternative solutions.

• Most of the methodology of standard physics depends on analysis of mathematical equations. In the case of theMaxwell-Dirac system, this is very difficult due to its nonlinearity. Thus, the use of numerical simulationmethods, emphasized here, can potentially provide important new insights. It is only very recently that thecomputational power to simulate large three-dimensional systems of this sort has become available.

• The specific idea of resolving the wave-particle duality with the wave-packet model has been almost completelydismissed since the late 1920's, due to the spreading issue and other apparent problems. However, given that weknow that the Schrödinger wave equation is incorrectly linear, and reflects a mix of multiple factors, it is entirelypossible to solve this spreading problem. But this has not been deeply explored in the field.

• The broad acceptance of quantum nonlocality is another strong reason that most people completely ignoresolutions of the form we're investigating.

Thus, it does not appear that anyone has previously investigated the specific set of ideas described here, andhopefully you'll be convinced from this chapter that there is sound reasoning and considerable promise behind theWELD approach, to continue reading.In the remainder of this chapter, the points introduced above are explicated in greater detail, and put into historicalperspective with the development of physical theories.

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The Computational Modeling Approach to Fundamental PhysicsAs we discussed in the overview above, the perspective of attempting to develop a computational model of anyphenomenon raises a number of important questions that can otherwise be overlooked in purely verbal or evenanalytical mathematical theorizing. In many different fields, computational models serve to bridge betweenlower-level mechanisms and the often-complex behavior that emerges from the unfolding interactions taking place atthis lower level. For example, computational models of neural networks in the brain can explain a wide range ofcomplex cognitive phenomena in terms of a small set of basic underlying information-processing mechanismssupported by the biological properties of neurons and their synaptic interconnections. This corresponds to theclassical reductionistic scientific theorizing -- reducing complexity at one level to greater simplicity at the levelbelow it. In the standard hierarchy of science, we think of biology being reduced to chemistry, which in turn reducesto physics.So what does physics reduce to? At some point, we have to posit a most fundamental, irreducible level, and expectthat everything else can emerge therefrom. The critical advantage of computational models in this endeavor is thatthey enable the reverse of reductionism, which can be called reconstructionism -- actually reconstructing emergentcomplexity from simpler lower-level mechanisms. A computational model can bring complex phenomena to life,which can otherwise be very difficult to analyze mathematically, or reason about purely verbally, pictorially, or viaother traditional means.As we discussed above, the most natural, simplest fundamental, irreducible level of representation in acomputational model of physics is a space-filling field, with local neighborhood interactions, which is essentially acellular automaton. If a physically accurate model can be developed in the form of a cellular automaton, whichcaptures all the known physical phenomena, then it would be difficult to conceive of a yet simpler model underlyingthat one. We describe the cellular automaton model in more detail next, followed by a discussion of the history ofthis form of field-based model.

The Cellular Automaton Model of Space and Time, and Local DeterministicInteractions

Figure 1.1: Illustration of a simple 2-dimensionalcellular automaton: space is divided into regular square

cells (a uniform, regular tiling of space), anneighboring states interact by influencing the state

update. Time updates synchronously, setting the fastestrate of propagation as cell width / time update.

A cellular automaton (CA) consists of a regular, uniform divisionof space into discrete cells, each of which has one or more statevalues, and each cell interacts only with its nearest neighbors (i.e.,locally) to update its state value over time (Figure 1.1, Figure 1.2).

Such a system was first described by Stanislaw Ulam in 1950, andhas been popularized in its two-dimensional form in "the game ofLife" by John Conway (described by Gardner, 1970). In this CA(widely available as a screensaver), there is a two-dimensionalgrid of square cells, with each cell having a single binary

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Figure 1.2: Neighborhood interactions in regular cubic tiling of space inthree-dimensions -- these interactions are used to compute the wave equation locally..

state value (0 = "dead" and 1 ="alive"). This state value updates indiscrete, simultaneous steps as afunction of the state values in the 8neighbors of each cell. If the sum ofthe neighbors' states is > 3 or < 2, thenthe cell is dead (0) on the next timestep (from "overcrowding" or"loneliness", respectively). Otherwiseif it has exactly 3 live neighbors and iscurrently dead, then it is "born" andgoes to 1, and if it was already "alive"then it remains so if it has 2-3 livingneighbors. As anyone who has seenthis system in operation knows, it iscapable of producing remarkablecomplexity from such simple, local,deterministic rules.

The CA framework provides thesimplest kinds of answers tofundamental questions about space, time, and the basic nature of physical laws (Zuse, 1970; Fredkin & Toffoli,1982; Fredkin, 1990). Space is real and fundamental in the form of the underlying cells -- it isn't just an emptyvacuum or a mathematical continuum. Instead, space is a discretized field. The discretization of space, as contrastedwith a true continuum, can be motivated by the levels of infinities associated with the Cantor sets -- a discrete spacecorresponds to the lowest level of infinity associated with the integer number line, and thus represents the simplestway of representing space. One still has an infinity to deal with, and this is plenty mind-blowing all by itself -- spaceand time continuing infinitely in all directions, forever -- but at least the further difficulty of an infinity of space ortime within any given segment can be avoided. And one can reasonably argue that the infinity of space and time ismore plausible than the notion of an edge -- like the old flat Earth models and the end of the world, it is just aninconceivable notion to imagine such a junction between existence and nonexistence. We discuss these issues furtherin the WELDBook/Cosmology chapter.

Time emerges naturally in its unique unidirectionality within the CA framework, simply as a discrete rate of changein the state values. And furthermore, the ratio of discrete spatial cell width to discrete rate of state update provides anatural upper limit to the rate at which anything can propagate within this system: i.e., the speed of light in avacuum. Thus, this principal postulate of special relativity that light has a fixed upper speed limit emerges as anecessary consequence of more fundamental assumptions about the nature of space and time in the CA framework.Furthermore, as we will see in the subsequent chapters, the basic wave equation can be computed using a simple,intuitive, local neighborhood interaction among cells in a CA-like system, and Maxwell's equations for theelectromagnetic field can be computed using primarily this basic wave equation. We discuss the more detailedfeatures of special relativity in relationship to the CA framework next, after first introducing the relevant concepts,but the main conclusion is that this framework predicts all of the features of special relativity, from first principlesbased on the discretization of space and time, together with wave dynamics.In summary, the CA framework is incredibly simple, elegant, and seemingly fundamentally consistent with the most basic facts of physics. To reiterate, if one could develop a viable physical theory within the general confines of this framework, it would seem to provide a preemptively simple and satisfying model of how nature works. It is hard to imagine how anyone could dispute the physical plausibility of such a model. Thus, it would seem that no effort

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should be spared in attempting to develop such a physical model, and every seeming obstacle in the way of such amodel should be treated with the utmost skepticism and scrutiny. Alas, this appears to be a significantly minorityview, not surprisingly so as we address the historical trajectory of quantum physics below.

History of Field Models, the Aether, and Special RelativityTheorizing in physics can be divided into two major epochs for our purposes -- the classical period which reached itszenith just prior to the development of quantum mechanics in the early 1900's, and the post-classical (and current)epoch of fully-developed quantum mechanics and the standard model, which started in the early 1930's. In between,there was an interesting period of transition -- the specific trajectory of theories and experiments in this transitionalperiod served to frame the nature of the mature quantum mechanics that emerged. One can't help but wonder howdifferently things might have turned out if this trajectory had been different -- the influence of Albert Einstein wasparticularly complex, as we'll see. Overall, it seems that he may have done more than any other individual to preventa WELD-like framework from emerging at that time, while at the same time, he was driven to develop a unified fieldtheory that sounds just like what we want to accomplish with WELD, and he famously rejected the standard quantummechanical theorizing.In the classical worldview, people believed that electromagnetic (EM) radiation, described by Maxwell's equations(which represent one of the crowning achievements of the classical epoch), propagated throughout space via theluminiferous aether -- some kind of mysterious, all-pervasive substance that provided a physical model for thephenomena described by Maxwell's equations. The classical worldview was thus dominated by the intuitivelysatisfying notion that local, deterministic physical laws, operating autonomously through some kind of real physicalmedium, could produce the observed behavior of nature. This is essentially identical to the CA framework describedabove.One harbinger of the end of the classical field model was the famous Michelson-Morley experiment of 1887, whichis widely regarded as disproving the existence of the aether. This experiment used patterns of interference from lightbeams traveling in different directions to test for any differences in the speed of light as a function of the relativemotion of the Earth to the aether. The idea was that if the aether is a fixed medium for light, the Earth must bemoving in some direction relative to this fixed medium (as a result of its orbit around the Sun, and the Sun throughthe galaxy, etc), and this difference should thus be measurable in terms of the differential speed of light in differentdirections. The experiment revealed no such differences -- light always travels at the same speed in every direction.

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Figure 1.3: The Lorentz Transformation, a central property of special relativity, whichcauses length in the direction of motion to shrink and time to expand (dilate) as a functionof relative speed, in just such a way as to preserve the observed speed of light regardlessof how fast one is going. The matter wave equation exhibits exactly this behavior, whichcompletely masks any fixed matrix in which such waves might be implemented. In this

example, a speeding light ray is observed at a given time t by an observer in a trainspeeding along at 86.6% of the speed of light , and by "us" sitting in

a stopped train (on a siding presumably). All the measurements in black are what weobserve in this stopped reference frame, while those in red are what the speeding train

guy observes. If we wait 100 nanoseconds (ns) ( seconds -- 100 times slowerthan the clock rate on a 1Ghz computer chip), then this light ray will have moved 30meters. However, from our stopped perspective, the speeding train will be partially

keeping up with the light ray, so that it will appear to have traveled only 4m relative to themoving train. Thus, in this stopped reference frame, where 100ns have passed for this

light to appear to have traveled 4m, we might naively assume that someone on thespeeding train would measure the speed of light as only -- oops! But theLorentz transformations of length and time exactly compensate. The length of the train inthe direction of motion shrinks in half, so that people on the train measure the 4m in the

stopped reference frame as 8m in the moving reference frame -- twice as long.Furthermore, time moves more slowly for the speeding train, such that the 100ns in our

reference frame is measured as only 50ns in the speeding train reference frame (at a staticreference point in the speeding train, which is the very back of the coal tender in this

example). The measurement of time is very strange in special relativity, because what isobserved as occurring at the same time (simultaneity) across different reference framesdepends on both time and location. Thus, when the light ray is measured at 8m ahead of

the back of the coal tender, this registers as only 26.8ns of elapsed time! If you divide this8m by that amount of time, it comes out to exactly the same speed of light as in the

stopped frame. The time transformation equation is: and theposition transformation is: , where t' and x' are as measured on the

speeding train and t, x are on the stopped one, and .

Then, in 1905, Einstein published hisfamous paper on special relativity,which elevated the constant speed oflight to the status of a fundamentalprinciple, along with the principle ofrelativity: that physics should appearthe same in any given (inertial)reference frame. This wasphilosophically the opposite of thediscredited classical notion of a fixedaether medium, which constitutes aprivileged reference frame where theaether itself is standing still. This newprinciple seemed much more appealingthan believing in an invisible aetherthat coincidentally seems to beunmeasurable. Hence, relativity soonbecame a bedrock of thinking inphysics.

However, it remains remarkablyunder-appreciated to this day thatspecial relativity is entirely compatiblewith the notion of a luminiferousaether, and indeed provides exactly theright explanation for why theMichelson-Morley experiment failed todetect it: because the speed of light is aconstant, the lengths of objects mustactually contract in their direction ofrelative motion, and time dilates, sothat even if you are racing very closeto the speed of light, almost keeping upwith a speeding light ray, you measurethe speed of this light to be the same assomeone standing still Figure 1.3.Specifically, because your measuringdevices (rulers) have all shrunk in thedirection of motion, distances appearlonger, and time dilation causes measured time intervals to appear shorter, with the net result that a moving observerobtains the same measured distance per unit time (i.e., speed) that someone standing still would measure. ThisLorentz transformation was already well established prior to Einstein's 1905 paper, based on measurements ofelectromagnetic phenomena.

