Quantum mechanics

"So, what is quantum mechanics? Even though it was discovered by physicists, it’s not a physical theory in the same sense as electromagnetism or general relativity. In the usual “hierarchy of sciences” – with biology at the top, then chemistry, then physics, then math – quantum mechanics sits at a level between math and physics that I don’t know a good name for. Basically, quantum mechanics is the operating system that other physical theories run on as application software (with the exception of general relativity, which hasn’t yet been successfully ported to this particular OS). There’s even a word for taking a physical theory and porting it to this OS: “to quantize.”"

"We cannot make apparatuses arbitrarily big. ... We might need some extension of quantum mechanics."

"Christian Imbert, to support my project and to act as my thesis advisor. He had advised me to go first to Geneva, to discuss my proposal with John Bell. I got an appointment without delay, and I showed up in John's office at CERN, quite nervous. While I explained my planned experiment, he listened silently. Eventually, I stopped talking, and the first question came: "Have you a permanent position?" After my positive answer, he started talking of physics, and he definitely encouraged me, making it clear that he would consider the implementation of variable analysers a fundamental improvement. Remembering this first question reminds me both of his celebrated sense of humour and of the general atmosphere at that time about raising questions on the foundations of quantum mechanics. Quite frequently there was open hostility, and in the best case, irony: "quantum mechanics has been vindicated by such a large amount of work by the smartest theorists and experimentalists; how can you hope to find anything with such a simple scheme, in optics, a science of the 19th century?" In addition to starting the experiment, I had then to develop a line of argument to try to convince the physicists I met (and among them some had to give their opinion about funding my project)."

"Quantum mechanics was, and continues to be, revolutionary, primarily because it demands the introduction of radically new concepts to better describe the world. In addition we have argued that conceptual quantum revolutions in turn enable technological quantum revolutions."

"No other theory of the physical world has caused such consternation as quantum theory, for no other theory has so completely overthrown the previously cherished concepts of classical physics and our everyday apprehension of reality. For philosophers, it has been a romping ground of epistemological adventure of pessimism about science's ability to expose ultimate truth. For physicists, it has required a confrontation with the nature of physical reality and a heady inhalation of new attitudes. For all scientists and technologists, it has been the key to advances in all fields of endeavor, from genetics to superconductivity. The extraordinary feature of quantum theory is that although we do not understand it, we can apply the rules of calculation it inspires, and compute properties of matter to unparalleled accuracy, in some cases with a precision that exceeds that currently obtained from experiment."

"… that what is proved, by impossibility proofs, is lack of imagination."

"I am a Quantum Engineer, but on Sundays I Have Principles."

"Quantum mechanics had never been wrong. And now we know that it will not be wrong even in these very tricky conditions."

"I'm quite convinced of that: quantum theory is only a temporary expedient."

"This particular question of locality is still open, in my opinion. I think we have not found a way of digesting this situation. I think we have not found a way of digesting this situation. We have the formulas of quantum mechanics, and they work extremely well; but I have not digested them. There certainly remains something to be said, some illumination to be found."

"I hesitated to think it might be wrong, but I knew that it was rotten. That is to say, one has to find some decent way of expressing whatever truth there is in it."

"The entire universe must, on a very accurate level, be regarded as a single indivisible unit in which separate parts appear as idealisations permissible only on a classical level of accuracy of description. This means that the view of the world being analogous to a huge machine, the predominant view from the sixteenth to nineteenth centuries, is now shown to be only approximately correct. The underlying structure of matter, however, is not mechanical. This means that the term "quantum mechanics" is very much a misnomer. It should, perhaps, be called "quantum nonmechanics"."

"For those who are not shocked when they first come across quantum theory cannot possibly have understood it."

