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FAQs
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How stealthy is a Arleigh Burke destroyer?
Thanks for the ATA! I served on 2 Burke Class DDGs.Burke Class DDGs employ several RCS or ‘Radar Cross Section’ reduction methods.First and foremost is that the hull and more especially the superstructure employ shaping techniques to present angles that will bounce or scatter radar return.Placement of equipment on deck is limited or able to be hidden/enclosed. The mast is angled, enclosed and made of aluminum. The 5 inch gun has been modified on later ships to include a reduced RCS turret.Equipment that cannot be placed elsewhere or hidden has special covers made of radar absorbent material. Similarly parts of the superstructure and any infrastructure that cannot be angled use RAM tile. Any of the RAM areas as well as identified hot spots are coated with special RAM paint. For instance the exhaust pipes are painted with a paint similar to what was used on the F117. Later ships removed the exhaust pipes altogether. Collectively these systems are called PCMS or Passive Counter Measures System.The result is a RCS that is about 70 percent smaller than the ships actual size depending up the platform trying to detect it.Aside from RCS reduction an area that is as important is ‘Emissions Control’. You can hide from radar completely and it will not help a bit if your electronic signature gives you away. The Burke Class excels at controlling emissions. Phased Array SPY-1 radars have a virtuous ability to control emissions by being able to digitally dial its echo, meaning the radar doesnt just use max energy all the time like an easily visible spotlight shooting up into a night sky. Burke can turn its radar off altogether and rely on its own electronic sniffing ECM sensors or through Cooperative Engagement and Tactical Databanks use other platforms sensors to get a picture. For example an E-2D Hawkeye is airborne, it can link with the Burke and now the Burke can see what it sees. Burke can even launch weapons using another platforms sensor picture.
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Why are African leaders and the world quiet about the imprisonment of Diane Rwigara and her family in Rwanda?
Because when you break the Rwandan laws, nobody can come to your rescue in Rwanda. What do you want the African leaders & the world to do? To tell Rwanda that some 35-year-old spoiled lady is above the law, just because she was pretending to run for political office? In case you are not tracking, this is not how Rwanda works.Didier Champion's answer to Why is Diane Rwigara and her family in custody?Rwanda is an independent country and does not receive lectures from the AU or the West. Anything you want to learn about her case, check out my answer on what I think happened and a series of mistakes that she made during her “ political” laughable campaign in Rwanda.This is another article from Johnson Busingye, the Rwandan Minister of Justice. It is an interesting read from the government perspective.Impunity? Not in Rwanda, that's why Diane Rwigara is behind barsThe highly emotional, agenda-driven social media space tends to blur reality, and often drives narratives far removed from facts on the ground.It is important to set straight the matter of Diane Rwigara and put it in the right context, particularly as a misguided campaign that has crossed the line into outright incitement to targeted violence against a particular group.Ms. Rwigara attempted to qualify as an independent candidate in the 2017 presidential elections. However, she failed to meet the requirements laid down in law, which include submitting 600 signatures of endorsement, with at least 20 from each of Rwanda’s 30 districts.Three other independent candidates fulfilled the criteria and were on the ballot, but Diane Rwigara was not. Ms. Rwigara did not contest the National Electoral Commission’s disqualification of her candidacy, nor did she challenge it in court.The NEC also found indications of systematic forgery in the documentation submitted by Diane Rwigara, particularly in the lists of signatures. Electoral fraud is a criminal offense, and the appropriate authorities accordingly commenced investigations.Rule of Law and IntegrityAt the close of investigations, the criminal investigation detectives believed they had evidence of serious crime.Among other things, media then reported unauthorized criminal break-ins into our National Identity Agency and access to ID details of people who would eventually surface on her list of seconders from districts.The Media again reported that many of these people were surprised at finding their names and identification information on those lists. The integrity of Rwanda’s elections is based on the integrity of voters' electronic data and the supporting infrastructure. Any attempt to compromise the system merits thorough investigation.Other suspicious activities involving Ms. Rwigara, her entourage and her family, also attracted media attention and later investigation by police. One example, reported widely in local and international media, and sections of the diplomatic community in Kigali, was the apparent disappearance of Ms. Rwigara, and demands for the government to account for her “disappearance”.Efforts by police to enter the Rwigara home to investigate this allegation were rebuffed by an employee of the family. Days later with pressure mounting, the police’s only option was to lawfully enter the barricaded compound as witnessed by media. Ms. Rwigara and her family were found alive and well in the house.They were taken in for questioning because they had failed to respond to several summonses, as part of the ongoing investigations, and later escorted back to their home.Judicial processMs. Rwigara, her mother, and sister were subsequently charged and presented in court. Charges were later dropped against her sister. Ms. Rwigara and her mother were denied bail because of the likelihood that they would use their substantial financial means to evade justice. The case then proceeded to court.The law presumes the innocence of suspects until proven guilty after trial.Ms. Rwigara and her mother were accorded full rights to legal representation, all the time required to prepare their respective defense, lawyers of their choice and other rights they are entitled to under the law. The next hearing is scheduled for September 24th.It is wrong to claim that Diane Rwigara is undergoing the judicial process described above because she contested the 2017 presidential election. She was a vanity candidate who had no chance of winning more than a handful of votes, and she posed no political challenge to any of Rwanda's established parties.It is also disingenuous to question Rwanda's credentials in women empowerment and gender equality. This policy remains firm in law and practice, and it will grow stronger. It's not just about women; it's about all Rwandans, men, and women. And it's based on the desire to sustain the practice of good politics, which is vital to Rwanda’s rapid and inclusive socio-economic development.Rwanda has recovered, reconciled its population, built unity and continues to register progress in every aspect because Rwandans have turned the page from a destructive era of impunity and entitlement.Where in the past some could and did get away with any kind of crime, today’s Rwanda is characterized by equality and the rule of law. Diane Rwigara is subject to the same rules of the game as any other citizen.In the Rwandan context, turning a blind eye to widely-reported impunity is simply not an option. Ms. Rwigara's rights will continue to be respected, and she will have her day in court. Any opening to impunity would erode Rwanda’s gains.The government I serve believes, as a matter of justice policy, that litigation should come to an end without undue delay, and without consideration of external pressure, so that the ends of justice are served.Hope this helps.Didier Champion
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Is wave-particle duality an illusion?
"Illusion" is an interesting choice of words. To acquire the kind of understanding I think you're after, let's back up a bit and see if we can excavate the foundation of this question. Let me start with a quote. “The voyage of discovery lies not in seeking new horizons, but in seeing with new eyes.” ~ Marcel Proust An examination of the double-slit experiment will give us a good introduction to the mystery you have singled out. But to make that examination worthwhile, we need to make sure that we are familiar with an important effect known as interference. [i]Interference applies universally to all interacting waves. A water wave, for instance, can be described as a disturbance in the shape of the water’s surface. This disturbance produces regions where the water level is higher and regions where it is lower than the undisturbed value. The highest part of each ripple is called a peak and the lowest part is called a trough. Typically waves involve periodic succession, peak followed by trough followed by peak and so on. In general, we can define a wavelength as the distance between identical parts of adjacent waves. Measurements from peak to peak, or trough to trough, for example, give the same value for wavelength.Figure 1 Peaks and troughs of wavesWhen waves interact in a medium, they interfere. For example, if we drop two rocks into spatially separated parts of a pond, their waves will interfere when they cross. (Figure 2) When a peak of one wave and a peak of another wave come together, the height of the water rises to a height equal to the sum of the two peaks. Similarly, when a trough of one wave and a trough of another wave cross, the depression of the water's surface dips to the sum of the two depressions. And when a peak of one wave crosses with a trough of another, the (at least partially) cancel each other out. The peak of one wave contributes a positive displacement while the trough of the other wave contributes a negative displacement. If the two waves have equal magnitude, then there will be perfect cancelation and the water's surface will be flat, just as it was before any wave existed.Figure 12-2 Constructive and destructive interference Keeping these rules of interference in mind, let’s turn our attention to light. If we take a laser emitting a single wavelength—a single color, and shine it on a screen that has a slit etched into it (Figure 3), what image should we expect to see on the wall behind the screen? [ii] Classically speaking, we would expect to see a stripe of light on the wall. (Classically means according to our four-dimensional intuition, or the rules of Euclidean geometry.) It turns out that this is what we see. In this sense light’s behavior correlates perfectly with our Euclidean intuition.Figure 12-3 Expected single slit projectionWhat image should we expect to see on the wall if we etch a second slit on our screen and cover the first slit with a black piece of tape? Well, our classical intuitions tell us to expect a line of light projected on the wall, just like we did before, except this line of light should be offset from the first. Again, this is exactly what we see when we perform the experiment. So far all of this is straightforward and conceptually trivial. But as it turns out, we are only one step away from a profound mystery. We discover this mystery by removing the piece of tape. To understand the impact of this mystery, ask yourself: What sort of projection do we expect to see on the wall when both slits are open?Classical intuition tells us that we should see two parallel bands of light on the wall (Figure 4).Figure 4 Expected double slit projectionBut this is where our classical training (our Euclidean intuition) lets us down. This is also where classical mechanics breaks down. When we perform this experiment, something completely counterintuitive happens, contradicting our Euclidean intuitions. A distinct interference pattern is projected on the wall (Figure 5).Figure 5 Actual double slit projection The bright and dark bands produced in this double-slit experiment are telltale signs that light propagates as a wave. [iii] Interference patterns are key signatures of waves. The problem is that this wavelike characteristic directly clashes with our observations of light’s particulate behavior. After all, photons are always found in point-like regions rather than spread out like a wave, and individual photons are always found to have very discrete amounts of energy. When measuring a wave, you would expect to find its energy spread out over a region instead of being concentrated in one location. So how are we supposed to make sense of this observation? What is going on?These diametrically opposed properties of light are verified facts. Contradictory as they may seem, they are here to stay. They have forced us to the seemingly paradoxical conclusion that light is both a wave and a particle. But how can this be? How can it be both? Although many scientists have found thewave-particle duality of light to be conceptually vague and schizophrenic, this description has persisted. In fact, after the wave-particle concept was adopted as an accurate description of light, it was extended to describe electrons and, eventually, all of matter. This transition was nothing short of a revolution.Up until 1910, atoms were simplistically viewed as miniature solar systems with the nucleus making up the “central star” and orbiting electrons being “planets”. [iv] The wave-particle duality of light and matter rejected this view and pointed to a signNowly different architecture for atoms. Of course, this conceptual transition did not take hold over night.In 1924, Prince Louis de Broglie found that in addition to their particle like character, [v] electrons also possessed a wavelike character. In 1927, Clinton Davisson and Lester Germer followed this up by firing a beam of electrons at a piece of nickel crystal, which acted as a barrier analogous to the one used in the double-slit experiment. A phosphor screen recorded the resultant pattern of electrons. [vi] When they examined the screen, they observed an interference pattern just like the one produced in the double-slit experiment, showing that even electrons have wavelike properties.These experiments shook the foundation of physics by threatening the structure of classical mechanics and destroying humanity’s intuitive framework of reality. But it didn’t stop there. The next step was to tune the beam of electrons down so that the electron gun fired just a single electron at a time. Similar experiments were later used with lasers wherein individual photons were fired seconds apart from each other. The results were mind-bending.Completely against expectation these experiments also produced interference patterns over time as the collection of electrons (or photons) continued to build (Figure 6).Figure 12-6 Over time individual photons construct an interference patternThese observations only added to the confusion. Waves are supposed to be a collective property—something that has no meaning when applied to separate, particulate ingredients. (A water wave, for example, involves a large number of water molecules.) So how can a single electron, or a single photon, be a wave? Furthermore, wave interference requires a wave from one place to interact with a wave from another place. So how can interference be relevantly applied to a single electron or photon? While we are considering such questions, we should also ask, if a single electron or photon is a wave, then what is it that is “waving”? [vii]To answer these questions, Erwin Schrödinger proposed that the stuff that makes up electrons might be smeared out in space and that this smeared electron essence might be what waves. If this idea was correct then we would expect to find all of the electron’s properties, spread out over a distance, but we never do. Every time we