Thus, we only need to modify our understanding of the properties of the aether, in accordance with the Lorentztransformation, to reconcile the appealing classical world view with the observed facts. But there are two obvious

WELDBook/Intro 12

problems with such an approach. First, the aether becomes essentially unmeasurable, and thus a belief in itsexistence would seem to be outside the scope of objective science. Second, the framework of special relativity has noneed for such a thing, and relativity provides such a nice compelling and self-contained world view, that there is nomotivation to retain this clunky, outdated notion of the aether. And thus, the notion of the aether drifted away like,well, so much aether.It turns out that the CA framework with waves that represent massive particles naturally produces the Lorentztransformation, and thus specifically predicts all of special relativity. As we will see in the next chapter, the simplestform of wave equation always propagates at a fixed speed (like light), but there is a very straightforward extension ofthat equation, involving the introduction of a mass term, that results in waves that can travel at any velocity belowthe speed of light, in proportion to the wavelength of the wave. This relationship between wavelength and speed isexactly as required by the Lorentz contraction, and thus we see fundamentally why the aether is invisible to us -- itemerges directly from the nature of wave dynamics. Therefore, instead of thinking of the Lorentz transformation asan ad-hoc way of reconciling the notion of an aether with the null results of the Michelson-Morley experiment, wecan see that it is a necessary consequence of wave-based physics. Historically, unfortunately, the matter waveequations were not developed until the 1920's, long after the aether had been relegated to a historical artifact, so thispoint was likely not very salient at that time. The historical trajectory of science does seem to be important.Although the elastic nature of space and time described by special relativity are taken for granted in modern physics,it is important to appreciate how fundamentally strange this is, if you think of matter consisting of hard littleparticles. In contrast, the wave-based model of matter produces this elasticity very naturally and automatically --matter is literally fluid and flexible. We will encounter this contrast again several times, for example in the context ofparticles being created out of raw energy, and destroyed back into a burst of energy -- this kind of fluidity is entirelycompatible with waves, and seems hard to reconcile with the notion of hard little particles.In summary, the CA framework provides a very different perspective on special relativity, showing how its essentialfeatures emerge from mechanistic considerations of how to most fundamentally perform physical interactions inspace and time. Although the mathematical results are entirely equivalent, the appealing mechanistic physical modelprovided by the CA framework makes it arguably more fundamental than the abstract principle of relativity.Indeed, relativity affords almost no guidance for thinking about how nature could operate autonomously at amechanistic level. The primary mathematical constructs in relativity involve converting values between differentreference frames using the Lorentz transformation equations, as in Figure 1.3. Without any primary, privilegedreference frame, the best you can do is take any two and convert between them. But what good is this as a physical,mechanistic model of nature? Do we imagine that nature is constantly converting between reference frames all thetime? Why would any one be chosen over another, and what would trigger a conversion, and what would be thepoint of doing any of this in the first place? In short, because the physics is the same in any reference frame, there isnothing to constrain the choice of any given one, and thus there is a fundamental indeterminism as a physical modelif all you have is the bare principle of relativity itself. Relativity is just a self-contained, internally consistentdescription of itself.The fact that the far stronger commitments to mechanism made by the CA framework align so well with what weknow about space and time, and naturally yield the postulates of special relativity as a consequence thereof, seemslike a much more striking coincidence. Therefore, we conclude that even though the mathematics are identical, andthus no experiment can distinguish between them (at the level of special relativity), CA wave dynamics (i.e., WELD)is more useful theoretical framework for understanding how nature actually functions. To further elaborate thisdistinction, we next consider more generally the distinction between autonomous physical models (exemplified byWELD) and calculational tools (exemplified by special relativity).

WELDBook/Intro 13

Calculational Tools vs. Autonomous Physical ModelsWe define an autonomous physical model as a theoretical and mathematical framework for representing physicalprocesses that can iteratively produce observed physical phenomena in a very general way, without requiring specialconfiguration or tuning for particular types of problems that need to be solved. In contrast, a calculational tool is atheoretical and mathematical framework that enables specific results to be calculated, often in one analytical step,typically requiring careful configuration of the equations tailored to the specific case in question. Each of these twodifferent kinds of theoretical frameworks in physics has different advantages and disadvantages for differentapplications, and a failure to distinguish between them can lead to myriad confusions, while keeping these categoriesin mind clarifies many issues. Special relativity is an example of a calculational tool -- it is a systematic set ofequations that enable one to calculate the results of experiments. This tool requires a person to configure and use itproperly -- it does not just crank away autonomously in the way that we imagine nature operating independent of ourown descriptions of it. In contrast, the cellular automaton is a paradigmatic example of a physical model -- itprovides a way of understanding how nature might actually operate, independently and autonomously cranking outthe phenomena we observe.The notion of autonomy provides a critical distinction between the two kinds of frameworks: whereas calculationaltools typically require lots of expert knowledge of how to represent a given physical situation, a physical model canjust iteratively crank away without any expert intervention, and accurately reproduce the known physics.Calculational tools can typically produce results in one step, whereas physical models require integration over manysteps, because they accurately reflect an underlying iterative physical process, and are thus typically more difficult toanalyze mathematically.To make these ideas concrete we consider a few examples:Newton's theory of gravitation (still widely used in practice) is a calculational tool that enables gravitational effectsto be computed in terms of the respective masses ( , ) and distance r between the centers of mass of twobodies:

•But this is not a physical model that could function autonomously, because the math requires one to somehow knowthe physical distances between relevant objects (and their respective masses), and not only is this a nonlocalcomputation, there are a potentially infinite number of other bodies that need to be taken into account. Because of thewell-known n-body problem with this kind of equation, one must carefully choose which entities to include in thecalculation, depending on the exact nature of the problem being solved. By contrast, one would expect that a general,autonomous physical model would compute gravitation directly from the collective effects of each individual atomwithin all the different celestial bodies in the universe, at which point the Newtonian computation is completelyunworkable and absurd.Einstein's general theory of relativity, on the other hand, shows how entirely local, speed-of-light propagation ofspacetime curvature, operating according to uniform functions at each location in space and time, can conveygravitational forces without any of the problems associated with the Newtonian calculational tool. It is a trueautonomous physical model of the first order: the mathematical constructs map directly onto physical processes thatare entirely plausible and compelling for what nature can be autonomously doing to produce the phenomenon ofgravitation.Coulomb's law for the strength of the electric field as a function of distances between charged particles is very similar to Newton's gravitational formula, and similarly represents a useful calculational tool, but is not a good model of how physics actually operates, for all of the same reasons. Similarly, the Coulomb gauge formulation of Maxwell's equations implies immediate action at a distance for the electrical potential, which is clearly incompatible with special relativity. It turns out that some nonlocalities in this framework actually cause the observed EM fields to still propagate at the speed of light, but one can still get into trouble using this gauge incorrectly (Brill & Goodman,

WELDBook/Intro 14

1967; Jackson, 2002; Onoochin, 2002).In contrast, Maxwell's equations in the Lorenz gauge provide a very appealing physical model of electromagnetism(EM), involving simple local wave propagation dynamics operating on the four-vector potential (we'll describe laterexactly what this math means -- for the time being, just appreciate how incredibly simple this equation appears, andalso appreciate that it describes a completely local interaction):

•In this physical model, EM waves naturally propagate at the speed of light, everything is automatically consistentwith the constraints of special relativity, and it is again easy to imagine how autonomous physics can happen likethis.Nevertheless, the reason that people still use Newton's gravitational equation instead of Einstein's equations, andprefer the Coulomb gauge over the Lorenz gauge, is that these frameworks are vastly simpler for calculating thekinds of experimental results people actually need in practical applications. Thus, it should be clear that bothcalculational tools and physical models play essential and complementary roles in the field, and should in no way beconstrued as mutually exclusive (even though people inevitably do). Even though physical models are often notconvenient frameworks for calculation, they play a crucial role in grounding and constraining physical intuition,which should then inform the application of calculational tools. In particular, calculational tools often containshortcuts and simplifications relative to the underlying physical model, and one can obtain nonsensical results ifthese are not appreciated (e.g., accidental violations of speed-of-light propagation in the Coulomb representation).Finally, it is remarkable that in so many instances, there are essentially equivalent complementary descriptions of thesame phenomena: a calculational tool and an autonomous physical model. We will see that the standard model, andstandard quantum mechanics more generally, has all the hallmarks of a calculational tool. And no correspondingautonomous physical model has yet to be described. This is the missing piece that the WELD framework is intendedto provide.

The Importance of Locality for Autonomous Physical ModelsThere is a clear pattern in the examples of autonomous physical models, as contrasted with calculational tools. All ofthe autonomous physical models leverage local propagation of signals according to simple laws, whereas all of thecalculational tools employ nonlocal equations. This is directly tied to their fundamental tradeoffs -- the calculationaltools need nonlocality to enable single-step calculations, whereas the physical models use local dynamics to enableiterative, autonomous calculations to work in the general case. It is difficult to imagine how an autonomous modelcould be nonlocal -- unless every entity interacts with every other one (complete nonlocality, which iscomputationally intractable), then it seems that there must be some kind of decision made to determine whichnonlocal interactions should take place in a given situation. How can this decision be made in a completelyautonomous manner? These problems with nonlocality will recur as we continue to examine more physical domains,and seems like a sufficiently general problem that it strongly motivates skepticism of nonlocal frameworks. We canprovisionally state that any nonlocal model must be a calculational tool, and there should be a correspondingphysical model that leverages local iterative computations to produce the same overall results. Again, WELD isintended to be this local physical model.

Waves vs. ParticlesHaving established some conceptual foundations of the WELD framework with respect to space, time and the nature of physical mechanisms, we now turn to the issues associated with a purely wave-based model of physics, with no trace of hard little particles. It is remarkable how little attention has been devoted to such a purely wave-based model, and it seems that we can attribute at least some of this to psychological issues. People can't help but project the features of our daily experience onto our conceptions of how physics should work. This is generally referred to

WELDBook/Intro 15

as anthropocentrism, and it is constantly creeping into physical theory. For example, we live in a world composed ofseemingly solid "matter" that moves through a transparent gas (air). This underlies the ubiquitous model of solidlittle particles of matter that somehow float through an empty vacuum of space. We'll see as we progress that thisparticle-based worldview creates innumerable problems and contradictions, and yet it seems difficult for people toconsider the obvious alternative: a pure wave ontology. Certainly there are some apparent problems with thewaves-only view, but none of them rise to the level of deep paradoxes and divergent mathematics that accompanythe particle-based approach.We start by dealing with the conceptual difficulties that were introduced by Einstein by proposing the photonparticle, which has an uncomfortable co-existence with the purely field-based model of electromagnetism providedby Maxwell's equations. This juxtaposition of particle and wave models for the same thing paved the way for a morepervasive wave-particle duality in the subsequent development of quantum mechanics. Again, we see the difficultiesthat Einstein seems to have created for a purely wave-based model. And unnecessarily so, it seems, as we find thatthe photon model can be dispensed with, by recognizing that it is just a misplaced accounting of quantized atomicproperties -- one can account for all the relevant phenomena using the classical EM field, interacting with aquantized atomic system (i.e., the semiclassical model of electrodynamics; Jaynes & Cummings, 1963; Jaynes,1973; Mandel, 1976; Grandy, 1991; Marshall & Santos, 1997; Gerry & Knight, 2005).Next, we see if the same disappearing act can be performed for the hard particle aspect of the electron -- this turnsout to be considerably more difficult, in part because almost nobody has seriously tackled this idea before.Nevertheless, none of the various challenges appear to be insurmountable, and indeed we end up with a compellingpurely wave-based model after addressing them. This process also illuminates many of the features of the standardquantum mechanics framework that unambiguously qualify it as a calculational tool, not an autonomous physicalmodel.

The WELD Model of Electromagnetism and the Photon Magic TrickWe begin by returning to the year of 1905, and another of Einstein's triumvirate of revolutionary papers from thatyear, that put another major nail in the coffin of the classical worldview. This paper provided a simple andcompelling model of the photoelectric effect, and introduced the notion of a particle-like doppelgänger of theelectromagnetic field, which we now call the photon (this term was not introduced until the 1920's). The centralpuzzle of the photoelectric effect is that only the frequency, not the amplitude, of light determines whether it willexcite an electron in an atom. You can shine a very bright light on an atom, but if the frequency of that light is belowthe critical value for that atom, then the electrons will remain in their ground state.Einstein's model borrowed some key ideas from Planck's 1901 paper on blackbody radiation, which introducedPlanck's constant h, and arguably represents the very first paper on quantum mechanics. In this paper, Planck wasable to accurately characterize the spectrum of thermal black-body radiation in terms of quantized oscillators whoseenergy was proportional to their frequency, but he made no particular commitment to the physical nature of thesediscrete oscillators, and regarded them initially as merely mathematical contrivances. Einstein in turn postulateddiscrete particle-like entities that carry energy in proportion to their frequency times Planck's constant:

•This accounts in principle for the photoelectric effect, because there is no longer any intensity in the energy equation-- only frequency. An intense beam just has more instances of these discrete photons, but if none of themindividually have sufficient energy to excite the atom, then the number of them doesn't matter. There is anassumption that only one photon can interact with an electron at a time, so energy from multiple photons cannotcontribute additively.This is the first instance of the paradoxical wave-particle duality that lies at the heart of quantum physics. This photon particle model has no clear relationship to the classical EM field described by Maxwell's equations, and this juxtaposition of two radically different descriptions of the same thing has left people in a quandary ever since. As

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with special relativity and Michelson-Morley, things might have been different if extant classical principles had notbeen so quickly rejected in favor of the beguilingly simple photon model. Like special relativity, this quantumphoton model is a calculational tool that does a great job of providing a simple description of the phenomena inquestion, and is extremely handy for calculating the results of experiments, but it really obscures the underlyingmechanisms at work.In particular, the photon model ascribes properties to EM radiation that are actually properties of the atomic system.If one instead recognizes that the atomic system, not the EM radiation itself, can have this discrete frequency-basedbehavior in its interaction with the continuous, classical EM field, then the quandaries and paradoxes start todisappear. As noted earlier, the situation is analogous to a common magic trick, where the magician draws attentionto one obvious system (e.g., the rabbit that seems to disappear), when in fact there is a less obvious system that isactually responsible (e.g., an extra pocket in the magician's hat, that holds the rabbit hidden from view). As weelaborate below, this sleight-of-hand that Einstein pulled out of his hat just over a hundred years ago snowballed intothe full strangeness of the standard theory of quantum mechanics.

Figure 1.4: Widely-cited dramatic example of resonance: the wild oscillations andsubsequent collapse of the Tacoma Narrows Bridge, due to wind driving the bridge to

flutter at a resonant frequency, creating an increasingly strong oscillation that drove it tocollapse. Resonant dynamics can explain the dependence of the photoelectric effect on the

frequency of the EM waves.

In contrast to the photon model, thesemiclassical explanation of thephotoelectric effect is based on apurely classical (Maxwell's equations)model of the EM field, which interactswith a quantized atomic system toproduce the observed frequencydependence on absorption (Jaynes &Cummings, 1963; Jaynes, 1973;Mandel, 1976; Grandy, 1991; Marshall& Santos, 1997; Gerry & Knight,2005). The basic intuition behind thesesemiclassical models is that electronsare locked into bound states in the atomic system, and a minimum resonant frequency is required to wedge them outof these states (Figure 1.4) -- any wave that is below this minimum frequency just doesn't resonate properly with theelectron, and it just passes right through. These bound electrons have discrete, quantized energy levels because theyobey wave equations, and essentially these waves must vibrate like drums or guitar strings, with an integral numberof wavelengths fitting within the overall space available in an atom.

By now, this standing wave model of atomic electrons is very well established, and is clearly responsible for thequantized overall behavior of atoms. The frequency dependence and quantized nature of the atomic system wouldhold if it interacted with anything -- it would be impossible for the EM field to behave other than in this discretizedmanner in its interactions with atoms. Thus, the notion of attributing the discreteness to this novel "photon" particlewould seem to be quite an extra ("quantum" if you will ;) leap. However, Einstein's photon model of 1905 predatedany understanding of the wave nature of electrons in atomic systems by roughly 20 years -- just another example ofhow the trajectory unfolded in an unfortunate way.Another clue that there is something fundamentally misplaced in the photon model is the presence of planck'sconstant h -- we will see that this constant arises directly from adding mass to the wave equations, where the wavestravel at speeds less than the speed of light (i.e., the Klein-Gordon and Dirac equations). Because light(electromagnetic radiation) has no mass, there is no reason for there to be such a constant associated with it, and theclassical EM equations have no place for this constant. On this basis alone, it seems clear that the photon energy isbased on the atomic sources and sinks of EM waves, not the EM field itself. But again, this association of h withmatter waves came later -- in 1905, it was just this mysterious brand-new constant that seemed to be solvingmysteries right and left.

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Although the photoelectric effect has a fairly compelling semiclassical explanation, there are other phenomena thatare harder to explain within this framework. For example, it is possible to have a system that emits a single "photon"of EM energy at a time, and this photon can then be detected later. Inevitably, it is only detected in one specificlocation. This seems like evidence for a localized little particle, and not a more broadly distributed wave. However,we must appreciate that the source of the EM field with sufficient energy to excite an atom is typically thespontaneous emission of photons from other atomic systems. This means that these photons were "created" by a kindof mirror image of the very same discrete process involved in detecting the photons -- this should impart a temporal,spatial, and energy-level discreteness to the EM radiation in the first place. It is essentially impossible to record thespecific profile of these EM waves, but it seems quite plausible that they have a spatiotemporal concentration that isin effect somewhat particle-like (we'll discuss this more below in terms of a wave packet). Thus, for all practicalpurposes, the EM field does behave like photons, but again all of this is attributable to the atomic sources and sinks,not the EM field itself. The exact nature of how the EM field carries energy and interacts with atomic systems viaemission and absorption remains somewhat mysterious, and developing a clear physical model of this process is amajor goal of the WELD approach.At the present time, it seems that the strongest defense of the photon model comes from statistical properties ofphoton emission (e.g., anticorrelations; Grainger et al., 1986; and antibunching; Hong et al., 1987). Semiclassicalaccounts of these phenomena have been provided, by leveraging an additional stochastic process associated with thehypothesized zero point field (Marshall & Santos, 1988, 1997), but this work has failed to overturn the status quobelief in photons, perhaps in part because of various important outstanding issues associated with this zero pointfield construct. We discuss these issues in greater detail later in the book.Overall, this semiclassical physical model requires much more complex calculations and conceptual frameworksthan the simple ideas and math associated with the photon model. This should be familiar, in terms of the dichotomybetween calculational tools (the photon model), and physical models (the semiclassical model). Thus, we fully agreethat the photon model is useful as a calculational tool, while stressing that it should not be taken as a physical model:there really is no reason to believe that such a thing as a photon actually exists in nature. In the next section, we findthat the notion of a photon in the standard model (i.e., quantum electrodynamics or QED) developed from the simplediscrete particle envisioned by Einstein (and probably by most of you), into quite a different thing entirely, which isin many ways the exact opposite.

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The QED Photon: An Infinite, Positionless Wave

Figure 1.5: The Fourier transform, which is the basis of the photon model in QED.A fourier transform converts a function from normal physical space into an

orthogonal basis space of sine waves parameterized according to their amplitude,phase and frequency. No position parameter is retained in Fourier space -- the sinewaves are infinite in extent. The QED model of the photon is, incredibly, one of

these Fourier sine waves, similarly infinite in extent, without any physicallocalization.

In QED, the photon is represented in Fourierspace, as an integer mode of the field -- i.e.,one Fourier component, with a specificfrequency and phase (Figure 1.5). Takenliterally, this means that the photon isdistributed across the entire universe,because a Fourier component is notlocalized in space. Fourier space is based onan orthogonal basis compared to the normalspace-time basis, and the very idea that aphysical entity could be described in termsof this mathematical abstraction seemsfrankly absurd as a physical model. Itcaptures nothing like a localized particle thatone might have otherwise imagined fromEinstein's photon model. Nevertheless, itclearly works fantastically well as acalculational tool, producing the mostaccurate predictions of any physical theory. It seems obvious, but widely unacknowledged in the field, and certainlyin the popular press, that this means that all the math just happens to work out a lot better in fourier space -- this kindof thing is done all the time in many areas of science, but nowhere else do you find people thinking that this meansthat the physical world is actually like Fourier space.

The mathematical connection between QED and Einstein's photon model is that a given Fourier mode has a single,uniform frequency, so it is easy to associate each mode with a specific energy according to the equation.Furthermore, Fourier space enables a convenient way of mathematically managing the creation and annihilation ofphotons. And this framework enables QED to capture the self field or radiation reaction of the EM field producedby the electron, acting back onto the very same electron, which transforms otherwise linear equations into nonlinearones, which are much more difficult to manage mathematically. It was apparently not possible to capture this selffield dynamic in the conventional QM framework prior to QED, and there is every indication that a great deal of thecelebrated accuracy of QED is due to its ability to handle this challenge. And it wasn't easy -- the many deepmathematical complexities associated with the renormalization and perturbation framework in QED reflect themathematical challenges associated with capturing this dynamic correctly. It is an amazing testament to theperseverance and creativity of the many scientists who helped to develop QED that they were able to work throughall these difficulties.We will show in this book how this self field can be captured in numerical simulations, which are essentiallynumerically simulating a cellular automaton-like version of the Maxwell-Dirac system, without any real difficulty orcomplexity whatsoever. Thus, the physical model of the radiation reaction may be much simpler in some ways thanthe calculational tool, although it remains to be seen if such a physical model can be used to derive numericalpredictions anywhere near as accurate as the QED model.

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The WELD Model of the Electron: Atomic Standing WavesWe now turn our attention to the electron, having dispensed conveniently with the photon by passing the quantumbuck from the electromagnetic field back over to the atomic system, where the electron is the primary actor that weconsider -- for our purposes the nucleus can be approximated by a static concentration of positive charge (heldtightly together by the strong nuclear force). This means that we must have a good physical model for electrons,particularly bound electrons that have been trapped by the positive charge of the atomic nucleus. We begin with thehistorical trajectory of classical models of the electron in the context of the atomic system, as they transitioned intothe quantum era. We'll see that the wave model of the electron solves a lot of mysteries, and seems entirelyappropriate for atomic systems. But more challenging problems arise when electrons escape from the atom and runfree -- this case will be addressed separately, in the process requiring a fuller treatment of the ideas andphenomenology of standard quantum mechanics.

The Death Spiral of Classical Point ElectronsThe dominant classical physical model of the atomic system in the early 1900's was the Rutherford model of 1911,with electrons as tiny points of charge and mass, orbiting a nucleus, much like planets orbiting the sun. This modelhad important failings, which the full development of quantum mechanics resolved, thus cementing the demise of theclassical worldview, and solidly establishing quantum mechanics. The major problem with the classical atom wasthat it is fundamentally unstable: the electron should emit EM radiation as it orbits around the nucleus, and thus loseenergy. As it loses energy, the orbit must get tighter, and eventually the electron should just collapse into thenucleus, just like one of those quarters you roll around in a gravity well at a science museum. Furthermore, as itsorbit gets tighter, it should emit higher frequency radiation, predicting a continuous and increasingly high frequencyemission spectrum. Instead, it was known that atoms emit consistent, discrete frequencies of radiation.In 1913, Niels Bohr provided an apparent solution to the problem, leveraging the emerging quantum ideas of Planckand Einstein. He postulated that electrons can only have orbits where the angular momentum is restricted to aninteger multiple of Planck's constant:

•

Although the reason for this restriction was not clear, it immediately made sense of a great deal of data, including theRydberg formula for hydrogen emission spectra. Interestingly, Bohr's initial model retained the classicalelectromagnetic field according to Maxwell's equations.

Matter WavesThe justification for Bohr's restriction on atomic orbits came in 1924, when Louis de Broglie proposed that electronshave a wave-like nature, and thus the only frequencies of electron wave vibration that are stable are standing waves.Standing waves must have an integer number of wavelengths -- retaining the orbiting electron model, this means thatthe electron orbits are restricted such that there are an integer number of such waves per orbit. Shortly thereafter, in1926, Erwin Schrödinger developed his famous wave equation, which then gave a complete mathematicaldescription of the behavior of bound electrons in atomic systems, which made sense of even more data than Borh'soriginal model.The experimental confirmation of de Broglie's matter wave hypothesis came in 1927 in an experiment by Davissonand Germer, who found that electrons moving through a crystal exhibit a diffraction pattern -- such a pattern canonly be produced by some kind of wave-like process, and calculations showed that the de Broglie wavelengthpredicted for the electrons fit the observed diffraction pattern quite well:

•

WELDBook/Intro 20

•

where p is the momentum, h is Planck's constant, and is the de Broglie wavelength. This is about .165 nanometersfor the electrons in the Davisson-Germer experiment (very tiny, but enough to produce a measurable diffractionpattern through the crystal).Both de Broglie and Schrödinger thought that these matter waves were real physical things, like light waves. deBroglie thought that the wave acted to guide the point electron around, in his pilot wave theory, which was laterdeveloped very extensively by David Bohm and colleagues as we discuss below (de Broglie-Bohm Pilot Waves]).Schrödinger initially had an even more radical view, which abandoned the point electron entirely -- he thought hiswave equation described a wave of charge density that is the actual electron. This is exactly the core idea behind theWELD model, and it appears to provide an extremely appealing model of the behavior of electrons within atomicsystems. Indeed, although apparently not widely appreciated, most electrons in an atomic system have zero angularmomentum: they aren't even orbiting the nucleus at all! Instead, they are more like waves on the surface of a drum,which just oscillate in place as standing waves within the atomic potential well, without any net motion at all. Wewill see this behavior emerges naturally out of the Dirac wave equation, in the Atoms Chapter. Thus, there is reallyno basis for any kind of particle model of a bound atomic electron. Currently, the most successful and widely-usedmathematical framework for understanding atomic electron behavior is called density functional theory, which isbased directly on the charge density model, without any point particles at all. However, interestingly enough, thismodel is universally regarded as purely a calculational tool, and thus nobody seems to consider the fundamentalimplications of a pure charge density model of atomic electrons.To summarize, if we think of electrons as pure waves of charge density, then we can readily explain why there arediscrete energy levels in atomic systems, associated with different integral wavelengths of standing waves.Furthermore, this fundamentally quantum behavior of atomic systems must hold regardless of how one might thinkabout the EM field -- there would seem to be absolutely no reason to impose a completely different description ofelectromagnetism in terms of discrete particle-like photons, when the standing-wave nature of the atomic systemseems to provide a full and complete explanation of the quantized nature of the photoelectric effect and other suchphenomena. Simplicity dictates that we retain the well-established wave nature of the EM field, and furthermoreadopt a pure wave model of the electron, making everything consistent and uniform and simple.

Breaking the Waves: Collapse of the Wave Model

Figure 1.6: A gaussian wave packet (a gaussianlocalized envelope times a sine wave) -- this is an

attractive model for a particle with wave-likeproperties, as long as it remains coherent and spatiallylocalized over time. The Schrödinger wave equation

and other standard linear wave equations do notmaintain this spatial localization, and instead lead toever-increasing spreading. Some form of nonlinearityis required -- this is at least the radiation reaction in

WELD, and possibly also gravitation will be required.