locate an electron, we find all of its mass and all of its charge concentrated in one tiny, point-like region. Max Born came up with a different idea. He suggested that the wave is actually a probability wave. [viii] Einstein tinkered with a similar idea when he hypothesized that these waves were optical observations that refer to time averages rather than instantaneous values. Inserting a probability wave (also called a state vector, or a wave function) as a fundamental aspect of Nature delivers another blow to our common-sense ideas about how things truly operate. It suggests that experiments with identical starting conditions do not necessarily lead to identical results because it claims that you can never predict exactly where an electron will be in a single instant. You can only define a probability that we will find it over here, or over there, at any given moment. Two situations with the same probabilistic starting conditions, say of a single particle, might not produce the same results, because the particle can be anywhere within that probability distribution. From a classical perspective, the discovery that the microscopic universe behaves this way is absolutely baffling. Nevertheless, it is how we have observed Nature to be.This leads us to a rather interesting precipice. It seems that the map we have been using to chart physical reality somehow dissolves when we look closely at it. The rules of four-dimensional geometry simply fail to accurately map Nature when we examine the smallest scales. Nature doesn’t strictly behave as our old Euclidean map dictates. Stumbling upon this discovery forces us to face a vital question. Is Nature ultimately and fundamentally probabilistic in a way that we may never understand, as many modern physicists have chosen to believe; or, is this probabilistic quality a byproduct of our reduced dimensional representation of Nature?After pondering these questions long and hard, some physicists have come to believe that the tapestry of spacetime is analogous to water: that the smooth appearance of space and time is only an approximation that must yield to a more fundamental framework when considering ultramicroscopic scales. As far as I can tell, however, up until now this point has only been entertained abstractly. Geometrically resolving a molecular structure for space might resolve our greatest quantum mechanical mysteries, but as of yet, no one has taken that final step. No one has developed a self-consistent picture from this geometric insight. No one has moved beyond the mathematical suggestion that spacetime is analogous to water, or interpreted the theoretical quanta of space as being physically real. Consequently, a framework that enables conceptualization of what is meant by the “molecules” or “atoms” of spacetime has not been developed.Eight decades of meticulous experiments have confirmed the predictions of quantum mechanics based on this wave function, or probability wave, description with amazing precision. “Yet there is still no agreed-upon way to envision what quantum mechanical probability waves actually are. Whether we should say that an electron’s probability wave is the electron, or that it’s associated with the electron, or that it’s a mathematical device for describing the electron’s motion, or that it’s the embodiment of what we can know about the electron is still debated.” [ix]Although quantum mechanics describes the universe as having an inherently probabilistic character, we don’t experience the effects of this character in our day-to-day lives. Why is this? The answer, according to quantum mechanics, is that we don't see quantum events like a chair being here now and then across the room in the next instant, because the probability of that occurring, although not zero, is absurdly miniscule. But what exactly makes the probability for large things to act, as electrons do, so small? At what scales do such effects become important? And, why should the macroscopic universe be so different from the microscopic universe?As if these newly uncovered characteristics of reality weren’t obscure enough, quantum physicists conceptually fuddle things further by suggesting that without observation things have no reality. They claim that until the position of an electron is actually measured the electron has no definite position. Before it is measured, the position exists only as a probability, and then suddenly, through the act of measuring, the electron miraculously acquires the property of position.Einstein acutely recognized the absurdity of this claim. When approached with this conjecture, he famously quipped, “Do you really believe that the moon is not there unless we are looking at it?” [x] To him everything in the physical world had a reality independent of our observations. Measurements that suggested otherwise were mere reflections of the incompleteness by which we currently map and comprehend physical reality. To many quantum physicists, however, the unobserved Moon’s existence became a matter of probability. To them, a discoverable, complete map of physical reality, with the ability to resolve an underlying determinism, became nothing more than a myth—a romantic dream.The mathematical projection of quantum mechanics can be statistically matched with our four-dimensional observations, but when it comes to a conceptual explanation of those observations, it completely lets us down. Intuitive explanations cannot be gleaned from a framework of physical reality that is assumed to be fundamentally probabilistic. By definition, randomness blurs causality. This vague description of physical reality keeps us from grasping a deeper truth by allowing what should be the most basic of concepts to drip into a realm of nonsense.As an example of the confusion that stems from swallowing the standard quantum mechanical interpretation “guts, feathers, and all,” consider the fact that a probabilistic treatment of quantum mechanics leads us to the conclusion that the double-slit experiment can be explained by assuming that a photon actually takes both paths. We can combine the two probability waves emerging from both slits to statistically determine where a photon will land on a screen. The result mimics an interference pattern.According to this, we can explain interference patterns by assuming that one photon somehow always manages to go through both slits, but is this really what is going on? Does a photon really travel along both paths? Can this count as an explanation if we have no coherent sense of what it means? You might notice that if we were to design our experiment with three slits, then we would have to consider whether or not the photon really travels all three routes. This question can be extended for as many slits as you like, but the fundamental conceptual problem remains the same.In order to solve this mystery, you may suggest that we place detectors in front of the slits to determine if the photons are actually going through both slits, or just one. When we do this, we always find that individual photons pass through one slit or the other—never both. But, when we measure the position of individual photons we no longer get an interference pattern and so the question retains its ambiguity. Some have taken this to mean that the act of observation forces wave properties to collapse into a particle, but how and why this theoretical collapse occurs still lacks explanation.Because probability waves are not directly observable and because photons (and electrons) are always found in one place or another when measured, we might be tempted to think that probability waves might not be real—that they were never really there. If that is true, then how are the interference patterns created? Surely these probability waves exist, but in what sense? What are they referencing? Why is it that whenever we know which path the photon takes, we get a classical image instead of an interference pattern? How does the detection of a photon, or an electron, change its behavior?To date, these questions have yet to be resolved. In fact, more clever experiments designed to solve these questions have only deepened the mystery. For example, let’s perform the double-slit experiment again, but this time let’s place devices in front of the slits, which mark (but do not stop or detect) the photons before they pass through the slits. This marking allows us to examine the photons that strike the screen and subsequently determine which slit they passed through. Thus we only gain knowledge of which path the photon takes after the path has been completed. For some reason, however, when we do this we find that the photons do not build up an interference pattern. They form a classical image (Figure 4).Once again, it seems that “which-path” information inhibits us from probing these ghostly waves. But is it really the fact that we gain the ability to determine which path a photon goes through—independent of when we gain that information—that disrupts the interference pattern? Or does our marking of the photon somehow disrupt its interference potential?To explore this question, we perform what’s known as the quantum eraser experiment. We start with the same set up we just described. Then we place another device between each slit and the screen, which completely removes the mark from the photon. We already know that the marked photons project a classical image. Will an interference pattern reemerge if we remove the effects of this mark—if we lose the ability to extract the which-path information?When we perform this experiment the interference pattern does return (Figure 7). Does this mean that photons somehow choose how to act, based on our knowledge of them? Or does it imply something even stranger—that the photons are always both particles and waves simultaneously? How are we to understand either conclusion?Figure 12-7 An interference pattern Another curiosity of Nature is known as the photoelectric effect. Philipp Lenard first discovered this effect through controlled experiments in 1900. When light shines on a metal surface, it causes electrons to be knocked loose and emitted. Knowing this, Lenard designed an experiment that allowed him to control the frequencyof the incoming light. During the experiment, he increased the frequency of the light—moving from infrared heat and red light to violet and ultraviolet. Greater frequencies caused the emitted electrons to speed away with more kinetic energy. After discovering this, Lenard reconfigured his experiment to allow him to control the intensity of the incoming light. He used a carbon arc light that could be made brighter by a factor of 1,000.Because both experiments involved increasing the amount of incoming light energy he expected to have identical results. In other words, because the brighter, more intense light had more energy, Lenard expected that the electrons emitted would have more energy and speed away faster. But that’s not what happened. Instead, the more intense light produced more electrons, but the energy of each electron remained the same. [xi]In response to these experiments Einstein suggested that light is composed of discrete packets called photons. Under this assumption, light with higher frequency would cause electrons to be emitted with more energy, and light with higher intensity, that is, a higher quantity of photons, would result in emission of more electrons—just as we observe.The problem with this solution (a solution that is now universally accepted among physicists) is that it doesn’t provide us with a clear description for what the light quanta are. Why does light come in quantized packets? Near the end of his life Einstein lamented over this problem in a letter to his dear friend Michele Besso. He wrote, “All these fifty years of pondering have not brought me any closer to answering the question, what are light quanta?” [xii] It’s been another fifty years and we seem as confused as ever over how it is that light is quantized into little discrete packets called photons.In the midst of these enigmas lies the uncertainty principle, which states that knowledge of certain properties inhibits knowledge of other complimentary properties. For example, the more accurately we determine the position of an electron, the less we can determine its momentum, and vise versa.Heisenberg tried to explain the uncertainty principle by appealing to the observer effect; claiming that it was simply an observational effect of the fact that measurements of quantum systems cannot be made without affecting those systems. [xiii] Since then, the uncertainty principle has regularly been confused with the observer effect. [xiv] But the uncertainty principle is not a statement about the observational success of current technology. It has nothing to do with the observer effect. It highlights a fundamental property of quantum systems, a property that turns out to be inherent in all wave-like systems. [xv] Uncertainty is an aspect of quantum mechanics because of the wave nature it ascribes to all quantum objects.If our current description of quantum mechanics is fundamental, if there is nothing beneath the state vector—a claim that defines the heart of the standard interpretation of quantum mechanics—then this uncertainty principle may be a sharp enough dagger to kill our quest for an intuitive understanding of physical reality. The corrosive power of the uncertainty principle, when mixed with our current paradigm, is poignantly illustrated by an old story involving Niels Bohr. According to the story, Bohr was once asked what the complementary quality to truth is. After some thought he answered—“clarity.” [xvi] Unlike classical mechanics, which describes systems by specifying the positions and velocities of its components, quantum mechanics uses a complex mathematical object called a state vector (also called the wave function [xvii]) to map physical systems. Interjecting this state vector into the theory enables us to match its predictions to our observations of the microscopic world, but it also generates a relatively indirect description that is open to many equally valid interpretations. This creates a sticky situation, because to “really understand” quantum mechanics we need to be able to specify the exact status of and to have some sort of justification for that specification. At the present, we only have questions. Does the state vector describe physical reality itself, or only some (partial) knowledge that we have of reality? “Does it describe ensembles of systems only (statistical description), or one single system as well (single events)? Assume that indeed, is affected by an imperfect knowledge of the system, is it then not natural to expect that a better description should exist, at least in principle?” [xviii] If so, what would this deeper and more precise description of reality be?To explore the role of the state vector, consider a physical system made of Nparticles with mass, each propagating in ordinary three-dimensional space. In classical mechanics we would use Npositions and N velocities to describe the state of the system. For convenience we might also group together the positions and velocities of those particles into a single vector V, which belongs to a real vector space with 6N dimensions, called phase space. [xix]The state vector can be thought of as the quantum equivalent of this classical vector V. The primary difference is that, as a complex vector, it belongs to something called complex vector space, also known as space of states, or Hilbert space. In other words, instead of being encoded by regular vectors whose positions and velocities are defined in phase space, the state of a quantum system is encoded by complex vectors whose positions and velocities live in a space of states. [xx]The transition from classical physics to quantum physics is the transition from phase space to space of states to describe the system. In the quantum formalism each physical observable of the system (position, momentum, energy, angular