Despite all the nice arguments in favor of the pure wave model ofthe electron, Schrödinger abandoned this model fairly quickly, andit has been left essentially dormant all these years, getting scantmention in the published literature. The most obvious problemwith this model arises when considering the behavior of electronsoutside of the atomic system -- it really seems as though theelectron behaves qualitatively differently inside vs. outside theatomic system, and all the wonderful wave-like properties of thebound electron seem to become major problems for describing thefree electron. Schrödinger initially adopted the wave packetmodel for the free electron, where a wave packet is a localizedconcentration of wave energy that seems to provide a potentiallyappealing resolution to the apparent contradictions between theparticle and wave nature of electrons (Figure 1.6). The localized

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nature of the wave packet captures the fact that particles are always seen in one specific location, but the waveproperties capture the interference and wave-like nature, e.g., as reflected in the Davisson-Germer experimentdescribed above.However, Schrödinger quickly realized that his wave equation, when applied to a free electron outside of the atomicsystem, tends to spread out over time, which would seem to imply that electrons would quickly become a broadsmear of charge. But this is incompatible with the observed phenomenology: whenever measured, electrons alwaysshow up well localized in some discrete location, always with the same mass and charge. For example, it hadrecently become possible in the early 1900's to observe the tracks of electrons in cloud chambers, and they alwaysseem to retain their point-like nature, following a clear particle-like track. More generally, this problem gets to theheart of the wave model: waves just seem too fragile and susceptible to diffusion, smearing, and splattering -- whydoesn't the electron leave little tracks of charge dust all over the place, or splatter into many separate droplets duringcollisions? There simply is no evidence of continuously varying charge or mass values for electrons, or fractions ofcharge getting left behind somewhere. Although it is theoretically possible to solve these problems within the contextof a wave equation through the introduction of nonlinearities, the linear Schrödinger wave equation definitely doesnot have the necessary properties to support a waves-only model of the particle. We will pick up this idea in moredetail below, developing the idea that a more accurate wave equation will exhibit the property of emergentlocalization -- it will automatically counteract the tendency of waves to spread, diffuse, and splatter, thus retainingthe intuitively appealing wave packet model of wave-particle duality.There was another problem that also caused Schrödinger to abandon his wave density model -- this is somewhatmore difficult to convey without explaining more about quantum mechanics (which we do later in the book), but ithas to do with the way that Schrödinger's equation is actually used to represent physical processes. The Schrödingerwave function for a single free electron is defined in thee-dimensional space, and it evolves over time. It seems likeit could be a physical thing. But if you want to model two interacting electrons, it turns out you need to use a waveequation defined over nine dimensional space ( ) -- the quantum state of two separate systems is the tensorproduct of the states of the two systems individually. Once this became clear, it was obvious that the wave functionis not plausible as a physical wave of charge in normal 3 dimensional space.If you don't know much about quantum mechanics, you might be a bit puzzled right now. Both of the aboveproblems seem like fundamental problems regardless of the world view you adopt. How could it make any sense inany framework to have some kind of physical process that is defined in a tensor product state space, which gets bigexponentially fast as you add more particles?

• size of quantum state where N is the number of particles, and s is the size of each particle's stateindividually.

The size of such a state space for even a tiny physical system with 100 or so particles is much larger than the numberof atoms in the universe! How would nature possibly accommodate such a thing? Also, isn't the spread of theelectron wave function over time a problem for any model -- what does this wave function really mean that it candescribe something physical that can get spread so thinly, and yet the tracks in the cloud chamber look like perfectclassical particles? There actually are not very satisfying answers to these questions in the standard quantummechanics (QM) worldview. But there is a self-consistent story, which we tell next.

Standard Quantum Mechanics: Probability WavesThe key insight that Bohr and Heisenberg developed in their Copenhagen Interpretation of QM in the late 1920's (which is still dominant today), is that Schrödinger's wave function could be described as a probability wave function, not a physical wave: the wave describes the probabilities of various experimental outcomes. Intrinsic to this view is the wave-particle duality, where the outcomes of experiments (i.e., measurements) involve the collapse of the wave function into a discrete particle state. Thus, in this framework the spreading Schrödinger wave function for the free electron indicates that the distribution of the probability of finding the electron gets wider and wider as time

WELDBook/Intro 22

moves on, but whenever you actually try to measure the electron, it shows up in some specific highly localizeddiscrete location.The other major properties of standard QM that were developed by Bohr and Heisenberg include the famousHeisenberg uncertainty principle, and the broader principle of complementarity (of which the wave-particleduality is one instance) -- the two are closely intertwined. The uncertainty principle can be derived directly fromwave mechanics as we'll see in the Matter Waves Chapter -- it basically says that as you try to resolve one propertyof the system to greater precision, you introduce greater uncertainty in another complementary property. Forexample, if you try to narrow down the position of a given particle to greater precision, you will necessarily increaseyour uncertainty about its momentum, because these are complementary properties. We will see that the momentum(speed) of a particle is a function of the frequency of the waves in its corresponding wave equation (as we mentionedearlier in the discussion of special relativity and the Lorentz contraction of length), and to have a single frequency atwork in a wave equation, the wave must exist over all of space -- that is, its position must be completely unknown.Conversely, to have a perfectly well-defined position, the wave state must have an infinite combination of alldifferent frequencies, superimposed in just the right way so everything cancels out except at one discrete point.Neither of these extremes are physically plausible, but the same principle holds at all intermediate levels -- increasesin spatial precision result in decreases in frequency (momentum) precision, and vice-versa.The fundamental problem with this probabilistic model, which has remained essentially unsolved to this day, is whatcauses this collapse of the wave function into a discrete outcome state, and why do you sometimes evolve the systemover time using Schrödinger's wave function, and then other times you have to collapse this function to get a discretemeasurement function? But if you ignore this tricky question for the moment (we pick it up again in the nextsection), the overall phenomenology of the probabilistic story is very consistent with empirical observations, and themath all works out nicely as well. Furthermore, it retains the intuitively appealing anthropocentric notion of adiscrete hard little particle in there somewhere, even if it has to coexist with a wave some of the time, in anuncomfortably underspecified way. So a large majority of physicists generally ignore these philosophical questions("shut up and calculate" is a common mantra), and the Copenhagen Interpretation adopts an "instrumentalist"philosophy that you simply cannot know what is happening in nature outside of what you actually measure in anexperiment -- every time you try to reason concretely about how nature unfolds when you're not looking at it, youend up confronting mind-numbing paradoxes. Looking at the seemingly endless churning and vague handwaving ofthe other fraction of physicists who attempt to engage in this philosophical debate indicates that these problemsremain as thorny as ever, with no clear satisfying solution. We review some of this in more detail later in the bookafter we cover the relevant phenomenology.At a very basic level, one must recognize that once you describe the wave function as non-physical and simply adevice for computing probabilities of experimental outcomes, the whole edifice of standard QM is obviously acalculational tool, and not a physical model. In this sense, one can actually embrace the Copenhagen Interpretationwholeheartedly -- the philosophy of the approach is very much that the theory does not describe what happens"under the hood", and you just have to take it on face value as a high-level description of how nature behaves. Oneshould not spend any time at all trying to think of the Schrödinger wave equation as a physically real wave, becauseit manifestly is nonphysical. It is just a calculational tool for computing probabilities of experimental outcomes. Likeany calculational tool, it will have its limitations, and it is not mutually exclusive with an actual physical model ofthe same phenomena. It is certainly valuable in developing such a physical model, as it provides very strongconstraints on how such a model must behave overall. But it also unfortunately places strong constraints on theimagination of scientists -- it drives certain patterns of thought and assumptions that can be difficult to overcome inpursuit of a true physical model of the quantum realm.The status of QM as a calculational tool is only more severe in the matrix formalism that was started by Heisenberg,and fully developed by Paul Dirac and John von Neumann -- this is the dominant mode of computation in QM today,and it further abstracts away from wave equations and amounts essentially to an abstract probability calculus. It is

WELDBook/Intro 23

just a way of transforming probability values represented as quantum state vectors -- one starts with an assumedinitial state, applies an operator that captures some assumed properties of a measurement apparatus, and ends up witha resulting state vector that gives the probabilities for each of a number of possible discrete measurement outcomes.Indeed, this probability calculus can be derived from several basic assumptions that have nothing to do with physicsat all (Hardy, 2001; Chiribella, D'Ariano, Perinotti, 2011). In sum, the standard matrix QM is a superb calculationaltool -- highly efficient and accurate for many situations, but it completely obscures any underlying physicalmechanism that might be going on.In the next sections, we develop a few of the properties of QM in greater detail, and then sketch out a framework forhow the WELD mechanisms could yield the known phenomena.

Schrödinger's Cat, Decoherence, and Wave Function CollapseThe thorny issue of wave function collapse can be illuminated somewhat by considering the question of why ourmacroscopic world seems so normal compared to the strange quantum world. This question is often raised in thecontext of the "Schrödinger's cat" thought experiment, where a macroscopic object (the cat) is postulated to exist in asuperposition of quantum states, being indeterminately alive and dead at the same time. Because the cat is placedinside a box, the opening of the box constitutes the first measurement of the cat's state, and the suggestion is thatprior to this measurement, the state of the cat was truly indeterminate -- it really was neither alive nor dead, butsomehow both at the same time (which is what a quantum superposition represents).This thought experiment is useful for recognizing how strange a superposition state really is, but the premise that amacroscopic object could exist in such a superposition state is wrong, which thus resolves the paradox at the outset.The reason a macroscopic object cannot exist in this quantum superposition state is that it has many constituentatoms, and the statistical odds of having all of these atoms in the necessary states of superposition such that theoverall object (e.g., Schrödinger's cat) can itself be in a superposition of two macroscopically different states is soincredibly improbable that it is effectively zero. It is exactly the same reason that coffee and milk never unmix -- it istheoretically possible, but the odds are effectively zero. Hence, our macroscopic world doesn't exhibit thestrangeness of the quantum world precisely because it is large (and also warm and full of noise, which create entropyconstantly pulling against any attempt to create improbably ordered states).At a more process-oriented level, the appreciation of quantum decoherence, developed in the work of Zurek andcolleagues, provides another perspective on the transition between the quantum and macroscopic realms. Essentially,the point is that quantum states of superposition and entanglement are very fragile, and interactions with theenvironment tend to lead these "coherent" states to decohere over time. With the recent push to build quantumcomputers, the problem of limiting decoherence has become a major goal -- the extreme difficulty of doing so is justthe other side of the coin for why quantum weirdness is not observable in the macroscopic everyday world. Thecommon techniques are extreme cold temperatures (very near absolute zero) and strong vacuums to reduce the pullof entropy.The most important fundamental consequence of decoherence is that it provides one explanation for what drives thecollapse of the wave function during a measurement event. Whenever a quantum state is measured, it must somehowinfluence a macroscopic object, and in this interaction, the massive wall of entropy that is the macroscopic objectoverwhelms the minute quantum weirdness present in the microscopic quantum system, thereby collapsing its wavefunction. Technically speaking, the underlying wave function still exists, but it becomes entangled with so muchnoise that it is effectively gone. Although there is still considerable debate about whether decoherence provides afully satisfactory account of wave function collapse, it is clearly an important conceptual tool to understand theactual phenomenology of QM. In the context of the WELD model, it doesn't solve the immediate problems of thespreading wave packet, and we'll have to make other substantial breaks from standard QM in any case.

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The Double-Slit Experiment

Figure 1.7: The double-slit experiment -- narrowopenings in the slits cause the wave to spread through

diffraction, and because of the different distancestraveled in the path from the two different slits to a

given point on the far screen P, the waves willexperience either constructive or destructive

interference, resulting in the wavy bands of light anddark as shown. The quantum paradox here is that this

pattern obtains even when a single particle is emitted ata time -- the particle only ever goes through one slit orthe other, but somehow the wave goes through both.

Figure from wikimedia commons by Lacatosias.

It has been said that the double-slit experiment (also known asYoung's experiment), contains the full mystery of quantummechanics, and we can use it to motivate some of the hardproblems we need the WELD model to solve. The double slitexperiment was around long before quantum mechanics, as a wayof generating interference patterns with waves (Figure 1.7).However, it gets weird in quantum physics when the intensity ofthe light, or beam of electrons or other particles, is reduced to thepoint where there is only a single particle passing through theapparatus at a time. One still observes the interference effect inthis case, demonstrating the paradoxical wave-particle dualityFigure 1.8. In the standard account, the basic phenomenology issimply restated as an "explanation": sometimes things act likeparticles and sometimes they act like waves, and don't expect toget a deeper understanding of why and when!In the case of light (i.e., single "photons"), we adopt the strongclaim as motivated above that the EM field provides the fullaccount of what is going on. Recall our magic trick analogy: thephoton exists not in the EM field but rather in the behavior of thesource and sink atomic systems. Thus, there can be absolutely nomystery in this case about the wave nature of the EM field itself,and its ability to produce interference patterns when passingthrough two slits. The fundamental mystery is then transportedback to why and how light quanta are emitted and absorbed in atomic systems in a way that appears so particle-like,which remains largely unaffected by the double-slit apparatus. A physical intuition is that the EM wave createdduring emission has an uneven distribution of intensity, where the highest intensity region is most likely to be wherethe "photon" is detected -- this creates a basis for particle-like localization within an otherwise wave-based system.Logically, there is a continuum between fully localized wave packets to fully uniform waves, and it is likely thatemitted radiation falls somewhere within this continuum, producing both wave and particle like effectssimultaneously. It is not clear that available experiments would be sensitive enough to detect these deviations fromuniformity, but if so, that would be an interesting prediction to test.The case of electrons and other more particle-like systems in the double-slit experiment is much more challenging tounderstand, even if we adopt the wave-packet model of the particle with emergent localization behavior. Theproblem is that we are caught between a rock and a hard place: if we want to avoid all the difficulties withelectrons-as-waves splattering and fractionating

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Figure 1.8: Results of adouble-slit experiment using

electrons, with increasing numbersof electrons recorded (11, 200,6000, 40000, and 140000). The

interference pattern emerges overtime, even though only single

electrons are detected on each trial.Figure from by Dr. Tonomura via

wikimedia commons.

all over the place, we really need to conceive of the electron wave packet goingentirely through one slit or the other. But if we do that, then we have noexplanation for what goes through the other slit to cause the interference effects.Thus, we continue to face the fundamental dilemma of how to reconcile theparticle-like and wave-like properties of things like electrons -- it seems impossibleto handle both within a single plausible physical model.One clear way to resolve this paradox is to retain the advantages of the coherent,particle-like localized wave packet model, but invoke a different wave medium inaddition that is responsible for the more distal effects associated with the quantumwave function, such as going through the other slit in the double-slit experiment.The only known such medium with the right properties would be the EM field, andwe develop just such an idea below. Outside of this approach of leveraging the EMfield, one would have to hypothesize a completely new field with wave dynamicsthat drives electron behavior -- it seems that one should make every attempt to seeif the existing EM field can do the necessary work, before resorting to making upan entirely new one.