momentum, etc.) has an associated linear operator acting in the space of states. (Vectors belonging to the space of states are called “kets.”) The question is, is it possible to understand space of states in a classical manner? Could the evolution of the state vector be understood classically (under a projection of local realism) if, for example, there were additional variables associated with the system that were ignored completely by our current description/understanding of it?While that question hangs in the air, let’s note that if the state vector is fundamental, if there really isn’t a deeper-level description beneath the state vector, then the probabilities postulated by quantum mechanics must also be fundamental. This would be a strange anomaly in physics. Statistical classical mechanics makes constant use of probabilities, but those probabilistic claims relate to statistical ensembles. They come into play when the system under study is known to be one of many similar systems that share common properties, but differ on a level that has not been probed (for any reason). Without knowing the exact state of the system we can group all the similar systems together into an ensemble and assign that ensemble state to our system. This is done as a matter of convenience. Of course, the blurred average state of the ensemble is not as clear as any of the specific states the system might actually have. Beneath that ensemble there is a more complete description of the system’s state (at least in principle), but we don’t need to distinguish the exact state in order to make predictions. Statistical ensembles allow us to make predictions without probing the exact state of the system. But our ignorance of that exact state forces those predictions to be probabilistic.Can the same be said about quantum mechanics? Does quantum theory describe an ensemble of possible states? Or does the state vector provide the most accurate possible description of a single system? [xxi]How we answer that question impacts how we explain unique outcomes. If we treat the state vector as fundamental, then we should expect reality to always present itself in some sort of smeared out sense. If the state vector were the whole story, then our measurements should always record smeared out properties, instead of unique outcomes. But they don’t. We always measure well-defined properties that correspond to specific states. Sticking with the idea that the state vector is fundamental, von Neumann suggested a solution called state vector reduction (also called wave function collapse). [xxii] The idea was that when we aren’t looking, the state of a system is defined as a superposition of all its possible states (characterized by the state vector) and evolves according to the Schrödinger equation. But as soon as we look (or take a measurement) all but one of those possibilities collapse. How does this happen? What mechanism is responsible for selecting one of those states over the rest? To date there is no answer. Despite this, von Neumann’s idea has been taken seriously because his approach allows for unique outcomes.The problem that von Neumann was trying to address is that the Schrödinger equation itself does not select single outcomes. It cannot explain why unique outcomes are observed. According to it, if a fuzzy mix of properties comes in (coded by the state vector), a fuzzy mix of properties comes out. To fix this, von Neumann conjured up the idea that the state vector jumps discontinuously (and randomly) to a single value. [xxiii] He suggested that unique outcomes occur because the state vector retains only the “component corresponding to the observed outcome while all components of the state vector associated with the other results are put to zero, hence the name reduction.” [xxiv]The fact that this reduction process is discontinuous makes it incompatible with general relativity. It is also irreversible, which makes it stand out as the only equation in all of physics that introduces time-asymmetry into the world. If we think that the problem of explaining uniqueness of outcome eclipses these problems, then we might be willing to take them in stride. But to make this trade worthwhile we need to have a good story for how state vector collapse occurs. We don’t. The absence of this explanation is referred to as the quantum measurement problem.Many people are surprised to discover that the quantum measurement problem still stands. It has become popular to explain state vector reduction (wave function collapse) by appealing to the observer effect, asserting that measurements of quantum systems cannot be made without affecting those systems, and that state vector reduction is somehow initiated by those measurements. [xxv] This may sound plausible, but it doesn’t work. Even if we ignore the fact that this ‘explanation’ doesn’t elucidate howa disturbance could initiate state vector reduction, this isn’t an allowed answer because “state vector reduction can take place even when the interactions play no role in the process.” [xxvi] This is illustrated by negative measurements or interaction free measurements in quantum mechanics.To explore this point, consider a source, S, that emits a particle with a spherical wave function, which means its values are independent of the direction in space. [xxvii] In other words, it emits photons in random directions, each direction having equal probability. Let’s surround the source by two detectors with perfect efficiency. The first detector D1should be set up to capture the particle emitted in almost all directions, except a small solid angle θ, and the second detector D2 should be set up to capture the particle if it goes through this solid angle (Figure 8).Figure 8 An interaction-free measurement When the wave packet describing the wave function of the particle signNowes the first detector, it may or may not be detected. (The probability of detection depends on the ratio of the subtended angles of the detectors.) If the particle is detected by D1 it disappears, which means that its state vector is projected onto a state containing no particle and an excited detector. In this case, the second detector D2will never record a particle. If the particle isn’t detected by D1 then D2 will detect the particle later. Therefore, the fact that the first detector has not recorded the particle implies a reduction of the wave function to its component contained within θ, implying that the second detector will always detect the particle later. In other words, the probability of detection by D2 has been greatly enhanced by a sort of “non-event” at D1. In short, the wave function has been reduced without any interaction between the particle and the first measurement apparatus.Franck Laloë notes that this illustrates that “the essence of quantum measurement is something much more subtle than the often invoked ‘unavoidable perturbations of the measurement apparatus’ (Heisenberg microscope, etc.).” [xxviii] If state vector reduction really takes place, then it takes place even when the interactions play no role in the process, which means that we are completely in the dark about how this reduction is initiated or how it unfolds. Why then is state vector reduction still taken seriously? Why would any thinking physicist uphold the claim that state vector reduction occurs, when there is no plausible story for how or why it occurs, and when the assertion that it does occur creates other monstrous problems that contradict central tenets of physics? The answer may be that generations of tradition have largely erased the fact that there is another way to solve the quantum measurement problem.Returning to the other option at hand, we note that if we assume that the state vector is a statistical ensemble, if we assume that the system does have a more exact state, then the interpretation of this thought experiment becomes straightforward; initially the particle has a well-defined direction of emission, and D2records only the fraction of the particles that were emitted in its direction.Standard quantum mechanics postulates that this well-defined direction of emission does not exist before any measurement. Assuming that there is something beneath the state vector, that a more accurate state exists, is tantamount to introducing additional variables to quantum mechanics. It takes a departure from tradition, but as T. S. Eliot said in The Sacred Wood, “tradition should be positively discouraged.” [xxix] The scientific heart must search for the best possible answer. It cannot flourish if it is constantly held back by tradition, nor can it allow itself to ignore valid options. Intellectual journeys are obliged to forge new paths.So instead of asking whether of not wave-particle duality is an illusion, perhaps we should ask whether wave-particle duality implies that the state vector is the most fundamental description of a quantum mechanical system, or if a deeper level description exists? That's an open question, and at the moment there are many possible answers — interpretations of quantum mechanics that are equally aligned with the empirical evidence. What's your answer?For more on this topic, and to discover how pilot-wave theory is elucidated by the assumption that the vacuum is a superfluid, see Einstein's Intuition, available in black and white softcover, full color softcover, full color hardcover, an iBook, and as an audiobook.[i] The discussion on interference and the double-slit experiment that follows is further developed by Brian Greene, (2004). The Fabric of the Cosmos: Space, Time and the Texture of Reality. New York: Knopf, pp. 84–84. Greene’s discussion was used as a general guide here.[ii] In order to show diffraction (a fuzzy border of light on the projected image) the slit must have a width that does not greatly exceed the wavelength of the color of the light that we have chosen.[iii] Light’s wave nature was first revealed in the mid-seventeenth century through experiments performed by the Italian scientist Francesco Maria Grimaldi, and was later expanded upon by experiments performed in 1803 by the physician and physicist Thomas Young. (1807). Interference of Light; Alan Lightman. A Sense Of The Mysterious. pp. 51–52, 71.[iv] Before the “planetary model” of the atom, physicists pictured the atom being a plum-shaped blob (the nucleus) with tiny protruding springs that each had an electron stuck to its end. When the atom absorbed energy it was thought that these electrons would jiggle (oscillate) on the ends of their springs. Consequently, any atom that was above its ground state of energy was understood to be an “excited atomic oscillator,” This depiction of the atom wasn’t overthrown until 1900. At that point in history the physical existence of atoms was still controversial. It was replaced by the planetary model, which in turn was replaced by the electron cloud model we use today—a model that was initiated in 1910 and was secured by 1930. Gary Zukav. The Dancing Wu Li Masters, pp. 49–50.[v] Electrons can be individually counted and you can individually place them on a drop of oil and measure their electric charge. Richard Feynman. (1988). QED, The Strange Theory of Light and Matter. Princeton University Press, p. 84.[vi] According to de Broglie’s doctoral thesis all matter has corresponding waves. The wavelength of the “matter waves” that “correspond” to matter depends upon the momentum of the particle. Specifically, , which falls into an important group of equations along with Planck’s equation ) and the ever famous . (λ, pronounced “lambda,” stands for wavelength, h is Planck’s constant, and pronounced ‘nu’ represents the frequency of a photon) From this equation we are told to expect that when we send a beam of electrons (something we might traditionally think of as a stream of particles) through tiny openings, like the spacing between atoms in a piece of nickel crystal, the beam will diffract, just like light diffracts. The only requirement here is that the spacing between the atoms of the material must be as small, or smaller, than the electron’s corresponding wavelength—just like the slits in our double-slit experiment. When we perform the experiment, diffraction and therefore interference, occurs exactly as wave mechanics predicts.[vii] Part of the problem here is that in keeping with our four-dimensional intuition we tend to assume a particle aspect in the double-slit experiment without accounting for nonlocality. By doing this we are technically violating Heisenberg’s uncertainty principle and missing the bigger picture.[viii] M. Born. (1926). Quantenmechanik der Stossvorgänge. Zeitschrift für Physik 38, 803–827; (1926). Zur Wellenmechanik der Stossvorgänge. Göttingen Nachrichten 146–160.[ix] Brian Greene. (2004), p. 91.[x] Albert Einstein quoted in Einstein by Walter Isaacson.[xi] Walter Isaacson. Einstein, pp. 96–97.[xii] Ibid.[xiii] Werner Heisenberg. The Physical Principles of the Quantum Theory, p. 20.[xiv] Masano Ozawa. (2003). Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Physical Review A 67 (4), arXiv:quant-ph/0207121; Aya Furuta. (2012). One Thing Is Certain: Heisenberg’s Uncertainty Principle Is Not Dead. Scientific American.[xv] L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y, Soudagar, & A. M. Steinberg. (2012). Violation of Heisenberg’s Measurement—Disturbance Relationship by Weak Measurements. Physical Review Letters 109 (10).[xvi] Steven Weinberg. Dreams Of A Final Theory, p. 74.[xvii] For a system of spinless particles with masses, the state vector is equivalent to a wave function, but for more complicated systems this is not the case. Nevertheless, conceptually they play the same role and are used in the same way in the theory, so that we do not need to make a distinction here. Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 7.[xviii] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. xxi.[xix] There are 6N dimensions in this phase space because there are N particles in the system and each particle comes with 6 data points (3 for its spatial position (x, y, z) and 3 for its velocity, which has x, y, zcomponents also).[xx] The space of states (complex vector space or Hilbert space) is linear, and therefore, conforms to the superposition principle. Any combination of two arbitrary state vectors and within the space of states is also a possible state for the system. Mathematically we write where & are arbitrary complex numbers.[xxi] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 19.[xxii] Chapter VI of J. von Neumann. (1932). Mathematische Grundlagen der Quantenmechanik, Springer, Berlin; (1955). Mathematical Foundations of Quantum Mechanics, Princeton University Press.[xxiii] It might be useful to challenge the logical validity of the claim that something can “cause a random occurrence.” By definition, causal relationships drive results, while “random” implies that there is no causal relationship. Deeper than this, I challenge the coherence of the idea that genuine random occurrences can happen. We cannot coherently claim that there are occurrences that are completely void of any causal relationship. To do so is to wisk away what we mean by “occurrences.” Every occurrence is intimately connected to the whole, and ignorance of what is driving a system is no reason to assume that it is randomly driven. Things cannot be randomly driven. Cause cannot be random.[xxiv] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 11.[xxv] Bohr preferred another point of view where state vector reduction is not used. D. Howard. (2004). Who invented the Copenhagen interpretation? A study in mythology. Philos. Sci. 71, 669–682.[xxvi] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 28.[xxvii] This example was inspired by section 2.4 of Franck Laloë’s book, Do We Really Understand Quantum Mechanics?, p. 27–31.[xxviii] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 28.[xxix] T. S. Eliot. (1921). The Sacred Wood. Tradition and the Individual Talent.
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Rwandan Diane Rwigara has been detained and faces 20 years in prison. What exactly was her crime?
Nobody is above the law in Rwanda.Rich or poor, young or old, male or female, wannabe politician or business person, the laws of Rwanda have to be respected. Plain and simple.I can hardly believe that I am answering a question about some lady who could not even get 600 required signatures out of millions of Rwandan voters. Failing to get 20 signatures from each of the 30 districts? How ill-prepared do you have to be disqualified to launch an independent candidacy in Rwanda?Had I been running, despite being a nobody in Rwanda, I could have gotten 5,000–6,000 signatures very easily.The lady had no idea of what she was doing. She failed to comply with local laws and regulations of the electoral commision. She was looking for attention in the western media. For better or worse, she has got it.Failed to provide enough signatures to support her candidacy to the Electoral Commission.When she failed, the falsified some of the signatures including those of dead people!When her candidacy was denied, she complained to the western media, crying for help. She thought she could get away on the “ woman” card!Later, due to her lack of experience, she started to talk to genocide deniers in Belgium and Europe about some phony advocacy for human rights. On this one, she did not know. But again, her ignorance was very apparent. In my opinion, the government could have dropped these charges.As things got tougher, she faked her disappearance and made the police force look for her. A few days later, she was discovered at her family house. She had been there all along. From there, things went worse for her.All in all, she lacked the experience and made very grave mistakes. She really should not be blaming anybody but herself and her advisors. Did she have any? I don’t know.I have a series of excellent answers on her criminal charges and why she was not ready for the tasks and challenges ahead when she decided to run for the highest office in the land without any qualification whatsoever.Didier Champion's answer to Why is Diane Rwigara and her family in custody?Didier Champion's answer to Why are African leaders and the world quiet about the imprisonment of Diane Rwigara and her family in Rwanda?This is a good article by Johnson Busingye, the Rwandan Minister of Justice about her criminal charges. It is the most comprehensible response from the government perspective.Impunity? Not in Rwanda, that's why Diane Rwigara is behind barsThe highly emotional, agenda-driven social media space tends to blur reality, and often drives narratives far removed from facts on the ground.Criminal ChargesIt is important to set straight the matter of Diane Rwigara and put it in the right context, particularly as a misguided campaign that has crossed the line into outright incitement to targeted violence against a particular group.Ms. Rwigara attempted to qualify as an independent candidate in the 2017 presidential elections. However, she failed to meet the requirements laid down in law, which include submitting 600 signatures of endorsement, with at least 20 from each of Rwanda’s 30 districts.Three other independent candidates fulfilled the criteria and were on the ballot, but Diane Rwigara was not. Ms. Rwigara did not contest the National Electoral Commission’s disqualification of her candidacy, nor did she challenge it in court.The NEC also found indications of systematic forgery in the documentation submitted by Diane Rwigara, particularly in the lists of signatures. Electoral fraud is a criminal offense, and the appropriate authorities accordingly commenced investigations.Rule of Law and IntegrityAt the close of investigations, the criminal investigation detectives believed they had evidence of serious crime.Among other things, media then reported unauthorized criminal break-ins into our National Identity Agency and access to ID details of people who would eventually surface on her list of seconders from districts.The Media again reported that many of these people were surprised at finding their names and identification information on those lists. The integrity of Rwanda’s elections is based on the integrity of voters' electronic data and the supporting infrastructure. Any attempt to compromise the system merits thorough investigation.Other suspicious activities involving Ms. Rwigara, her entourage and her family, also attracted media attention and later investigation by police. One example, reported widely in local and international media, and sections of the diplomatic community in Kigali, was the apparent disappearance of Ms. Rwigara, and demands for the government to account for her “disappearance”.Efforts by police to enter the Rwigara home to investigate this allegation were rebuffed by an employee of the family. Days later with pressure mounting, the police’s only option was to lawfully enter the barricaded compound as witnessed by media. Ms. Rwigara and her family were found alive and well in the house.They were taken in for questioning because they had failed to respond to several summonses, as part of the ongoing investigations, and later escorted back to their home.Judicial processMs. Rwigara, her mother, and sister were subsequently charged and presented in court. Charges were later dropped against her sister. Ms. Rwigara and her mother were denied bail because of the likelihood that they would use their substantial financial means to evade justice. The case then proceeded to court.The law presumes the innocence of suspects until proven guilty after trial.Ms. Rwigara and her mother were accorded full rights to legal representation, all the time required to prepare their respective defense, lawyers of their choice and other rights they are entitled to under the law. The next hearing is scheduled for September 24th, 2018.It is wrong to claim that Diane Rwigara is undergoing the judicial process described above because she contested the 2017 presidential election. She was a vanity candidate who had no chance of winning more than a handful of votes, and she posed no political challenge to any of Rwanda's established parties.It is also disingenuous to question Rwanda's credentials in women empowerment and gender equality. This policy remains firm in law and practice, and it will grow stronger. It's not just about women; it's about all Rwandans, men, and women. And it's based on the desire to sustain the practice of good politics, which is vital to Rwanda’s rapid and inclusive socio-economic development.Rwanda has recovered, reconciled its population, built unity and continues to register progress in every aspect because Rwandans have turned the page from a destructive era of impunity and entitlement.Where in the past some could and did get away with any kind of crime, today’s Rwanda is characterized by equality and the rule of law. Diane Rwigara is subject to the same rules of the game as any other citizen.In the Rwandan context, turning a blind eye to widely-reported impunity is simply not an option. Ms. Rwigara's rights will continue to be respected, and she will have her day in court. Any opening to impunity would erode Rwanda’s gains.The government I serve believes, as a matter of justice policy, that litigation should come to an end without undue delay, and without consideration of external pressure, so that the ends of justice are served.Hope this helps.Didier Champion
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What is wave-particle duality?
Warning: Wave-particle duality gave birth to the mind-numbing world of quantum mechanics. Understanding it may mean upturning everything you believe about the world so that you are free to climb to a different perspective where it all makes sense. If you're willing to take that challenge on, then keep reading.“The voyage of discovery lies not in seeking new horizons, but in seeing with new eyes.” ~ Marcel ProustAn examination of the double-slit experiment is a great place to start. To make that examination worthwhile, we need to make sure that we are familiar with an important effect known as interference. [i]Interference applies universally to all interacting waves. A water wave, for instance, can be described as a disturbance in the shape of the water’s surface. This disturbance produces regions where the water level is higher and regions where it is lower than the undisturbed value. The highest part of each ripple is called a peak and the lowest part is called a trough. Typically waves involve periodic succession, peak followed by trough followed by peak and so on. In general, we can define a wavelength as the distance between identical parts of adjacent waves. Measurements from peak to peak, or trough to trough, for example, give the same value for wavelength.Figure 1 Peaks and troughs of wavesWhen waves interact in a medium, they interfere. For example, if we drop two rocks into spatially separated parts of a pond, their waves will interfere when they cross. (Figure 2) When a peak of one wave and a peak of another wave come together, the height of the water rises to a height equal to the sum of the two peaks. Similarly, when a trough of one wave and a trough of another wave cross, the depression of the water's surface dips to the sum of the two depressions. And when a peak of one wave crosses with a trough of another, the (at least partially) cancel each other out. The peak of one wave contributes a positive displacement while the trough of the other wave contributes a negative displacement. If the two waves have equal magnitude, then there will be perfect cancelation and the water's surface will be flat, just as it was before any wave existed.Figure 12-2 Constructive and destructive interference Keeping these rules of interference in mind, let’s turn our attention to light. If we take a laser emitting a single wavelength—a single color, and shine it on a screen that has a slit etched into it (Figure 3), what image should we expect to see on the wall behind the screen? [ii] Classically speaking, we would expect to see a stripe of light on the wall. (Classically means according to our four-dimensional intuition, or the rules of Euclidean geometry.) It turns out that this is what we see. In this sense light’s behavior correlates perfectly with our Euclidean intuition.Figure 12-3 Expected single slit projectionWhat image should we expect to see on the wall if we etch a second slit on our screen and cover the first slit with a black piece of tape? Well, our classical intuitions tell us to expect a line of light projected on the wall, just like we did before, except this line of light should be offset from the first. Again, this is exactly what we see when we perform the experiment. So far all of this is straightforward and conceptually trivial. But as it turns out, we are only one step away from a profound mystery. We discover this mystery by removing the piece of tape. To understand the impact of this mystery, ask yourself: What sort of projection do we expect to see on the wall when both slits are open?Classical intuition tells us that we should see two parallel bands of light on the wall (Figure 4).Figure 4 Expected double slit projectionBut this is where our classical training (our Euclidean intuition) lets us down. This is also where classical mechanics breaks down. When we perform this experiment, something completely counterintuitive happens, contradicting our Euclidean intuitions. A distinct interference pattern is projected on the wall (Figure 5).Figure 5 Actual double slit projection The bright and dark bands produced in this double-slit experiment are telltale signs that light propagates as a wave. [iii] Interference patterns are key signatures of waves. The problem is that this wavelike characteristic directly clashes with our observations of light’s particulate behavior. After all, photons are always found in point-like regions rather than spread out like a wave, and individual photons are always found to have very discrete amounts of energy. When measuring a wave, you would expect to find its energy spread out over a region instead of being concentrated in one location. So how are we supposed to make sense of this observation? What is going on?These diametrically opposed properties of light are verified facts. Contradictory as they may seem, they are here to stay. They have forced us to the seemingly paradoxical conclusion that light is both a wave and a particle. But how can this be? How can it be both? Although many scientists have found the wave-particle duality of light to be conceptually vague and schizophrenic, this description has persisted. In fact, after the wave-particle concept was adopted as an accurate description of light, it was extended to describe electrons and, eventually, all of matter. This transition was nothing short of a revolution.Up until 1910, atoms were simplistically viewed as miniature solar systems with the nucleus making up the “central star” and orbiting electrons being “planets”. [iv] The wave-particle duality of light and matter rejected this view and pointed to a signNowly different architecture for atoms. Of course, this conceptual transition did not take hold over night.In 1924, Prince Louis de Broglie found that in addition to their particle like character, [v] electrons also possessed a wavelike character. In 1927, Clinton Davisson and Lester Germer followed this up by firing a beam of electrons at a piece of nickel crystal, which acted as a barrier analogous to the one used in the double-slit experiment. A phosphor screen recorded the resultant pattern of electrons. [vi] When they examined the screen, they observed an interference pattern just like the one produced in the double-slit experiment, showing that even electrons have wavelike properties.These experiments shook the foundation of physics by threatening the structure of classical mechanics and destroying humanity’s intuitive framework of reality. But it didn’t stop there. The next step was to tune the beam of electrons down so that the electron gun fired just a single electron at a time. Similar experiments were later used with lasers wherein individual photons were fired seconds apart from each other. The results were mind-bending.Completely against expectation these experiments also produced interference patterns over time as the collection of electrons (or photons) continued to build (Figure 6).Figure 12-6 Over time individual photons construct an interference patternThese observations only added to the confusion. Waves are supposed to be a collective property—something that has no meaning when applied to separate, particulate ingredients. (A water wave, for example, involves a large number of water molecules.) So how can a single electron, or a single photon, be a wave? Furthermore, wave interference requires a wave from one place to interact with a wave from another place. So how can interference be relevantly applied to a single electron or photon? While we are considering such questions, we should also ask, if a single electron or photon is a wave, then what is it that is “waving”? [vii]To answer these questions, Erwin Schrödinger proposed that the stuff that makes up electrons might be smeared out in space and that this smeared electron essence might be what waves. If this idea was correct then we would expect to find all of the electron’s properties, spread out over a distance, but we never do. Every time we locate an electron, we find all of its mass and all of its charge concentrated in one tiny, point-like region. Max Born came up with a different idea. He suggested that the wave is actually a probability wave. [viii] Einstein tinkered with a similar idea when he hypothesized that these waves were optical observations that refer to time averages rather than instantaneous