Most approaches that even contemplate the physical basis of quantum mechanics (e.g., the pilot wave theorydescribed below) simply posit the existence of the quantum wave function without considering its possiblerelationship to other known physical fields. This is presumably because, in the standard model, the wave function isso seemingly strange in its extreme high dimensionality and other peculiar properties, that nobody would considerthe possibility that the observed effects could be mediated by the plain old EM field. Furthermore, even electricallyneutral particles are known to have quantum wave functions. Nevertheless, we will see that we can potentially dealwith all of these issues, thereby providing a very parsimonious explanation of the quantum wave function in terms ofknown mechanisms.

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The WELD Free-Electron ModelOver the next few sections, we develop the major components of our model of the free electron, leveraging someimportant ideas from the work of de Broglie and Bohm on the pilot wave approach, and the idea that the EM fieldends up playing a critical role in quantum wave-based phenomena. The core idea is that the Schrödinger waveequation in standard QM summarizes multiple distinct physical processes, one of which is associated with theirreducible uncertainty in the initial configuration of a given system, and another of which is the effects ofoscillations in the EM field that the electron itself generates.

de Broglie-Bohm Pilot Waves

Figure 1.9: Trajectories for particles in the double-slit experiment computedaccording to the de Broglie-Bohm pilot wave model. The interference effects canbe seen as relatively localized bumps in the trajectories, corresponding to steep

gradients in the Schrödinger wave equation. Critically, the underlying trajectoriesare considered to exist at all points even if you don't happen to observe them.

A very different approach to QM that stilluses Schrödinger's wave equation, and endsup being identical to the standard approachin terms of experimental predictions, is thepilot wave theory initially proposed by deBroglie. Somewhat amazingly, David Bohmwas able to develop this approach into afull-fledged account of QM that haswithstood considerable scrutiny, and thisapproach provides a much more concretephysical model for what is going on underthe hood. The key idea is to embrace thewave-particle duality explicitly andcompletely, and not as a complementaryaspect of nature, but rather in terms of twosimultaneously present and interactingproperties of nature. The particle aspect ofthe wave-particle duality is influenced bythe wave function to travel in trajectoriesthat reflect the appropriate wave-baseddynamics, producing the observed wave-likeinterference effects, etc.

In this framework, it is possible to have particles that follow contiguous trajectories through space, without jumpingdiscontinuously from one location to another, or springing into existence only when you look at them (Figure 1.9,Figure 1.10). This is a huge advance over the purely agnostic, instrumental Copenhagen model, because it allows usto understand the nature of the seemingly probabilistic behavior of the quantum realm. Specifically, a majorcomponent of the randomness derives from uncertainty in the initial state

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Figure 1.10: Reconstructed trajectories of photons in a double-slit experimentusing a weak measurement technique that allows aggregate trajectory informationto be reconstructed over many repeated samples that are post-sorted according to a

weak additional modulation of the system -- these are not individual particletrajectories. There is a striking correspondence to the predictions of the de

Broglie-Bohm model. Figure from Kocsis et al, 2011.

Figure 1.11: Spread in the probability wave function due to uncertainty in initialconditions for identically prepared states (each arrow indicates a different actual

trajectory from a different repetition of the experiment) -- this form of wavefunction spreading, captured by the Schrödinger wave equation, is perfectly

sensible. However, the individual particles or wave packets on each run need notexhibit this same kind of spreading -- they can remain tightly localized. This

distinction is captured by the de Broglie-Bohm pilot wave model, and it indicatesthat Schrödinger's wave equation is a composite of multiple different factors, oneof which is purely epistemological, reflecting a basic lack of knowledge about theinitial conditions, and the compounding of the effects of this ignorance over time.

of the system, as opposed to the wildlystochastic behavior of individual particles(Figure 1.11). Thus, the probabilitydistributions reflect the aggregate paths thatdifferent particles can take, given that wedon't really know how they start out, due tofundamental limitations on how well onecan measure things that are so tiny thatevery measurement causes a majordisturbance on the thing being measured.

This is a purely epistemological componentto the quantum wave function, reflecting thelimits on our knowledge, not an actualphysical process. Furthermore, it provides anice explanation for why the wave functionmust inexorably spread out over time --uncertainty always increases the longer yougo without measuring something, assumingthere is some initial uncertainty to beginwith. Thus, wave function spreading is alsoepistemological in this case, not physical,consistent with our idea that the physicalwave packets can remain localized. Lastly,this understanding of what a portion of thewave function represents helps to clear upsome of the mystery regarding the wavefunction collapse process: whenever you getaround to making a measurement, you'll beable to reduce a lot of the uncertainty thathad been building up since you lastmeasured the system -- again, this is not aphysical collapse but just an epistemologicalone.

There remains however a physically realcontribution of the quantum wave functioneven after the epistemological componenthas been factored out. This can be seen inthe double-slit experiment, in terms of theinterference effects from the waves thatwent through the other slit (Figure 1.9,Figure 1.10). These interference effectscannot be attributable to any kind of compounding of uncertainty. In the de Broglie-Bohm framework, thetrajectories of the particles are shaped by a quantum force, which is proportional to the gradient (local slope) of theSchrödinger wave equation. This predicts specific relatively localized bumps in the trajectories wherever theparticles cross a particularly steep portion of the wave -- that the empirically reconstructed trajectories in an actualexperiment seem to show these same bumps appears to be a compelling confirmation of this prediction (Figure 1.10).

WELDBook/Intro 28

The specific form of the quantum force in this theory, and its resulting relatively focal effects on trajectories,provides a nice way of seeing how the epistemological and physical components of the overall Schrödinger waveequation can be decomposed. The overall density and spreading of the wave represents the epistemologicalcomponent, and the little gradient bumps represent what needs to be accounted for by a physical model of thequantum wave effects.The de Broglie-Bohm model falls short of a compelling physical model, because it continues to rely heavily on thestandard Schrödinger wave equation, and does not explicitly draw the epistemological vs. physical distinction thatwe make here. It adopts the same high-dimensional configuration space framework as in standard QM, and indeedBohm really embraced this high dimensional model, regarding there to be one undivided quantum state for the wholeuniverse, which acts nonlocally to communicate influences of all particles on all others. We'll see later that a majorreason for adopting such a framework is the apparent nonlocality of quantum physics, but the evidence for thisnonlocality is actually considerably weaker than is generally recognized, enabling us to pursue a purely local wavemodel. The pilot wave model has a preferred representation in positional (location) space, and it actually has to"defer" to the standard Schrödinger wave function for many other variables (e.g., spin), raising the question as tohow much actual work the particle aspect of the theory is doing.Nevertheless, this model plays an essential role in the WELD framework, by providing a framework for fractionatingthe epistemological and physical aspects of the overall quantum wave function, and generally supporting the notionthat reality does exist even when you aren't looking at it! By taking these ideas to a greater extreme, dispensing withthe high-dimensional Schrödinger waves entirely, and replacing the hard particles with localized wave packets (thathave a nice physical basis for spin and other quantum properties), we think the WELD approach can offer a morefully compelling physical model.

Electromagnetic Oscillations Mediate Quantum Wave Effects

Figure 1.12: Illustration of oscillations in EM field(radiating circles) generated by the vibration of the

wave packet of charge, whose wavelength isproportional to the velocity according to the matterwave equation, exactly as in the Schrödinger wave

equation.

To account for the physical aspects of the quantum wave equationfor an electron, we pursue the most parsimonious model, which isthat the physical phenomena emerge naturally from theMaxwell-Dirac dynamics, as waves within the EM field thatpropagate at the speed of light throughout space, with the deBroglie wavelength associated with the velocity of the electron(Figure 1.12). In the double-slit experiment, it is these EM wavesthat go through the other slit that the electron does not go through,and produce the observed interference. In support of this model,we note several things:• Electrons produce an EM field which generates e.g., the

electrostatic force associated with their negative charge. This isautomatically generated in the Maxwell-Dirac system as aresult of the charge density represented by the Dirac wave.

• The Dirac equation for the electron exhibits an oscillation at thede Broglie wavelength, proportional to the velocity of theelectron -- this is the same wavelength as the Schrödinger wavefunction. This oscillation should entrain a correspondingoscillation in the EM field being emitted by the electron.

• Given the importance of resonance in understanding EM-electron wave interactions, it seems likely that thefrequency of these EM oscillations will determine their ability to influence electron motion -- hence the primaryinfluence on a given electron will be precisely its own EM wave oscillations, which are guaranteed to be at thecorrect frequency. This captures the general principle from quantum mechanics that the particle can only interfere

WELDBook/Intro 29

with itself.• The de Broglie frequency is typically extremely high relative to normal EM radiation, and is thus effectively

invisible. It is also likely to be a very small amplitude ripple that would not be sufficient to excite any kind ofdetector system -- they are just strong enough to slightly tweak electron trajectories.

• This EM-mediated quantum wave function would obtain for any composite object that is made from chargedcomponents, even if the overall composite object is electrically neutral, because it is the oscillations in the EMfield, not the raw level of the field, that matters. Thus, even neutrons or neutral mesons or other particlescomposed of charged quarks would have this property. This is important because it has been established thatneutrons do exhibit wave-like interference effects. The only things that would not would be neutrinos, which arefundamentally neutral and not compositely neutral. They are sufficiently difficult to measure that it is unlikelythat definitive evidence of their quantum wave functions exist, but this theory would state that it does not.

Overall, this appears to provide a plausible physical model for the non-epistemological component of the quantumwave function, that should emerge naturally out of the Maxwell-Dirac system at the heart of the WELD framework.This EM wave naturally propagates within the standard 3-dimensional physical space, avoiding the problemsassociated with the high-dimensional configuration-space model. It currently remains to be seen whether thisdynamic actually does emerge naturally in the system, but it at least provides a promising avenue to pursue at thispoint.

Further Issues with the Schrödinger Wave EquationTo summarize the situation at this point, we can conclude that Schrödinger's wave equation represents a non-physicalcalculational tool that provides a good high-level description of the quantum world. Most attempts to understand theunderlying reality behind this description nevertheless retain the Schrödinger wave equation, which seems like afundamental error. We argue that this equation reflects a mixture of distinct factors, including the epistemologicaluncertainty in initial conditions, and a physical EM-mediated oscillation that alters trajectories slightly to produceinterference effects. That these different factors can be described using the single, linear Schrödinger equation mustbe regarded as a remarkable coincidence, but these coincidences seem rather to be the norm instead of the exceptionin physics. For example, the Bohr model of the hydrogen atom is manifestly inaccurate in assuming particles withactual dynamical orbits around the nucleus, but it gives the same results as the more accurate standing wave model.And we described several other examples of calculational tools and physical models that provide complementarydescriptions of the same thing.To further substantiate the idea that Schrödinger's equation is not appropriate for a physical model, it is easy to showthat it is clearly inaccurate as a model of the electron, for several reasons. As mentioned earlier in the section onQED, the Schrödinger equation is missing any account of how the electromagnetic field emitted by the electroncomes back and influences the very same electron itself. This is known as the radiation reaction or self field. Theproblem is, once you include this radiation reaction (if you can -- it is not easily done in many frameworks), theequations become nonlinear. This has many consequences. First, it ruins the simplicity of the QM calculational tool,which depends critically on the linearity of the Schrödinger equation. Nonlinear equations make math much moredifficult. On the plus side, it provides a critical opportunity to eliminate the exponentially huge state spaces of thestandard framework: the only way to capture interactions between particles using linear equations is through thehigh-dimensional spaces -- if you just have two linear Schrödinger equations in regular 3-dimensional space, theywill just move right through each other due to the property of linear superposition (which we will explore in the nextchapter). In contrast, nonlinear waves have the potential to interact in complex ways in regular 3-dimensional space.Also, Schrödinger's equation is not accurate at relativistic speeds, and more generally the entire framework ofstandard QM treats space and time in a way that is incompatible with relativity: a universal coordinate system andsingle clock are woven into the fabric of the framework. Various arguments have been made about how there are nostrong violations of relativity in actual practice, but at a very basic level, the equations directly violate the principles

WELDBook/Intro 30

of relativity.Finally, any theory that adopts a notion of point particles has a fundamental problem with the self field, because thisfield becomes infinitely strong as you get infinitesimally close to the electron. In contrast, a distributed charge-massmodel of the electron solves this problem handily -- there is a finite density of charge per location (e.g., cell in thecellular automaton), and this gives rise to a finite EM field, which then interacts in obvious well-described ways withthe charge density.