values.Inserting a probability wave (also called a state vector, or a wave function) as a fundamental aspect of Nature delivers another blow to our common-sense ideas about how things truly operate. It suggests that experiments with identical starting conditions do not necessarily lead to identical results because it claims that you can never predict exactly where an electron will be in a single instant. You can only define a probability that we will find it over here, or over there, at any given moment. Two situations with the same probabilistic starting conditions, say of a single particle, might not produce the same results, because the particle can be anywhere within that probability distribution. From a classical perspective, the discovery that the microscopic universe behaves this way is absolutely baffling. Nevertheless, it is how we have observed Nature to be.This leads us to a rather interesting precipice. It seems that the map we have been using to chart physical reality somehow dissolves when we look closely at it. The rules of four-dimensional geometry simply fail to accurately map Nature when we examine the smallest scales. Nature doesn’t strictly behave as our old Euclidean map dictates. Stumbling upon this discovery forces us to face a vital question. Is Nature ultimately and fundamentally probabilistic in a way that we may never understand, as many modern physicists have chosen to believe; or, is this probabilistic quality a byproduct of our reduced dimensional representation of Nature?After pondering these questions long and hard, some physicists have come to believe that the tapestry of spacetime is analogous to water: that the smooth appearance of space and time is only an approximation that must yield to a more fundamental framework when considering ultramicroscopic scales. As far as I can tell, however, up until now this point has only been entertained abstractly. Geometrically resolving a molecular structure for space might resolve our greatest quantum mechanical mysteries, but as of yet, no one has taken that final step. No one has developed a self-consistent picture from this geometric insight. No one has moved beyond the mathematical suggestion that spacetime is analogous to water, or interpreted the theoretical quanta of space as being physically real. Consequently, a framework that enables conceptualization of what is meant by the “molecules” or “atoms” of spacetime has not been developed.Eight decades of meticulous experiments have confirmed the predictions of quantum mechanics based on this wave function, or probability wave, description with amazing precision. “Yet there is still no agreed-upon way to envision what quantum mechanical probability waves actually are. Whether we should say that an electron’s probability wave is the electron, or that it’s associated with the electron, or that it’s a mathematical device for describing the electron’s motion, or that it’s the embodiment of what we can know about the electron is still debated.” [ix]Although quantum mechanics describes the universe as having an inherently probabilistic character, we don’t experience the effects of this character in our day-to-day lives. Why is this? The answer, according to quantum mechanics, is that we don't see quantum events like a chair being here now and then across the room in the next instant, because the probability of that occurring, although not zero, is absurdly miniscule. But what exactly makes the probability for large things to act, as electrons do, so small? At what scales do such effects become important? And, why should the macroscopic universe be so different from the microscopic universe?As if these newly uncovered characteristics of reality weren’t obscure enough, quantum physicists conceptually fuddle things further by suggesting that without observation things have no reality. They claim that until the position of an electron is actually measured the electron has no definite position. Before it is measured, the position exists only as a probability, and then suddenly, through the act of measuring, the electron miraculously acquires the property of position.Einstein acutely recognized the absurdity of this claim. When approached with this conjecture, he famously quipped, “Do you really believe that the moon is not there unless we are looking at it?” [x] To him everything in the physical world had a reality independent of our observations. Measurements that suggested otherwise were mere reflections of the incompleteness by which we currently map and comprehend physical reality. To many quantum physicists, however, the unobserved Moon’s existence became a matter of probability. To them, a discoverable, complete map of physical reality, with the ability to resolve an underlying determinism, became nothing more than a myth—a romantic dream.The mathematical projection of quantum mechanics can be statistically matched with our four-dimensional observations, but when it comes to a conceptual explanation of those observations, it completely lets us down. Intuitive explanations cannot be gleaned from a framework of physical reality that is assumed to be fundamentally probabilistic. By definition, randomness blurs causality. This vague description of physical reality keeps us from grasping a deeper truth by allowing what should be the most basic of concepts to drip into a realm of nonsense.As an example of the confusion that stems from swallowing the standard quantum mechanical interpretation “guts, feathers, and all,” consider the fact that a probabilistic treatment of quantum mechanics leads us to the conclusion that the double-slit experiment can be explained by assuming that a photon actually takes both paths. We can combine the two probability waves emerging from both slits to statistically determine where a photon will land on a screen. The result mimics an interference pattern.According to this, we can explain interference patterns by assuming that one photon somehow always manages to go through both slits, but is this really what is going on? Does a photon really travel along both paths? Can this count as an explanation if we have no coherent sense of what it means? You might notice that if we were to design our experiment with three slits, then we would have to consider whether or not the photon really travels all three routes. This question can be extended for as many slits as you like, but the fundamental conceptual problem remains the same.In order to solve this mystery, you may suggest that we place detectors in front of the slits to determine if the photons are actually going through both slits, or just one. When we do this, we always find that individual photons pass through one slit or the other—never both. But, when we measure the position of individual photons we no longer get an interference pattern and so the question retains its ambiguity. Some have taken this to mean that the act of observation forces wave properties to collapse into a particle, but how and why this theoretical collapse occurs still lacks explanation.Because probability waves are not directly observable and because photons (and electrons) are always found in one place or another when measured, we might be tempted to think that probability waves might not be real—that they were never really there. If that is true, then how are the interference patterns created? Surely these probability waves exist, but in what sense? What are they referencing? Why is it that whenever we know which path the photon takes, we get a classical image instead of an interference pattern? How does the detection of a photon, or an electron, change its behavior?To date, these questions have yet to be resolved. In fact, more clever experiments designed to solve these questions have only deepened the mystery. For example, let’s perform the double-slit experiment again, but this time let’s place devices in front of the slits, which mark (but do not stop or detect) the photons before they pass through the slits. This marking allows us to examine the photons that strike the screen and subsequently determine which slit they passed through. Thus we only gain knowledge of which path the photon takes after the path has been completed. For some reason, however, when we do this we find that the photons do not build up an interference pattern. They form a classical image (Figure 4).Once again, it seems that “which-path” information inhibits us from probing these ghostly waves. But is it really the fact that we gain the ability to determine which path a photon goes through—independent of when we gain that information—that disrupts the interference pattern? Or does our marking of the photon somehow disrupt its interference potential?To explore this question, we perform what’s known as the quantum eraser experiment. We start with the same set up we just described. Then we place another device between each slit and the screen, which completely removes the mark from the photon. We already know that the marked photons project a classical image. Will an interference pattern reemerge if we remove the effects of this mark—if we lose the ability to extract the which-path information?When we perform this experiment the interference pattern does return (Figure 7). Does this mean that photons somehow choose how to act, based on our knowledge of them? Or does it imply something even stranger—that the photons are always both particles and waves simultaneously? How are we to understand either conclusion?Figure 12-7 An interference pattern Another curiosity of Nature is known as the photoelectric effect. Philipp Lenard first discovered this effect through controlled experiments in 1900. When light shines on a metal surface, it causes electrons to be knocked loose and emitted. Knowing this, Lenard designed an experiment that allowed him to control the frequency of the incoming light. During the experiment, he increased the frequency of the light—moving from infrared heat and red light to violet and ultraviolet. Greater frequencies caused the emitted electrons to speed away with more kinetic energy. After discovering this, Lenard reconfigured his experiment to allow him to control the intensity of the incoming light. He used a carbon arc light that could be made brighter by a factor of 1,000.Because both experiments involved increasing the amount of incoming light energy he expected to have identical results. In other words, because the brighter, more intense light had more energy, Lenard expected that the electrons emitted would have more energy and speed away faster. But that’s not what happened. Instead, the more intense light produced more electrons, but the energy of each electron remained the same. [xi]In response to these experiments Einstein suggested that light is composed of discrete packets called photons. Under this assumption, light with higher frequency would cause electrons to be emitted with more energy, and light with higher intensity, that is, a higher quantity of photons, would result in emission of more electrons—just as we observe.The problem with this solution (a solution that is now universally accepted among physicists) is that it doesn’t provide us with a clear description for what the light quanta are. Why does light come in quantized packets? Near the end of his life Einstein lamented over this problem in a letter to his dear friend Michele Besso. He wrote, “All these fifty years of pondering have not brought me any closer to answering the question, what are light quanta?” [xii] It’s been another fifty years and we seem as confused as ever over how it is that light is quantized into little discrete packets called photons.In the midst of these enigmas lies the uncertainty principle, which states that knowledge of certain properties inhibits knowledge of other complimentary properties. For example, the more accurately we determine the position of an electron, the less we can determine its momentum, and vise versa.Heisenberg tried to explain the uncertainty principle by appealing to the observer effect; claiming that it was simply an observational effect of the fact that measurements of quantum systems cannot be made without affecting those systems. [xiii] Since then, the uncertainty principle has regularly been confused with the observer effect. [xiv] But the uncertainty principle is not a statement about the observational success of current technology. It has nothing to do with the observer effect. It highlights a fundamental property of quantum systems, a property that turns out to be inherent in all wave-like systems. [xv] Uncertainty is an aspect of quantum mechanics because of the wave nature it ascribes to all quantum objects.If our current description of quantum mechanics is fundamental, if there is nothing beneath the state vector—a claim that defines the heart of the standard interpretation of quantum mechanics—then this uncertainty principle may be a sharp enough dagger to kill our quest for an intuitive understanding of physical reality. The corrosive power of the uncertainty principle, when mixed with our current paradigm, is poignantly illustrated by an old story involving Niels Bohr. According to the story, Bohr was once asked what the complementary quality to truth is. After some thought he answered—“clarity.” [xvi] Unlike classical mechanics, which describes systems by specifying the positions and velocities of its components, quantum mechanics uses a complex mathematical object called a state vector (also called the wave function [xvii]) to map physical systems. Interjecting this state vector into the theory enables us to match its predictions to our observations of the microscopic world, but it also generates a relatively indirect description that is open to many equally valid interpretations. This creates a sticky situation, because to “really understand” quantum mechanics we need to be able to specify the exact status of and to have some sort of justification for that specification. At the present, we only have questions. Does the state vector describe physical reality itself, or only some (partial) knowledge that we have of reality? “Does it describe ensembles of systems only (statistical description), or one single system as well (single events)? Assume that indeed, is affected by an imperfect knowledge of the system, is it then not natural to expect that a better description should exist, at least in principle?” [xviii] If so, what would this deeper and more precise description of reality be?To explore the role of the state vector, consider a physical system made of N particles with mass, each propagating in ordinary three-dimensional space. In classical mechanics we would use N positions and N velocities to describe the state of the system. For convenience we might also group together the positions and velocities of those particles into a single vector V, which belongs to a real vector space with 6N dimensions, called phase space. [xix]The state vector can be thought of as the quantum equivalent of this classical vector V. The primary difference is that, as a complex vector, it belongs to something called complex vector space, also known as space of states, or Hilbert space. In other words, instead of being encoded by regular vectors whose positions and velocities are defined in phase space, the state of a quantum system is encoded by complex vectors whose positions and velocities live in a space of states. [xx]The transition from classical physics to quantum physics is the transition from phase space to space of states to describe the system. In the quantum formalism each physical observable of the system (position, momentum, energy, angular momentum, etc.) has an associated linear operator acting in the space of states. (Vectors belonging to the space of states are called “kets.”) The question is, is it possible to understand space of states in a classical manner? Could the evolution of the state vector be understood classically (under a projection of local realism) if, for example, there were additional variables associated with the system that were ignored completely by our current description/understanding of it?While that question hangs in the air, let’s note that if the state vector is fundamental, if there really isn’t a deeper-level description beneath the state vector, then the probabilities postulated by quantum mechanics must also be fundamental. This would be a strange anomaly in physics. Statistical classical mechanics makes constant use of probabilities, but those probabilistic claims relate to statistical ensembles. They come into play when the system under study is known to be one of many similar systems that share common properties, but differ on a level that has not been probed (for any reason). Without knowing the exact state of the system we can group all the similar systems together into an ensemble and assign that ensemble state to our system. This is done as a matter of convenience. Of course, the blurred average state of the ensemble is not as clear as any of the specific states the system might actually have. Beneath that ensemble there is a more complete description of the system’s state (at least in principle), but we don’t need to distinguish the exact state in order to make predictions. Statistical ensembles allow us to make predictions without probing the exact state of the system. But our ignorance of that exact state forces those predictions to be probabilistic.Can the same be said about quantum mechanics? Does quantum theory describe an ensemble of possible states? Or does the state vector provide the most accurate possible description of a single system? [xxi]How we answer that question impacts how we explain unique outcomes. If we treat the state vector as fundamental, then we should expect reality to always present itself in some sort of smeared out sense. If the state vector were the whole story, then our measurements should always record smeared out properties, instead of unique outcomes. But they don’t. We always measure well-defined properties that correspond to specific states. Sticking with the idea that the state vector is fundamental, von Neumann suggested a solution called state vector reduction (also called wave function collapse). [xxii] The idea was that when we aren’t looking, the state of a system is defined as a superposition of all its possible states (characterized by the state vector) and evolves according to the Schrödinger equation. But as soon as we look (or take a measurement) all but one of those possibilities collapse. How does this happen? What mechanism is responsible for selecting one of those states over the rest? To date there is no answer. Despite this, von Neumann’s idea has been taken seriously because his approach allows for unique outcomes.The problem that von Neumann was trying to address is that the Schrödinger equation itself does not select single outcomes. It cannot explain why unique outcomes are observed. According to it, if a fuzzy mix of properties comes in (coded by the state vector), a fuzzy mix of properties comes out. To fix this, von Neumann conjured up the idea that the state vector jumps discontinuously (and randomly) to a single value. [xxiii] He suggested that unique outcomes occur because the state vector retains only the “component corresponding to the observed outcome while all components of the state vector associated with the other results are put to zero, hence the name reduction.” [xxiv]The fact that this reduction process is discontinuous makes it incompatible with general relativity. It is also irreversible, which makes it stand out as the only equation in all of physics that introduces time-asymmetry into the world. If we think that the problem of explaining uniqueness of outcome eclipses these problems, then we might be willing to take them in stride. But to make this trade worthwhile we need to have a good story for how state vector collapse occurs. We don’t. The absence of this explanation is referred to as the quantum measurement problem.Many people are surprised to discover that the quantum measurement problem still stands. It has become popular to explain state vector reduction (wave function collapse) by appealing to the observer effect, asserting that measurements of quantum systems cannot be made without affecting those systems, and that state vector reduction is somehow initiated by those measurements. [xxv] This may sound plausible, but it doesn’t work. Even if we ignore the fact that this ‘explanation’ doesn’t elucidate howa disturbance could initiate state vector reduction, this isn’t an allowed answer because “state vector reduction can take place even when the interactions play no role in the process.” [xxvi] This is illustrated by negative measurements or interaction free measurements in quantum mechanics.To explore this point, consider a source, S, that emits a particle with a spherical wave function, which means its values are independent of the direction in space. [xxvii] In other words, it emits photons in random directions, each direction having equal probability. Let’s surround the source by two detectors with perfect efficiency. The first detector D1should be set up to capture the particle emitted in almost all directions, except a small solid angle θ, and the second detector D2 should be set up to capture the particle if it goes through this solid angle (Figure 8).Figure 8 An interaction-free measurement When the wave packet describing the wave function of the particle signNowes the first detector, it may or may not be detected. (The probability of detection depends on the ratio of the subtended angles of the detectors.) If the particle is detected by D1 it disappears, which means that its state vector is projected onto a state containing no particle and an excited detector. In this case, the second detector D2 will never record a particle. If the particle isn’t detected by D1 then D2 will detect the particle later. Therefore, the fact that the first detector has not recorded the particle implies a reduction of the wave function to its component contained within θ, implying that the second detector will always detect the particle later. In other words, the probability of detection by D2 has been greatly enhanced by a sort of “non-event” at D1. In short, the wave function has been reduced without any interaction between the particle and the first measurement apparatus.Franck Laloë notes that this illustrates that “the essence of quantum measurement is something much more subtle than the often invoked ‘unavoidable perturbations of the measurement apparatus’ (Heisenberg microscope, etc.).” [xxviii] If state vector reduction really takes place, then it takes place even when the interactions play no role in the process, which means that we are completely in the dark about how this reduction is initiated or how it unfolds. Why then is state vector reduction still taken seriously? Why would any thinking physicist uphold the claim that state vector reduction occurs, when there is no plausible story for how or why it occurs, and when the assertion that it does occur creates other monstrous problems that contradict central tenets of physics? The answer may be that generations of tradition have largely erased the fact that there is another way to solve the quantum measurement problem.Returning to the other option at hand, we note that if we assume that the state vector is a statistical ensemble, if we assume that the system does have a more exact state, then the interpretation of this thought experiment becomes straightforward; initially the particle has a well-defined direction of emission, and D2 records only the fraction of the particles that were emitted in its direction.Standard quantum mechanics postulates that this well-defined direction of emission does not exist before any measurement. Assuming that there is something beneath the state vector, that a more accurate state exists, is tantamount to introducing additional variables to quantum mechanics. It takes a departure from tradition, but as T. S. Eliot said in The Sacred Wood, “tradition should be positively discouraged.” [xxix] The scientific heart must search for the best possible answer. It cannot flourish if it is constantly held back by tradition, nor can it allow itself to ignore valid options. Intellectual journeys are obliged to forge new paths.So instead of asking whether of not wave-particle duality is an illusion, perhaps we should ask whether wave-particle duality implies that the state vector is the most fundamental description of a quantum mechanical system, or if a deeper level description exists? That's an open question, and at the moment there are many possible answers — interpretations of quantum mechanics that are equally aligned with the empirical evidence. My intuition is that a deeper level description of reality exists (something like Bohmian Mechanics yet deeper—like Superfluid vacuum theory).*This response is a modified excerpt from my book 'Einstein's Intuition'. Page on einsteinsintuition.com[i] The discussion on interference and the double-slit experiment that follows is further developed by Brian Greene, (2004). The Fabric of the Cosmos: Space, Time and the Texture of Reality. New York: Knopf, pp. 84–84. Greene’s discussion was used as a general guide here.[ii] In order to show diffraction (a fuzzy border of light on the projected image) the slit must have a width that does not greatly exceed the wavelength of the color of the light that we have chosen.[iii] Light’s wave nature was first revealed in the mid-seventeenth century through experiments performed by the Italian scientist Francesco Maria Grimaldi, and was later expanded upon by experiments performed in 1803 by the physician and physicist Thomas Young. (1807). Interference of Light; Alan Lightman. A Sense Of The Mysterious. pp. 51–52, 71.[iv] Before the “planetary model” of the atom, physicists pictured the atom being a plum-shaped blob (the nucleus) with tiny protruding springs that each had an electron stuck to its end. When the atom absorbed energy it was thought that these electrons would jiggle (oscillate) on the ends of their springs. Consequently, any atom that was above its ground state of energy was understood to be an “excited atomic oscillator,” This depiction of the atom wasn’t overthrown until 1900. At that point in history the physical existence of atoms was still controversial. It was replaced by the planetary model, which in turn was replaced by the electron cloud model we use today—a model that was initiated in 1910 and was secured by 1930. Gary Zukav. The Dancing Wu Li Masters, pp. 49–50.[v] Electrons can be individually counted and you can individually place them on a drop of oil and measure their electric charge. Richard Feynman. (1988). QED, The Strange Theory of Light and Matter. Princeton University Press, p. 84.[vi] According to de Broglie’s doctoral thesis all matter has corresponding waves. The wavelength of the “matter waves” that “correspond” to matter depends upon the momentum of the particle. Specifically, , which falls into an important group of equations along with Planck’s equation ) and the ever famous . (λ, pronounced “lambda,” stands for wavelength, h is Planck’s constant, and pronounced ‘nu’ represents the frequency of a photon) From this equation we are told to expect that when we send a beam of electrons (something we might traditionally think of as a stream of particles) through tiny openings, like the spacing between atoms in a piece of nickel crystal, the beam will diffract, just like light diffracts. The only requirement here is that the spacing between the atoms of the material must be as small, or smaller, than the electron’s corresponding wavelength—just like the slits in our double-slit experiment. When we perform the experiment, diffraction and therefore interference, occurs exactly as wave mechanics predicts.[vii] Part of the problem here is that in keeping with our four-dimensional intuition we tend to assume a particle aspect in the double-slit experiment without accounting for nonlocality. By doing this we are technically violating Heisenberg’s uncertainty principle and missing the bigger picture.[viii] M. Born. (1926). Quantenmechanik der Stossvorgänge. Zeitschrift für Physik 38, 803–827; (1926). Zur Wellenmechanik der Stossvorgänge. Göttingen Nachrichten 146–160.[ix] Brian Greene. (2004), p. 91.[x] Albert Einstein quoted in Einstein by Walter Isaacson.[xi] Walter Isaacson. Einstein, pp. 96–97.[xii] Ibid.[xiii] Werner Heisenberg. The Physical Principles of the Quantum Theory, p. 20.[xiv] Masano Ozawa. (2003). Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Physical Review A 67 (4), arXiv:quant-ph/0207121; Aya Furuta. (2012). One Thing Is Certain: Heisenberg’s Uncertainty Principle Is Not Dead. Scientific American.[xv] L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y, Soudagar, & A. M. Steinberg. (2012). Violation of Heisenberg’s Measurement—Disturbance Relationship by Weak Measurements. Physical Review Letters 109 (10).[xvi] Steven Weinberg. Dreams Of A Final Theory, p. 74.[xvii] For a system of spinless particles with masses, the state vector is equivalent to a wave function, but for more complicated systems this is not the case. Nevertheless, conceptually they play the same role and are used in the same way in the theory, so that we do not need to make a distinction here. Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 7.[xviii] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. xxi.[xix] There are 6N dimensions in this phase space because there are N particles in the system and each particle comes with 6 data points (3 for its spatial position (x, y, z) and 3 for its velocity, which has x, y, zcomponents also).[xx] The space of states (complex vector space or Hilbert space) is linear, and therefore, conforms to the superposition principle. Any combination of two arbitrary state vectors and within the space of states is also a possible state for the system. Mathematically we write where & are arbitrary complex numbers.[xxi] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 19.[xxii] Chapter VI of J. von Neumann. (1932). Mathematische Grundlagen der Quantenmechanik, Springer, Berlin; (1955). Mathematical Foundations of Quantum Mechanics, Princeton University Press.[xxiii] It might be useful to challenge the logical validity of the claim that something can “cause a random occurrence.” By definition, causal relationships drive results, while “random” implies that there is no causal relationship. Deeper than this, I challenge the coherence of the idea that genuine random occurrences can happen. We cannot coherently claim that there are occurrences that are completely void of any causal relationship. To do so is to wisk away what we mean by “occurrences.” Every occurrence is intimately connected to the whole, and ignorance of what is driving a system is no reason to assume that it is randomly driven. Things cannot be randomly driven. Cause cannot be random.[xxiv] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 11.[xxv] Bohr preferred another point of view where state vector reduction is not used. D. Howard. (2004). Who invented the Copenhagen interpretation? A study in mythology. Philos. Sci. 71, 669–682.[xxvi] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 28.[xxvii] This example was inspired by section 2.4 of Franck Laloë’s book, Do We Really Understand Quantum Mechanics?, p. 27–31.[xxviii] Franck Laloë. Do We Really Understand Quantum Mechanics?, p. 28.[xxix] T. S. Eliot. (1921). The Sacred Wood. Tradition and the Individual Talent.
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Is Nobel physicist Wilczek likely to see his 'time crystal' proven as exhibiting perpetual motion? In theory, it seems to work.