QED Not Quite to the RescueThe framework of quantum electrodynamics (QED) resolves some of these problems with standard QM. It directlyincludes the radiation reaction, and many of its spectacularly accurate corrections to standard QM can be traced backto the effects of the self field. It also is based on the relativistic version of Schrödinger's equation: the Dirac equation,and it specifically includes the process of particle creation and annihilation, which are important relativisticphenomena implied by the famous formula. However, it suffers horribly from infinities that can be tracedback to the point particle infinities from the self field. These were resolved through a process called renormalization,but this is very inelegant and remains a major blemish in the theory. More generally, the theory is based oncomputing infinite sums in a perturbation-based framework (i.e., an infinite series expansion like the Taylor seriesthat may be familiar from calculus), and is thus quite cumbersome. As noted earlier, QED is also basedfundamentally on a fourier space representation that is manifestly non-physical, and it also relies on virtual particles,which are clearly nonphysical as well. These virtual particles can be seen as a mathematical device for capturing theself-field interactions, but also reflect some contribution of background noise that may be present in the EM field, aswe discuss in the next section.In short, QED is clearly yet another calculational tool with many non-physical elements, but it nevertheless showsthat the nonlinearities in the radiation reaction are important, and demonstrates that another, quite different modelcan underlie the simpler standard QM framework. This gives some basis for optimism that a third qualitativelydifferent physical model, the WELD model, can also produce the standard QM behavior as well.

The Zero Point Field and Stochastic ElectrodynamicsMuch has been made of the apparent randomness at the heart of QM. For example, Einstein's oft-repeated quote that"God does not play dice". Empirically, is is certain that nature produces highly variable behavior for experimentalsetups that attempt to control the initial conditions to the greatest extent possible. One important source of thisvariability, discussed above in the context of the de Brogile-Bohm pilot wave model, is the simple inability tocontrol the initial conditions beyond some limit, because you can never establish any kind of known reference pointin the microscopic realm. Everything is subject to the uncertainty principle (which follows directly from everythingbeing fundamentally wave-based), and it is simply impossible to measure or manipulate microscopic systemswithout introducing yet more uncertainty into them.But there is also reason to believe that the EM field is flooded with a level of background noise that could play animportant role in many physical processes. In the QED framework, the vacuum is not actually treated as emptyspace, but is rather the province of the zero point field (ZPF), which has a non-zero level of energy. This can bederived from the uncertainty principle: if a system had zero energy sitting in a confined space (i.e., the bottom of anEM potential well), it would have a definite momentum and position at the same time, which is forbidden. Empiricalevidence for this ZPF comes potentially from the Casimir effect, which is a tiny force measured between two parallelneutral metal plates brought very close together -- the region between these plates should exclude longerwavelengths of the ZPF, and thus have lower energy than the outside region, producing a net force. However, it isalso possible that this force reflects a radiation reaction effect, as it can be derived from QED on that basis alone(Jaffe, 2005).

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The stochastic electrodynamics (SED) and stochastic optics models (Marshall & Santos, 1988; Marshall & Santos,1997; de la Pena & Cetto, 1996) incorporate the ZPF as actual random oscillations in the classical EM field(described by Maxwell's equations), and show how such a field could produce various phenomena such as photonantibunching statistics, which have been taken as one of the last elements of definitive support of the quantumphoton model over the semiclassical approach (a classical EM field interacting with a quantized atomic system).The major problem associated with all of these ZPF models is that the amount of energy in the ZPF would beastronomically huge. Also, it would seem to predict a higher level of spurious photon detection events than isactually observed, although there may be a reasonable solution to this latter problem (Marshall & Santos, 1997). Thesemiclassical theorist Jaynes suggested that instead of imagining that this ZPF fills all of space, it may just reflectnoise emitted by atomic and molecular systems, which will be most intense in the immediate vicinity of thesesources, and fall off dramatically outside of them -- this could potentially eliminate the problem of the huge energylevel -- it would just be a small additional contribution to the observed mass values of atomic systems. In thiscontext, the oscillations in the EM field proposed as the physical component of the quantum wave function may playan important role in the stochastic processes described by the SED model.In any case, going forward, we will keep the possible contributions of EM background noise in mind, as a potentialadditional factor in producing the randomness observed at the quantum level, and a potential contributor to thetrajectories and interactions of electrons in the EM field.

Summary of the WELD Electron ModelWe can now situate the WELD approach to the electron in the above context. WELD fundamentally includes theradiation reaction through the coupled interactions of the Maxwell-Dirac equations, and is thus based on a nonlinearsystem. By using the Dirac equations instead of the Schrödinger waves, it is relativistically accurate, in ways that areconsiderably enhanced by not having to deal with the concept of a hard little particle. For example, waves provide amore natural physical model of particle creation and annihilation, because particles are just emergent waveconfigurations anyway, so it isn't hard to imagine these configurations being created and destroyed. In contrast, it isdifficult to imagine how all the particles get created and destroyed if they are supposed to have these hard, inviolateproperties. Richard Feynman was apparently once asked by his father if the photon was in the atom before it gotemitted. He didn't have a good answer.Due to the nonlinearity of the Maxwell-Dirac system, it is possible for WELD to remain entirely within standard3-dimensional space, while still capturing the complex interactions between multiple particles.We derive a key insight from the de Broglie-Bohm pilot wave model, that a portion of the probabilistic behaviorcaptured in standard QM is attributable to uncertainty in the initial state, and it is this uncertainty that spreads overtime with the Schrödinger wave function. Another portion is due to EM oscillations generated by moving wavepackets, which can cause self-interference effects such as those captured in the double-slit experiment. Furthermore,accumulated background noise in the EM field as discussed above may play an important role as well in shaping(shaking!) the particle trajectories and interactions, to produce additional sources of randomness.A major assumption of the framework is that various forces conspire to produce emergent localization of the wavepacket, counteracting its natural tendency to spread out over time. This then captures the particle-like, localizednature of electron. At the present moment, this is the least well developed aspect of the framework -- it may eveninvolve the influence of gravitation, which can be relatively stronger at the very short length scales associated withquantum phenomena, and need not be particularly strong to counteract the weak tendency of wave packet spread.Despite having a few unresolved features at the present moment, the WELD model provides reasonable physicalanswers to many of the paradoxes that plague the standard model. One of the most important features of the WELDmodel is that it is based on local neighborhood interactions to propagate wave equations -- we next turn to whetherthis property is viable given the current purported evidence for nonlocality in quantum physics.

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Locality vs. Quantum Nonlocality and the Nature of theMeasurement ProcessFinally, we are ready to address the most significant hurdle that the WELD approach must surmount, which is thewidespread belief that quantum physics is fundamentally nonlocal, in ways that defy any sensible physical model.This nonlocality would appear to completely rule out any endeavor along the lines we are pursuing, and is thus amajor reason why it has not been taken seriously by the physics community.The primary line of reasoning underlying this widespread belief in nonlocality traces back to a paper that Einsteinwrote with Podolsky and Rosen in 1935, known as the EPR paper, about the strange implications of quantumentanglement. Entanglement occurs when two particles interact locally in a way that causes their states to becomeentangled (intertwined). The clearest example is when two particles are created from the destruction of anothersource particle -- conservation laws dictate that the sum of the various state values of these particles must equal thoseof the source particle, and this linkage requires that the two particles be entangled. John S. Bell developed the EPRideas further in 1964 using a set of mathematical inequality relationships that really seemed to establish thefundamental nonlocality of quantum mechanics. Bell's inequalities provide a way to experimentally test whether QMis actually nonlocal, and most of these empirical tests appear to support this nonlocality conclusion. However, all ofthese tests have important limitations, which are characterized as "loopholes". If you read the mainstream literatureon this topic, these loopholes are almost universally discounted as the crazy machinations of diehard "realists" whocan't come to terms with the cold hard facts of the quantum world.If you look more closely at the papers describing these loopholes, most of them are actually presenting detailedphysical models based on well-constrained parameters for the devices used in the relevant experiments, andreproducing the observed results without a shred of nonlocality (Marshall, Santos, Selleri, 1983; Marshall & Santos,1985; Thompson, 1996; Adenier & Khrennikov, 2003; Santos, 2005; Aschwanden et al., 2006; Adenier &Khrennikov, 2007). Furthermore, a recent paper shows positive evidence for the main loophole (Adenier &Khrennikov, 2007). Despite the eminently reasonable nature of these purely local models, the fundamental problemis that the results they produce happen to coincide with the predictions from standard nonlocal QM. So again we findan inconvenient coincidence in nature -- if these local models represent the actual truth of the matter, why is natureso perverse in having loopholes that happen to produce the values that the nonlocal model would predict!? At leastone paper has argued that in fact this may be due to some judicious parameter setting on the part of theexperimenters -- nobody runs these experiments blind, and there are always a number of corrective factors that needto be estimated -- it is perhaps not too surprising that the parameters end up aligning with the expected result(Thompson, 1996).Thus, the empirical situation is essentially an inconclusive mess, and given that the limitations on the experimentaldevices appear to be rather fundamental and unlikely to be resolvable, it may be quite some time, or perhaps never,before a definitive experimental proof of nonlocality can be achieved. Santos (2005) argued that the ever-increasingtime passing without such a definitive proof provides increasing support for the localist alternative. Furthermore, heargues that the claim of nonlocality is so momentous and impossible to understand physically, that it should requirethe absolute strongest kind of experimental proof. Certainly this kind of evidence is not yet available. Thus, weconclude overall that local physical models are still tenable, and given how many other implausible properties we'reaccumulating about the standard QM model, it seems unwise to put much faith in its predictions in all respects. It isclearly an abstract calculational tool, and we know that such tools inevitably have limitations -- the predictions ofnonlocality appear to be one in this case. We'll develop a very specific characterization of this limitation below.The other major issue that comes up in this context is understanding the nature of the measurement process in quantum physics. The question is: what is the "true" state of the system before we measure it? Are measurements really just revealing the true inviolate underlying state of the system, or is the measurement outcome actually more of a product of the state of the measuring device interacting with the state of the thing being measured, in potentially complex ways? If you adopt a particle worldview, it is tempting to say that the particle has a very specific set of

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properties at all times, and measurements reveal these. This is called a realist viewpoint (which seems overlygeneral), or a hidden variables theory (with the hidden variables being this definite state that existed prior to themeasurement). Unfortunately, realism gets dragged along with the localist viewpoint, as in a "local realist"perspective, in all the discussions, creating a lot of extra confusion. We can blame this on Einstein, who believed inparticles (e.g., his photons), and took a strong local realist stance.This is unfortunate because our local but wave-based worldview does not require such a strong realist stance: wavescan definitely exist in states of superposition where there really isn't any definite property, and certainly for positionand momentum we can easily see that these are always spread out and imprecise, at least to some degree.Furthermore, waves are very squishy and malleable, so we would fully expect the measurement process tofundamentally transform the wave state itself. If you try to localize a wave, you will definitely squeeze it and changeits momentum (frequency) distribution. This transformative effect of the measurement process is captured in thestandard QM model in terms of the idea that there is no reality outside of the measured reality -- the measurementprocess in some sense creates the reality right there on the spot, and anytime you try to talk about what the state ofthe system was before that measurement, problems arise.

Correlations and Measurements: Starting with Socks

Figure 1.13: Bertlmann's socks as an illustration of a purely predeterminedcorrelation -- correlations in QM are not of this sort -- the process of measurementitself alters the state, creating a correlation in measurements of that state. Figure

from J. S. Bell, 1987

In our treatment of this topic, we focus onthe primary domain where nonlocality hasbeen tested, which involves the generationof entangled pairs of "photons", which arethen subjected to separate measurementprocesses after they fly apart from eachother for some time. The evidence fornonlocality amounts to correlations in theseseparate measurements, which indicate thatsomehow a measurement on one photon isaffecting the outcome of the othermeasurement, despite a significant physicalseparation. Standard QM predicts thiscorrelation because it treats these twoentangled photons as if they were really justa single entity -- they share a commonquantum state. Thus, anything that affectsone aspect of this state must necessarilyaffect the other. This model of entangledparticles sharing a common quantum state is necessary because they start out sharing some properties by virtue ofbeing entangled, and thus if you find out something about one of them, it makes sense that you should knowsomething about the other. J. S. Bell emphasized that this kind of quantum entanglement is not of the intuitivelyobvious sort, as illustrated by the example of Bertlmann's socks (Figure 1.13):

“ ...The philosopher in the street, who has not suffered a course in quantum mechanics, is quite unimpressed byEinstein–Podolsky–Rosen correlations. He can point to many examples of similar correlations in everydaylife. The case of Bertlmann’s socks is often cited. Dr. Bertlmann likes to wear two socks of different colours.Which colour he will have on a given foot on a given day is quite unpredictable. But when you see that thefirst sock is pink you can be already sure that the second sock will not be pink. Observation of the first, andexperience of Bertlmann, gives immediate information about the second. There is no accounting for tastes, butapart from that there is no mystery here. And is not the EPR business just the same?...” (J. S. Bell, 1987).