For almost all practical purposes, space is homogeneous and isotropic. Philip Warren AndersonBasic Notions of Condensed Matter Physics ( 1984 )Look, I am going to make a hypothesis :: Frank Wilczek is playing a massive joke on all of us, to see if we've gone collectively crazy. He is one of the great physicists of the last century. Saying the words perpetual motion machine was meant, I think, as a marketing gimmick - that worked, through the noise of Twitter and Wired. The man is a genius. He wears awesome T shirts full of math and physics wisdom and humor.. He also says funny things while also saying quite profound things. And he totally looks like what you would expect from Paul Giamatti's uncle. I made that up. As far as, I know, he is not Paul Giamatti's uncle. However, that does not mean that his papers will not lead to something incredibly awesome. Here is why.Why does spontaneous time symmetry breaking not imply a perptual motion machine?A perpetual motion machine of the first kind in common lore is a device that accomplishes more work than is put into it. A perpetual motion machine of the second kind extracts work from a thermal bath, like Maxwell's demon. Wilczek is referring to the first kind. The limiting case of the second kind was resolved in a paper on the thermodynamics of computation by Charles H. Bennett - IBM Research, where Bennett calculated the entropic cost of the erasure of memory. An analogous phenomena is persistent currents in normal metals, where non-superconducting electrons can flow through resistive metals without dissipation when their wave functions have the appropriate boundary conditions. The Jack Harris Lab at Yale did a beautiful experiment demonstrating the phenomena of persistent currents in aluminum, measuring them on silicon cantilevers through their angular mechanical signatures instead of through their magnetic signatures via SQUIDS. Persistent Currents in Normal Metal RingsDid Jack create a time crystal? Maybe. There is a sense in which something is moving in the experiments and Jack measures that movement, persistently. But, perhaps a more correct statement would be to say that he observed a momentum crystal. Did Jack observe perpetual motion?Well, yes, sort of, but you could not power anything with it, though, because the persistence is in the ground state. None of the above involves perpetual motion, in the sense of a perpetual motion machine, of the first kind, because you cannot extract any work from the systems - they are already in their lowest energy state. Another way to think about conservation of energy and time crystals is to note, analogously, you cannot extract infinite momentum from a space crystal, even though conservation of momentum in a space crystal is not strictly conserved and only conserved under modular arithmetic - that is, mod the inverse of the lattice spacing. That is the summary. ----Here is a proposal to investigate the physics in the paper. Does an atom exist with an electronic ground state with non-zero angular momentum that is not rotationally symmetric?We know that atoms exist that have ground states - lowest energy states - that have non-zero angular momentum, in analogy with persistent currents in normal metals. The main difference between Jack's experiment and Wilczek's proposal is that Jack did not break rotational symmetry. As far as we know, persistent currents in normal metals actually depend on not breaking that symmetry, by extending the wavefunction of the free electron in the metal symmetrically around the ring. Think of a circle. Now, rotate the circle a bit. Looks the same. Now, put a dot on the circle. Rotate the circle a bit. Looks different. That dot can be used to track the motion precisely. But, of course, not too precisely, because their exists an uncertainty relation between measuring space and momentum. You could imagine using a different material for the ring that had interactions between the electrons appropriate and strong enough - or even tunable by a magnetic field - to produce a soliton ( the dot), or some rotational symmetry breaking, like a p wave, in the ground state. Then, you could measure the soliton or whatever moving around the ring, persistently or not. That would also be a time crystal, in the sense Wilczek defined it, just the solid state version rather than the cold atom version. The uneven distribution around the ring would create a wobble behavior, like an imbalanced spinning plate, that would certainly show up in the resonance coupling to the cantilever. The problem with localizing anything into a soliton is that you might lose the global boundary conditions necessary for the persistent current. That is the real issue here, mathematically. In the normal metal ring, the electron wave function wraps around the ring and the current is enforced by the requirement that the wavefunction be continuous where the electron meets itself on the other side. The question is whether or not you can have some stable kink as you wrap around the ring while maintaining the persistent boundary conditions. I do not know of any principle that says by creating a soliton, which itself depends on special boundary conditions, you also need to lose the boundary conditions that allow for persistent currents. If it exists, it's probably a theorem in topology, either way. We already know and observe momentum crystals, which yield perpetual or persistent motion, all the time in quantum coherent phenomena like superconductors, superfluids and coherent electron persistent motion in normal metals. If you think of a spatial crystal lattice being a system collapsing around a single spatial vector that defines the lattice, then these persistent flow quantum coherent phenomena all are momentum crystals where the system of particle collapses around a single momentum vector that defines the flow. All the electron pairs that compose a superconductor, for example, flow together with the same momentum. That crystallization in momentum space gives the superconductor the rigidity to flow without dissipation, just as a solid like copper exhibits a certain rigidity. That is, of course, relevant because the quantum mechanical model used by Wilczek is basically the same model used to describe superconductivity, macroscopically. Also interesting to note that the other mathematical models studied in the papers show striking resemble to PT symmetric quantum mechanical models of Carl Bender, if one were to complexity them by adding a complex real space variable in addition to the higher derivatives of momentum. Physics Video Archive COLLOQ_BENDERI think that is an extremely promising way to look at these models, since they are the discrete ( reflection ) symmetry versions of the proposals that want to break continuous time and spatial symmetry separately, but maintain some remaining combined symmetry. In the PT symmetric models, an extremely precise mathematical relationships is developed between systems that have balance gain and loss and systems that do not, related to the PT symmetry itself being broken or unbroken. Such systems have been realized in many experiments, quantum and classical, and have subtle and critical boundary condition relationships. Finally, PT symmetric models are deeply related to the more general CPT symmetry, which is essential for Lorentz invariance. The proposal by Wilczek is strikingly reminiscent at a schematic level of CPT Violation Experiments. By the way, I have a time crystal for you that exhibits perpetual motion and periodicity in time. Light. Photons have a well defined frequency and never rest. Speaking of light, note that though Wilczek was inspired by the Lorentz symmetry between time and space to look for time crystals, none of his models are relativistic. They cannot be, in the manner he is investigating time crystals, because all the models are non-relativistic with non-linear dispersion relations.---- FUTURE RADIO EDIT :: Almost everything below that is not referenced is pure speculation. Read for enjoyment, not for physical accuracy. All lot above this line is speculative. I am going to continue to edit and learn about this area, because it is a fascinating area of physics. I might do that in a blog, and get more detailed with the mathematics. The answer is redundant in some places and certainly incorrect or poorly written in others, but I wanted to get it up so you could enjoy and learn from pieces of it; and hopefully, explore some of the questions yourself with more powerful tools and analogies. You should also check out Carver Mead's book Collective Electrodynamics: Quantum Foundations of Electromagnetism: Carver A. Mead: 9780262133784: Amazon.com: Books because it takes as its logical foundation the following coherent quantum phenomena. 1911 Superconductivity1933 Persistent Current in Superconducting Ring1954 Maser1960 Atomic Laser1961 Quantized Flux in Superconducting Ring1962 Semiconductor Laser1980 Integer Quantum Hall Effect1981 Fractional Quantum Hall Effect1995 Bose-Einstein Condensate2009 Persistent Currents in Normal Metal Rings ----Four dimensional crystallography is a different path to investigate the idea ::Ordinary crystallography deals with regular, discrete, static arrangements in space. Of course, dynamic considerations— and thus the additional dimension of time—must be introduced when one studies the origin of crystals (since they are emergent structures) and their physical properties such as conductivity and compressibility. The space and time of the dynamics in which the crystal is embedded are assumed to be those of ordinary continuous mechanics. In this paper, we take as the starting point a spacetime crystal, that is, the spacetime structure underlying a discrete and regular dynamics. A dynamics of this kind can be viewed as a “crystalline computer.” After considering transformations that leave this structure invariant, we turn to the possible states of this crystal, that is, the discrete spacetime histories that can take place in it and how they transform under different crystal transformations. This introduction to spacetime crystallography provides the rationale for making certain definitions and addressing specific issues; presents the novel features of this approach to crystallography by analogy and by contrast with conventional crystallography; and raises issues that have no counterpart there. Tommaso ToffoliA pedestrian’s introduction to spacetime crystallography ( 2004 )Lets use the same analogy that Wilczek used to come up with the idea of time crystals by looking at spatial crystals. Here's the key analogical observation to make ::Solids spontaneously break the continuous symmetry of space down to periodic discrete symmetry, yet we cannot extract infinite momentum from them, even though momentum is not strictly conserved in the solid. Noether's theorem tells us that in mechanical and quantum mechanical systems describable by a Lagrangian, any symmetry transformation that leaves the Lagrangian invariant leads to a conservation law. Continuous time translation symmetry yields conservation of energy. Continuous space translation symmetry yield conservation of momentum. Continuous rotation translation symmetry yields conservation of energy. Sometimes, however, that symmetry is broken naturally, as in a solid state crystal. As Wilczek says, "When a physical solution of a set of equations displays less symmetry than the equations themselves, we say the symmetry is spontaneously broken by that solution." Similarly, a time crystal does not imply that we can extract infinite energy from the system even if the system spontaneously breaks the continuous symmetry of time down to periodic discrete symmetry. As Wilczek says, " ... one interesting case, that will concern us here, is of the lowest energy solutions of a time-independent,conservative, classical dynamical system. If such a solution exhibits motion, we will have broken time translation symmetry spontaneously ... Speaking broadly, what we’re looking for seems perilously close to perpetual motion." [ emphasis mine ]A crystal lattice formed by atoms in a solid is a great example of spontaneous symmetry breaking. The fundamental equations describing the dynamics of the nuclei and electrons of the atoms have continuous time, space and rotational symmetry. However, at low enough average energy ( related to temperature ), elemental atoms may form solutions to these equations that do not exhibit that full symmetry. Specifically, a solid state lattice exhibits discrete rather than continuous translation symmetry such that conversation of momentum is no longer strictly conserved, but rather only conserved modulo a specific value related to the inverse of the lattice spacing. For example ...At 2,835 degrees Kelvin, Copper atoms transition from a gas state to a liquid state. At 1,357.77 K, copper atoms will solidify naturally into a face centered cubic lattice crystal structure of the cubic crystal system. The type of lattice a particular atom will solidify into is determined by its electronic structure; however, the group theory of crystallography mandates that only, starting with the 14 Bravais lattice and keeping one point of the lattice fixed, one obtains the 32 Point groups. If the latter are combined with translations, one obtains the 230 Space groups (ascertained in 1891). Image :: The Bauhinia blakeana flower on the Hong Kong flag has C5 symmetry; the star on each petal has D5 symmetry. A beautiful book on symmetry is The Symmetries of Things by the great mathematician John Horton Conway. What happens in a solid is that [ a ] the symmetry breaking results in a "rigidity" of the system in space and [ b ] the dynamics particles flowing through that solid - electrons or phonons, for example - only conserve momenta under modular arithmetic. What do I mean by that?The easiest way to see what is happening to conservation of momentum in a crystal that break spontaneously breaks spatial symmetry is to look at a Bloch wave, which simply describes the wave function of a particle such as an electron in any periodic potential, like that found in a solid state crystal.First, lets temporarily remove the lattice atoms completely and just analyze free space. Say you took an electron in free space and applied an electric field. The electron would accelerate and gain momentum and energy. Note that you are not creating a perpetual motion machine. The electric field comes from somewhere and you had to do work to create it. If you remove the electric field at some point, the electron will continue to move with the same momentum and energy for eternity, precisely because free space is homogeneous and isotropic. That means, if you shift free space a little in time or space, or rotate free space slightly, nothing changes about free space. It's like if you moved an infinite line a little to the left or right. It looks exactly the same. Well, a particle moving along a line is exactly the same as a line moving along a particle. Momentum conservation reduces to tautology if you think about it correctly. If something is symmetric, it does not change. If something is conserved, it does not change. By Noether's Theorem, free space being homogenous and isotropic means all physical systems conserve momentum, energy and angular momentum. Just because the particle moves forever - perpetually - after you've removed the electric field does not make it a perpetual motion machine, either. It's actually just Newton's first law of motion, dressed up in a little more sophistication. Now, lets put the face centered cubic arrangement of atoms of copper, or whatever, and assume they fill all of the universe. A giant block of solid copper. Now, apply an electric field. Remember again that we had to create the electric field, so we are putting work into the system. For those in the know, I am about to describe Bloch oscillations, which clearly demonstrate the modular arithmetic of momenta in solid state crystals. As you apply an electric field on the electron in the copper lattice, the momentum of the electron increases. However, the crystal lattice structure puts an upper limit on the momenta that is the inverse of the lattice spacing. Lets say in appropriate units that upper limit is 12. After applying the extremely weak electric field for 1 hour, the momentum of the electron is now 1; and so on. Now imagine the clock you are using to measure time. When you signNow 12, you start back again at zero. That's modular arithmetic. And that's what happens to momentum in a solid. Actually, a better way to think of the clock is starting at minus 6 at the bottom, zero at the top and plus six approaching the bottom clockwise. The momentum of a particle in a solid literally goes from plus six to minus six instantly due to the symmetry breaking of the lattice. That is because momentum is only conserved mod 12. So, plus and minus six are equivalent. However, there is absolutely no way to exploit that momentum jump to extract infinite momentum outside the solid because from the perspective of the lattice plus and minus 6 are smoothly connected in momentum space, which takes the shape of a 3-torus for a cubic lattice. ( By the way, a circle is a 1-torus and a torus is a 2-torus. )That is, you cannot simply apply an electron field to silicon and copper and extract infinite momentum in a perpetual motion machine. Intel and Samsung would have a field day with that, if you could, and your Apple iPhone would power your city. What you can do is interpret the seemingly large momentum shift as an interference scattering effect of the electron wave function off the periodic lattice, recalling that on the atomic scale, electron dynamics behave according the quantum mechanical wave equations. And, of course, the lattice nuclei are much much heavier than the electrons, so the electrons hitting the lattice is like a ball bouncing off a wall. Modular arithmetic is extremely useful and powerful in number theory. For me, it's fascinating to see it arise in quantum mechanics as a result of discrete symmetry in Bloch waves. Now, lets play some games here.Ironically, the relativistic notion of mixing time and space through Lorentz transformation was used as a motivation for the work. However, the theory of special relativity requires a linear relationship between energy and momentum. That allows linear transformations between energy and momentum to occur and allows energy and momentum to be combined into a single, highly compact energy-momentum four vector. At low energy, you can expand out any relativistic equation with the speed of light in the denominator of any terms and extract non-relativistic physics by ignoring those terms, since their effect will be very small. What you end up with is a relationships between energy and momentum that is parabolic rather than linear, if no interactions between particles or other objects in the theory add any further complexity. The papers take as a starting point a relationship between energy and momentum - a dispersion relation - that is both non-linear, as noted, and exhibits a cusp singularity. The dispersion relation looks a swallow's tail, like the shape of the swallowtail butterfly in the images above at the beginning of the answer. The curve shows a crossing where the body of the butterfly rests. They have a parabolic term and a quartic term. Guess what the dispersion relation of Bloch waves are?The cosine function. The cosine function is non-linear and periodic. Guess what the first two terms Taylor series expansion of a cosine function yields up to an overall constant?A parabolic with a negative coefficient and a quartic term with a positive coefficient. The same form as in Wiczek's papers. Guess how you get from electric field to magnetic fields in electromagnetism? Lorentz transformations. The basis of the spacetime physics that inspired Wilczek to write his papers. And note that the primary example used in his papers is a particle oscillating around a circular lattice in a weak magnetic field. I am playing with the idea that Wilczek "discovered" the "time version" of Bloch oscillations. And, just as Bloch waves in a solid ( aka a "space crystal" ) do not violate conservation of momentum in a manner that enables a perpetual motion machine, Wilczek waves in a "time crystal" do not violate conservation of energy in a manner that enables a perpetual motion machine. I do not even think it's appropriate to call them the time version, in the experiment being proposed in cold atoms. The appropriate name for the experiment being proposed would be magnetic Bloch oscillations. A space-time crystal actually implies that the lattice atoms disappear for a well-defined time step; just as in a space crystal, matter disappears for a well defined spatial step called the lattice spacing in a well defined crystallographic arrangement. Have we found a system that breaks continuous time translation symmetry such that matter blinks in and out of existence periodically? I do not think we have. That would be a true time crystal, in my mind.That system would require a quantum field that oscillates in time between a ground state with a mass gap and a ground state that is gapless. Such a system would also not allow you to build a perpetual motion machine, even though it violates conservation of matter and energy.That is, you could not extract energy by coherently scattering from a time crystal just as you cannot extract momentum by coherent scattering off a space crystal. Furthermore, given the analogy with Bloch oscillations, which is nearly mathematically equivalent to the example used by Wilczek, a system that exhibits periodic motion in the ground state is not actually that surprising. Actually, it turns out that what Wilczek is saying is even less surprising when you think about superconductivity in the right way. Superconductors are essentially crystals in momentum space. Just as atoms condense to a specific spatial lattice vector in solids that are "rigid," electron pairs condense to a specific momentum lattice vector in superconductors, yielding persistent currents that are, in their own way, "rigid." That observation is, in fact, how London developed his London equations of superconductors. A superconducting condensate exhibits a persistent current because the condensate collapses to a momentum vector, which implies motion. That motion may be angular, around a ring and periodic with a magnetic field. So, not only is Wilczek simply describing the magnetic version of Bloch oscillations in his papers; he is also simply describing the persistent currents of superconductors. The requirement he posits to break a cylindrical spatial symmetry of a persistent current condensate in order to then break time symmetry by making the motion in the ground state more salient does not actually make any difference. In non relativistic quantum mechanics, we have real space and momentum space, which are simply related by Fourier transforms. The reason you cannot isolate the location of a superconductor condensate is because the Fourier transform of a single momentum vector is completely and evenly spread out in real space. Conversely, in a solid state lattice, the momentum distribution is relatively spread out. If you want to create a time crystal in the sense Wilczek is after, you have to be in the relativistic regime. However, to be in the relativistic regime, you need a linear dispersion relation. But, the only way that you can create a time crystal in the way Wilczek wants to is by being in a highly non-linear, non-relativistic regime. What would be interesting is if someone could describe and experimentally realize a state that naturally interpolated back and forth between a solid ground state, collapsed on a spatial vector, and a superconducting ground state, collapsed on a momentum vector, in a closed, non-relativistic quantum mechanical system that was both naturally conservative and time independent.You could then watch the momentum and space vectors of the state collapses and expand, periodically in time. It actually turns out that someone has done that, in a sense,Greiner - Mott Insulator to Superfluid transitionbut that transition was still driven rather than occurring naturally in a conservative, time independent system. Perpetual motion machines are out. Time crystals have not been created. What specifically is going on in the time crystal papers that is interesting?The basic mathematical problem that arises in Wilczek's papers is that the energy is multivalued in the momentum. That, actually, is a fascinating area of physics. There is, I should mention, an entire book on multivalued quantum fields ::Multivalued Fields: In Condensed Matter, Electromagnetism, and Gravitation: Hagen Kleinert: 9789812791719: Amazon.com: Booksbut, I have not yet read it. I've been meaning to for a while. Any book with a Riemann surface on the cover with detailed mathematical descriptions of superconductors and gravity in the interior should be read by people like me.So, I will, now, within the decade. In fact, the quantum mechanical equation to be solved is the non-linear, non-relativistic Schrödinger equation that is used in Ginzburg–Landau theory to describe the Cooper pair condensate in superconductors in a single wavefunction. The non-linearity of the theory results from the emergent physics of superconductivity and leads to topological objects like flux vortices, as discovered by Alexei Alexeyevich Abrikosov. The theory includes a momentum term that is parabolic and a momentum term that is quartic when related to energy. The mathematical qualities of the coefficients of these terms matter greatly. The non-linear theory is emergent because it evolves via a process of renormalization from a completely linear quantum mechanical theory of electrons interacting with each other via repulsive Coloumb forces and with phonons - excitations of the underlying solid state lattice. At low enough temperatures, the interactions between the electrons and the phonons effectively switch the interactions between the electrons to be attractive rather than repulsive. The electrons pair up to form bound states that are bosons, the electromagnetic field mediating the interaction between electrons attains a mass gap and the boson condense into a collective state describable by the theory mentioned above. Topologically, Wilczek's swallowtail curve looks like the curve on the cover of the book Elliptic Tales: Avner Ash, Robert Gross: Amazon.com: Kindle Store. It's very similar to the curves found in Jack Huizenga's answer ::Given two low-degree polynomials defined on the integers, how can one find the integers which are in the range of both polynomials?In that answer, Jack gives a procedure for analyzing the intersection of two curves :: complexify, projectify ( to infinity and beyond), and normalize ( that is, smooth over the singularities).You might immediately object to apply anything like that procedure to analyzing a Hamiltonian system. If you are a physicist you know that the Hamiltonian of a quantum system must be Hermitian - that is, both real and probability conserving. However, as Carl Bender shows us in [quant-ph/9809072] PT-Symmetric Quantum Mechanics, we can relax that mathematical condition and replace it with a physical condition of PT symmetry and find some interesting results. The PT symmetry physical condition relaxes the constraint that the Hamiltonian is real; for example, [math] H = p^2 + i x^3 [/math]is PT symmetric, but obviously not Hermitian since it is complex. That is a hugely powerful constraint to relax and opens up an entire new world of mathematics to explore. You can actually see the mathematics that Bender is revealing to us in any power of the momentum. That is, he already solved Wilczek's problem, by the process - complexify, projectify, normalize. That work started with something known as the Yang Lee edge singularity. I do not know what that is, yet. Why do I care?Wilczek's class on topological quantum physics at MIT was by far my favorite course while I was in graduate school at Harvard. I wrote a paper on trying to extend Alexei Kitaev's K-theory classification in [0901.2686] Periodic table for topological insulators and superconductors to strongly interacting topological condensed matter systems using the success of the Seiberg–Witten invariants that survive strong coupling in supersymmetric QCD as a guide, which can be embedded in string theory [hep-th/9611190] Introduction to Seiberg-Witten Theory and its Stringy Origin. What Seiberg Witten theory describes is the electromagnetic dual of a superconductor. In fact, it describes a condensation of magnetic monopoles that allow electric flux tubes to form as a simplified model of QCD, as opposed to the condensation of electron ( pairs ) that allow magnetic flux tubes to form in a real superconductor. The face they used complex curve theory to solve their equations always fascinated me. Why?I wanted to somehow use the idea of a coobordism to track how the structure of the theory evolved under the tuning of the interaction strength; and, to show that certain invariant quantities survived that the tuning of the interaction strength in the topological electronic systems. The topological invariants would tell you if two different topological phases were connected through a strongly interacting regime, which would otherwise be hidden you by traditional analytic calculations involving an expansion in a small parameter. Seiberg-Witten theory is one of the few strongly interacting theories that is completely soluble, due to the strong supersymmetry in the theory. My paper completely failed to do that. He still gave me an "A" in the class, though everything I said was complete nonsense. I think he is returning the favor to the rest of us now. Kitaev later wrote a paper accomplishing what I had hoped to accomplish in [0904.2197] The effects of interactions on the topological classification of free fermion systems. Actually, that paper only identified a problem in the previous classification with small interactions. But, the problem of understanding topological phases still remains largely a mystery, though recent progress was made by Xiao-Gang Wen, now at the Perimeter Institute, in his paper [1106.4772] Symmetry protected topological orders and the group cohomology of their symmetry group. That's important because, from everything we know about M / string theory and topological quantum field theory ( which by the way has no dynamics and a Hamiltonian of zero ) understanding black holes and quantum gravity requires a deep understanding of topological phases. Wilczek's analysis showing up in the news gave me a different idea, one related to my M theory ideas here ::What do theoretical physicists think of Mark Morales' answer about M-theory?Whatever the case, I cannot wait to see someone create a Calabi Yau manifold in their laboratory hologram.Postscript :: If you followed my link above, you'll see that I proposed a general shift in mathematical approach to M theory. Along those lines, I found a good introduction to Elliptic Curves and Cryptography from Josh Alman ::Good introduction to elliptic curves?
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