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This kind of correlation is of the predetermined sort. But the problem in QM is that nothing can be said of the stateof the system prior to a measurement. Thus, counterintuitively, QM says that although the two photons share someproperty, this property is undefined until it is measured. Hence, the correlation cannot be of this simplepredetermined sort like Bertlmann's socks -- instead it is as if Bertlmann is initially wearing white socks on both feet(white being an undefined equipotential color state, reflecting the superposition of all the different colors), whichmagically change to specific colors only when you look at them. If you happen to look at his left foot first, and thesock appears red, then you know the other one cannot be red. This is the (anti)correlation in the measurementoutcomes mentioned above.So far, we could imagine that his socks were somehow still predetermined to change to specific colors when youlook at them, and so the correlation could still be determined in advance. But the quantum situation is weirder thanthat. Instead, it turns out that the parameters of your measurement apparatus play a big role in determining whichcolors you observe. To strain the analogy further, imagine that you have to throw some kind of chemical on his socksto get them to change color, and the color that results is an interaction between which of the different chemicals youchoose, and some kind of stochastic parameters on the socks themselves. Thus, if you throw lemon juice at the leftsock, it will either turn yellow or blue. But if you throw soda water at it, it will either turn red or green. Which oneactually happens is essentially random for all you can tell with a single measurement -- if you do this experimentmany times, on average you get 50% of each option with each chemical (yellow, blue, red, green). But, the weirdthing is that without doing anything to the other sock, you can predict what the measurement will be on it. If you gotyellow, then the other sock must be blue. If you got red, the other must be green. What Bell showed is that it isactually impossible to pre-program the socks to produce these correlations, without somehow having the onemeasurement actually affect the outcome of the other measurement in some way (nonlocality). It may not appearobvious that this is impossible, so we'll work it out in more detail now in a more realistic case.

Beyond Socks: Photon EntanglementThe key move to making this all more understandable is to consider the simpler case of a single particle on whichyou perform two successive measurements, instead of trying to work out what happens with two separate particlesand two separate measurements. QM says these two situations should actually be identical, so we can take advantageof this and just look at the simpler case. After working through this exercise, our conclusion will be that this idea thatthese two situations should always be identical is absurd and completely nonphysical in many cases, especially thoseinvolving photons traveling apart at the speed of light. We then attempt to better understand why QM makes such anabsurd assumption, and argue that the mathematics of the framework are simply insufficiently flexible to represent amore reasonable physical model of this situation. Hence, we find evidence that the calculational tool of quantummechanics is leading us astray. Then, we review the alternative "loophole" models.Our experiment involves making two successive polarization measurements (M1 and M2) on a photon withunknown initial polarization (Figure 1.14a). Polarization refers to the rotational axis of oscillation of the light --unpolarized means that all axes are uniformly represented, while polarized means that a single axis carries most ofthe energy. A polarization measurement is made by placing a photodetector behind a polarizing filter. If we happento know the exact polarization angle of a light source (call it ), a classical EM result known as Malus's law statesthat the polarization filter will allow amount of light through, where is the angle of the filter. Thisis 100% if the angles are the same, and 0% if they are 90 degrees apart, and somewhere in between otherwise. Youcan try it out yourself by tilting your head while looking at your laptop or any other LCD screen with polarizedsunglasses on.Using these facts, we can calculate what would happen with our two measurements. The result of the first measurement (M1) is always going to be completely random, because the polarizations of the photons are unknown and unlikely to be biased in any way, and the angle of the polarization detector used for M1 is totally arbitrary. However, at the next measurement (M2), we can make some very strong predictions. If M2 is set to the same angle

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as M1, then there should be a 100% coincidence rate between the two detectors. That is, if M1 registers a detectionevent, then M2 should as well, and vice-versa. And if M2 is 90 degrees off of M1, there should be a 0% coincidencerate. And for any angle in between, the probability of M2 firing given that M1 did should be based on the respectiveangles of the detectors.Here is the first critical point: these predictions are true regardless of how you rotate the polarizing detectors (thispolarization detector angle is the equivalent of the chemical you throw on the socks in the above example). The onlyway for this to be the case is if the first measurement actually rotates the polarization of the light to align with itsfilter. Otherwise, if instead you thought that the photon had some specific polarization angle that remainedunchanged by M1, the results of M2 would be given by , where is this true "source" polarizationangle. This case would clearly have absolutely no relationship to the angle on M1. This difference of a cosinerelationship between the two detector angles vs. an independent relationship between the two detector angles is thebasis for Bell's inequalities, which simply quantify this difference in a way that is amenable for empirical tests.

Figure 1.14: Demonstration that polarization actually rotates the "photons" in light -- thepolarized lens closest to the camera is oriented perpendicular to the polarization of theLCD screen, and thus blocks nearly all of that light. However, the other lens interposedbetween it and the screen rotates the light so that it can then make it through the lens, as

seen in their overlapping region.

In this case of two sequentialmeasurements, it is trivial to conductthis experiment yourself and see theresults (Figure 1.14). Just take twopolarized sunglasses and rotate themrelative to each other while looking ata polarized light source (e.g., LEDscreen). The critical test is to put thefirst (M1) filter at a 45 degree anglerelative to the polarization of thesource, and then put the second (M2)filter exactly perpendicular to thesource's polarization. If you hold themin a partially overlapping manner, youcan see that no light coming directlyfrom the source gets through thesecond lens, but the light comingthrough the first M1 filter "rescues"(rotates) it and allows light to passthrough.From this experiment, it is obvious that the "measurement process", at least for light waves, does not immaculatelyreveal the "true" polarization state, but rather reflects an interaction between the incoming light wave properties andthe properties of the measuring device. The measuring device imposes a good bit of its own "reality" onto the state ofthe light wave. In QM terminology, this means that the measurement is contextual (Shimony, 1984; Gudder, 1970;Khrennikov, 2001; Rovelli, 1996). As noted above, it is difficult to imagine any measurement taking place on a wavethat would not be contextual in this way. This means that we cannot adopt a strong realist perspective: if we didn'talready know the polarization of the initial source, and it was random, there is no way we can know what the statewas prior to the measurement, because the measurement destroys that information. We only know it was highlyunlikely to have been 90 degrees orthogonal to the orientation of our polarization filter (no filter actually achieves100% blockage).

The key implication of this contextual measurement process is that the quantum correlations arise becausemeasurements affect the system being measured.

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Figure 1.15: Illustration of two successive measurements vs. twoentangled measurements in case of polarization (illustrated as angle of

sine wave oscillation). In the two successive case, the "quantum"correlations arise because the measurement device alters the thing being

measured. In the two separate measurements on entangled photons,there is no obvious physical reason why a measurement in one case

would affect the other, given that they are physically separated and haveno way of interacting. This kind of nonlocal interaction is strictly

impossible for a classical EM field.

Now we get to the second critical part: We take our results from the two sequential measurements, and attempt toapply them, as QM says we should, directly to the case of two separate but entangled photons (call them A and B)(Figure 1.15b). The theory says (with a straight face), that if we perform polarization measurement M1 with a givenangle on photon A, it must somehow influence things such that measurement M2 on B obeys the very same equationas for two sequential measurements: . Yes, the angle of the M1 polarizer must somehow influencethe behavior of the measurement process on B. Even if A and B have had enough time to fly arbitrarily far apart(e.g., in principle to the opposite ends of the universe). It gets better: this is all supposed to happen absolutelyinstantly. No time delay at all.This is a good candidate for the most fantastical, absurd prediction in the history of science, and nearly everyone inquantum physics swallows it whole. It is a completely non-physical, non-local, non sequitur. The physics of twosequential measurements on one photon versus two separate measurements on two separate photons are entirelydifferent, and it is just not clear why anyone would think they should correspond to the exact same thing. This seemslike a classic case of the calculational tools of QM being misapplied, and a cross-check with some kind of physicalmodel would quickly reveal the error. But because there is no accepted physical model in QM, there really isn't asuitable fallback position, and so people just seem to accept what the calculational tools tell them.

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In the WELD framework, we can apply the classical EM physical model to the two photon entanglement case, and itis patently obvious that no such entanglement phenomena should or could be observed. As we just described above,the physics behind the two successive measurements is completely obvious and sensible -- you can visualize thepolarization of the light waves rotating around as they pass through the M1 filter, thus affecting the results for theM2 measurement. The extension to two separate photons makes absolutely no sense -- how can a local interactionbetween a polarization filter and a light wave possibly affect a similar such interaction separated by an arbitrarydistance, when the two light waves have been traveling apart at the speed of light!? There is simply no way withinthe classical EM framework for light waves to continue to interact once they start heading in different directions --everybody's moving at the speed of light, and nothing sticks around in between to mediate any kind of connectionbetween them. Furthermore, EM waves do not even have any way of interacting with each other -- there is nophysical basis for any kind of "signal" to be sent from one EM wave to another -- the only medium for such a signalwould be the EM field itself, and it just passes right through due to linear superposition. Thus, there is no plausiblemechanism that could mediate an entanglement state in the first place, at least according to the classical EM model.

Experimental Tests of Bell-type Inequalities and the LoopholesA number of experiments using entangled photon sources with separate measurements of polarization (as describedearlier) have been conducted, and their results appear to confirm the QM entanglement predictions (Aspect,Dalibard, & Roger, 1982; Aspect, Grainger, & Roger, 1982,Tittel et al, 1998). In one case, the two measurementswere separated by 10km (Tittel et al., 1998)! The major "loophole" for the experiments based on photons is knownas the detection/fair sampling loophole, which basically states that the QM predictions depend on the detectorsreporting a fair sample of the photons that are generated from the source, and enough of them to make sure that allthe relevant statistics are being counted. Well, it turns out that even the best current photodetectors can only detectup to 30% of the photons, and furthermore, there are strong physical reasons to believe that the polarization anglestrongly influences the detection probability, violating the fair sampling assumption. Detailed models of this sort canreproduce the observed data quite accurately, for a variety of experimental configurations (Marshall, Santos, &Selleri, 1983; Marshall & Santos, 1985; Thompson, 1996; Adenier & Khrennikov, 2003; Santos, 2005; Aschwandenet al., 2006; Adenier & Khrennikov, 2007). Interestingly, one of these analyses (Adenier & Khrennikov, 2007)shows that accepting the fair sampling assumption produces results that violate the "no signaling" property of thestandard QM prediction, strongly implicating that fair sampling has been violated.As for the other major loophole, amusingly enough called the "locality" loophole, it pertains to experiments onmassive particles, which are apparently the only ones that can practically close the detection loophole (with rates >90%; Rowe et al., 2001). If locality is considered a loophole, something is seriously wrong with the term "loophole".And the distinction between massive and massless (photons) that determines which "loophole" applies is anythingbut arbitrary, counter to the implication often suggested in discussions of the loopholes. Two massive entangledparticles can always communicate via light-speed interactions (e.g., EM waves) by virtue of the Lorentz contractioneffects of special relativity, which ensure that even when massive objects are moving near the speed of light, lightstill moves at the speed of light relative to them. Indeed, in the Rowe et al (2001) experiment, the two atoms inquestion were strongly interacting via a Coulomb (EM) force, over a very short distance. Furthermore, there areother problems associated with these experiments related to errors in the measurement angles (Santos, 2009).Thus, again, one cannot help but conclude that any reasonable person who appreciated the true importance of theconstruct of locality for understanding how nature actually works, would recognize that these experiments providewoefully ambiguous support for the standard QM model of entanglement, and indeed could be seen as providingincreasingly strong support against the standard view, given the increasing passage of time without a more definitiveexperiment that overcomes the "loopholes" (Santos, 2005).In the next section, we revisit the assumptions that lead to the QM description of entanglement, and consider how thecalculational tool of QM may prevent a more accurate description of the underlying physical processes in terms of

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the distinction between massive and massless particles.

Quantum Entanglement RevisitedWhy does QM predict this bizarre entanglement phenomenon in the first place, and is there some way to generalizethe theory that would accommodate a strong locality constraint on entanglement? The central features ofentanglement that need to be captured in any framework are that the states of the two particles are unknown, and yetthere are strong constraints on their states (either they must be the same or opposite, depending on the specific casein question). Representing exactly this kind of situation for a single particle is the forté of QM: the unknown state isrepresented by a superposition of multiple possible states, and the strong constraints come from the basicconservation laws built into QM, which define how states are affected by measurements. Interestingly, themathematics of QM can be derived very generally from certain kinds of conservation laws, suggesting that thestandard QM formalism is really just an abstract probability calculus, with these strong conservation laws, whichmanifest as a requirement for continuous reversibility (Hardy, 2001), or in the purification postulate (Chiribella,D'Ariano, Perinotti, 2011).One mental image for this is that the measurement process in QM is really just about rotating things around in statespace, e.g., on the surface of a sphere -- you never lose (or gain?) any information about the system in question, youjust rotate that information around on different axes. If we go back to our polarization detectors, these just rotate thepolarization state of the photon around to different angles, but do not fundamentally alter the magnitude of thepolarization property itself. In contrast, if the measurement process did not rotate the polarization state of the photon,then it would be possible to setup a sequence of measurements that eliminate the polarization state entirely --- itwould end up with no measurable polarization at all! Hence, the contextuality of the measurement process is reallyjust a manifestation of this conservation principle that lies at the heart of QM. Another potentially useful image is aball of mercury -- you can squeeze it into many different shapes, but it fundamentally conserves its overallproperties. If you try to measure how tall it is, that squeezing process may cause it to squirt out in the horizontaldimension, and vice-versa. This captures the fundamental uncertainty principle -- squeezing things one way causesthem to squirt out in other ways, meaning that you can't measure both properties simultaneously.All of this makes sense for the state of a single coherent entity (a "particle"'), which is generally indivisible andreally should always behave like that tight little ball of mercury. But does it make sense for two separate entangledparticles? Mathematically, QM represents the two entangled particles just like a single unknown particle, becausethat is presumably the only way to capture the appropriate properties of the state being in a superposition and yetstrongly constrained. This raises the possibility that entanglement is a kind of mathematical accident of thelimitations of the calculational framework -- it just cannot represent this state accurately.But what would an accurate, local, physically-plausible model of entanglement look like? Perhaps the simplestmodel is just that the extent of local interaction determines the amount of continued entanglement that takes place. Inthe most obviously entangled case, you have particles that remain in close physical proximity and are thuscontinuously entangled -- it seems clear here that a first measurement M1 on particle A would likely produce strongdisruption of the state of particle B by perturbing the waves in the local vicinity, such that a second measurement M1on B would very plausibly be affected by M1, exactly as the standard QM entanglement model holds. In this case,the underlying physical model accords well with the assumptions required from the calculational tool, andeverything is consistent. The case of entangled photons moving away from each other at the speed of light representsthe other extreme, which could be described as formerly entangled, and is simply not representable within the QMformalism. Hence all the confusion surrounding this erroneous case. In between, one might imagine some kind ofcontinuum, where some degree of continued interaction produces some level of correlation in the measurements, butnot as strong as one would expect from the continuously entangled case.Interestingly, the idea that physical locality drives entanglement is an important component of the standard QM model already, in terms of the source of entanglement in the first place: creating it requires local physical interaction.

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Therefore, it doesn't seem to be a particularly radical suggestion that the continued maintenance of entanglementshould also depend on continued physical proximity. It is not clear how to mathematically integrate this localityconstraint into the QM formalism, but given that it is merely a calculational tool, it is to be expected that there arethings that it cannot accommodate.It is important to recognize that even with the extreme formerly entangled case, there is no violation of theconservation laws. We do have to assume that each particle does have its zero-sum share of the properties of thesource particle when the entanglement is created, but because the measurement process imposes its own mark on theoutcome, we don't need to maintain that the two separate outcomes sum exactly to one anymore. This is equivalent tosaying that the lack of correlation that we expect to see in this case does not imply some fundamental lack ofconservation.

The EPR ParadoxThe EPR paper (Einstein, Podolsky, & Rosen, 1935) attempted to reveal a deep paradox in the standard QM model,and thus indicate that it was an incomplete model of the quantum realm. This paradox is that the entanglementscenario appears to allow one to determine more information than would otherwise seem possible about the state of aparticle, by performing separate measurements on each of two entangled particles, instead of two sequentialmeasurements on a single "particle." As initially formulated, this paradox was erroneous from the standard QMperspective, because EPR assumed that the two measurements would not affect each other, and yet that M1 on Awould nevertheless tell you something precisely about B. This is having your quantum cake and eating it too -- theonly way M1 can tell you something definitive about B is if it actually affects B in exactly the same way it affects A.Thus, once M1 affects B and thus M2, then it really is identical to two sequential measurements, and there is noparadox.Conveniently, the spectrum outlined above does nothing to introduce a new paradox. Never do we adopt theuntenable assumption of hidden states that simultaneously determine all measurements -- each measurement is aninteraction (i.e., contextuality). For the formerly entangled case of photons, the outcome of M1 on A doesn't tell youvery much about what is going to happen with M2 on B -- in the case of polarization you really only know that A(and thus B) is not polarized 90 degrees relative to the angle on M1 -- it could be 89 or 91 or any other polarity (andthis assumes perfect polarizers which is never possible in reality). Thus, the "heritage" information is much weakerthan the continuously entangled case, and much weaker than what was envisioned in the EPR hidden variables.

Summary of Foundation for the WELD ModelWe have now covered many of the most important foundational issues that motivate and shape the WELD approach.Here, we provide a synthetic summary of the overall picture.First, consistent with the cellular-automaton approach, we think of space as a real entity filled with tiny cubes thatcontain state variables, which are updated through local neighborhood interactions, most of which are variants of thesecond order wave equation. This framework automatically produces the central features of special relativity,including a fixed upper limit on the speed of propagation (one cube per unit time, otherwise known as the speed oflight), and the Lorentz contraction (which happens automatically in the matter wave equation), that ensures that allobservers measure the same speed of light even if they are moving very fast. This Lorentz contraction also causes theunderlying reference frame of the space cubes to be completely invisible.The two coupled, interacting wave equations we consider here are Maxwell's equations for the classicalelectromagnetic field (in the Lorenz gauge, where it is literally just a simple second-order wave equation operatingon the electromagnetic potential), and the Dirac equation for the electron, which we consider to describe a wave ofcharge and mass. These each require four state values, and their first temporal derivatives, for a total of 16 distinctstate values. The Dirac charge produces a corresponding disturbance in the Maxwell field, which in turn pushes the

WELDBook/Intro 40

Dirac charge around -- the system naturally models the radiation reaction or self-field, right along with the naturalinfluence of other EM waves on the electron charge field. Many of the important phenomena captured by the QEDmodel can be understood as emanating from this radiation reaction as well, providing a basis for thinking that theWELD model could produce the same well-verfied results as QED.The coupled Maxwell-Dirac system has many attractive properties as a physical model, including conservation ofcharge and total energy, even though it operates completely locally. Thus, we can be assured that nothing unphysicallike losing charge or energy can occur in this model, even when it might be somewhat difficult to trace at amacroscopic level where everything is going.Although we have just summarized the entire scope of the actual WELD model in terms of physical mechanisms,there are many other interpretational issues of importance to relate this model to the standard thinking in the field.We regard the photon model as a convenient calculational tool, with properties that are attributable to the quantizednature of bound electrons in atomic systems, not to any particle-like nature to the EM field. Thus, we are confidentbased on the considerable successes of existing semiclassical approaches (Jaynes & Cummings, 1963; Jaynes, 1973;Mandel, 1976; Grandy, 1991; Marshall & Santos, 1997; Gerry & Knight, 2005) that the Maxwell field will besufficient to capture all known physics associated with EM radiation.Standard quantum mechanics (QM) based on the linear Schrödinger equation is also a convenient calculational tool,but we know for certain that the relevant wave equation must be overall nonlinear due to the radiation reaction,which is missing from the Schrödinger picture. Furthermore, the standard QM model is not relativistically correct,and uses a single common reference frame and clock. We hypothesize that by using a nonlinear coupled system(Maxwell-Dirac), we can capture the interactions of multi-particle states using strictly a physical 3-dimensionalspace, instead of the exponentially explosive configuration space required by the linear Schrödinger equation.The fully wave-based model of the electron seems entirely appropriate for atomic systems, where electrons typicallydo not have any angular momentum and thus do not "orbit" the nucleus in any sense. Instead, they are bestunderstood as standing waves oscillating within the potential well created by the nuclear charge, and the "electroncloud" (e.g., as captured by the highly successful density functional theory) is an entirely appropriate model foratomic electrons. Thus, simulating the atomic system is the "low hanging fruit" of the WELD approach.The physical model of free electrons is considerably more complicated, because the available data suggests that freeelectrons are more particle-like than a diffuse wave of charge would seem to support. Thus, we hypothesize thatemergent localization must occur, counteracting the natural tendency of waves to spread out over time, and retainingthe localized wave packet model originally envisioned by Schrödinger. To reconcile this model with the standardQM description based on the Schrödinger wave equation, we argue that the Schrödinger wave treated as aprobability wave encompasses at least two distinct sources of variability: one is the variability due to the intrinsicinability to know the initial conditions of particles, and the other is due to EM oscillations produced by moving wavepackets, which can physically impact the actual particle trajectory. This mixing of multiple underlying factors, one ofwhich is clearly entirely epistemological, is another reason to think of the standard QM model as a purelycalculational device for computing results of experiments, and not as any kind of model of the underlying physicalprocess. The quote in the overview by Jaynes, about the "omelette" mixing together epistemological and physicalfactors, summarizes this situation nicely.Finally, we argue that all of the empirical tests of true nonlocality in quantum physics, due to entangled states, aredeeply flawed and subject to compelling alternative explanations in terms of the physical properties of the detectorsused in these experiments. Furthermore, the mathematical framework of standard QM does not seem to be capable ofaccurately describing what seems to be the most appropriate description of these experiments: a formerly-entangledpair of particles that are in an unknown state -- it can only either represent these as having the same or differentquantum states, and neither has the right properties. Thus, one can regard the strong prediction of nonlocality bystandard QM as a clear limitation of the calculational tool, not a necessary reflection of the underlying physicalsystem. This exception does not contradict any other valid applications of standard QM, except this peculiar case of

WELDBook/Intro 41

formerly-entangled particles that are now far apart and have no local way of interacting further. All of these issuesare strongly obscured by the necessary use of high-dimensional configuration space representations in standard QM,which are always nonlocal in nature, making the entangled-at-a-distance case seem entirely ordinary andunexceptional.

Overview of the BookHere is a quick overview of the organization of the remainder of the book. The first chapter, WELDBook/Wavesdevelops the most basic second-order wave equation, and explores the properties of these waves at some length, tobuild up a fundamental understanding of some of the more counter-intuitive properties of waves. Then,electromagnetic (EM) waves are covered in depth in WELDBook/EM -- we can use our basic second-order waveequations to simulate Maxwell's equations, using the Lorenz gauge. This approach greatly simplifies theunderstanding of EM phenomena, letting the numerical simulation do all the hard work, while you enjoy nicevisualizations of various EM phenomena. Next, we extend the basic wave equation to include a mass term inWELDBook/Matter, which provides the basis for matter waves, (i.e., the Klein-Gordon equation) which obey theclassic Newtonian laws of motion, as well as the relativistic versions of these when velocities approach the speed oflight, while also exhibiting a number of uniquely quantum phenomena (such as the relationship between wavelengthand energy/momentum). One can understand a huge swath of physics by fully comprehending just this one simpleequation, including the Hamiltonian (conservation of energy) approach to deriving equations of motion, etc. Afterexploring all of these phenomena, we then extend the Klein-Gordon equation to couple with the EM field, andconserve charge, resulting in the (second-order) Dirac wave equation WELDBook/Dirac, which provides an accuratedescription of the electron. With these basic equations under our belt, we then apply the coupled Maxwell-Diracsystem to a number of specific phenomena, such as atomic systems, free electrons, etc.

References[1] http:/ / en. wikipedia. org/ wiki/ Maxwell's_equations#Units_and_summary_of_equations[2] http:/ / en. wikipedia. org/ wiki/ Standard_Model[3] http:/ / en. wikipedia. org/ wiki/ Special_relativity[4] http:/ / en. wikipedia. org/ wiki/ Cardinality_of_the_continuum

Article Sources and Contributors 42

Article Sources and ContributorsWELDBook/Intro  Source: http://grey.colorado.edu/WELD/index.php?oldid=211  Contributors: Oreilly, Pauli

Image Sources, Licenses and ContributorsFile:fig_ca_2d.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_ca_2d.png  License: unknown  Contributors: OreillyFile:fig space cubes fec lapl.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_space_cubes_fec_lapl.png  License: unknown  Contributors: OreillyFile:fig_lorentz_xform.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_lorentz_xform.png  License: unknown  Contributors: OreillyFile:fig_tacoma_narrows_bridge.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_tacoma_narrows_bridge.png  License: unknown  Contributors: OreillyFile:fig_fourier_transform.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_fourier_transform.png  License: unknown  Contributors: OreillyFile:fig_wave_packet_raw.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_wave_packet_raw.png  License: unknown  Contributors: OreillyFile:fig_double_slit_expt.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_double_slit_expt.png  License: unknown  Contributors: OreillyFile:fig_double_slit_expt_electrons.jpg  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_double_slit_expt_electrons.jpg  License: unknown  Contributors: OreillyFile:fig double slit debroglie bohm.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_double_slit_debroglie_bohm.png  License: unknown  Contributors: OreillyFile:fig double slit kocsis et al 11.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_double_slit_kocsis_et_al_11.png  License: unknown  Contributors: OreillyFile:fig schrodinger spread init cond.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_schrodinger_spread_init_cond.png  License: unknown  Contributors: OreillyFile:fig wave packet wave radiation.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_wave_packet_wave_radiation.png  License: unknown  Contributors: OreillyFile:fig bertlmanns socks.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_bertlmanns_socks.png  License: unknown  Contributors: OreillyFile:fig double polarization expt.jpg  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_double_polarization_expt.jpg  License: unknown  Contributors: OreillyFile:fig qm entanglement photons.png  Source: http://grey.colorado.edu/WELD/index.php?title=File:fig_qm_entanglement_photons.png  License: unknown  Contributors: Oreilly