"If relativity is about the geometrical structure of space-time, what is quantum mechanics about? There are a surprising variety of answers to this question: that quantum mechanics is about energy being quantized in discrete lumps or quanta, or about particles being wavelike, or about the universe continually splitting into countless co-existing quasi-classical universes, with many copies of ourselves, and so on. A rather more mundane answer, with quite remarkable implications, has emerged over the past thirty years or so from the study of the difference between classical information and quantum information: quantum mechanics is about new sorts of probabilistic correlations in nature, so about the structure of information, insofar as a theory of information in the sense relevant to physics is essentially a theory of probabilistic correlations."

"It is a poorly-kept secret that the grandfathers of quantum mechanics, Bohr, Oppenheimer, Heisenberg, Einstein, de Broglie, Jeans, but in particular Schrödinger were fascinated and inspired by Vedic cosmology."

"We shall see how the two foundations of twentieth-century physics - quantum theory and relativity - both force us to see the world very much in the way a Hindu, Buddhist or Taoist sees it .."

"Scientists can use quantum mechanics with perfect confidence. But it’s a . We can set up a physical situation, and make predictions about what will happen next that are verified to spectacular accuracy. What we don’t do is claim to understand quantum mechanics. s don’t understand their own theory any better than a typical smartphone user understands what’s going on inside the device."

"The power of the new quantum mechanics in giving us a better understanding of events on an atomic scale is becoming increasingly evident. The structure of the helium atom, the existence of half-quantum numbers in band spectra, the continuous spatial distribution of photo-electrons, and the phenomenon of radioactive disintegration, to mention only a few examples, are achievements of the new theory which had baffled the old."

"Already in 1948, observations... agreed with quantum mechanics, not with local realism."

"The current probabilistic interpretation of the quantum theory leads in its general lines to exact conclusions. But since it denies every possibility of a precise image of the development of phenomena in space and time, it continues to be surrounded by a certain obscurity. It is not at all certain that it furnishes a complete description of physical reality : scientists as eminent as Planck, Einstein and Schrödinger have always expressed doubts on this subject. The idea of Prof. Bohm that it may be necessary to introduce new 'levels' of physical reality deeper and more hidden than those revealed by current experience therefore seems perfectly defensible to me. For my part, returning after a number of years to certain ideas that I had considered previously when I was developing the first bases of wave mechanics, I have examined this question in the light of the conceptions of Prof. Bohm and in collaboration with certain young scientists at the . In particular, I have asked myself whether it would not be possible to find an interpretation which, while retaining all the results given by probabilistic quantum physics, would permit us to obtain a more clear and more intelligible image of micro-physical facts."

"Classical mechanics has been developed continuously from the time of Newton and applied to an ever-widening range of dynamical systems, including the electromagnetic field in interaction with matter. The underlying ideas and the laws governing their application form a simple and elegant scheme, which one would be inclined to think could not be seriously modified without having all its attractive features spout. Nevertheless it has been found possible to set up a new scheme, called quantum mechanics, which is more suitable for the description of phenomena on the atomic scale and which is in some respects more elegant and satisfying than the classical scheme. This possibility is due to the changes which the new scheme involves being of a very profound character and not clashing with the features of the classical theory that make it so attractive, as a result of which all these features can be incorporated in the new scheme."

"I have observed in teaching quantum mechanics, and also in learning it, that students go through an experience similar to the one that Pupin describes. The student begins by learning the tricks of the trade. He learns how to make calculations in quantum mechanics and get the right answers, how to calculate the scattering of neutrons by protons and so forth. To learn the mathematics of the subject and to learn how to use it takes about six months. This is the first stage in learning quantum mechanics, and it is comparatively painless. The second stage comes when the student begins to worry because he does not understand what he has been doing. He worries because he has no clear physical picture in his head. He gets confused in trying to arrive at a physical explanation for each of the mathematical tricks he has been taught. He works very hard and gets discouraged because he does not seem to be able to think clearly. This second stage often lasts six months or longer. It is strenuous and unpleasant. Then, unexpectedly, the third stage begins. The student suddenly says to himself, “I understand quantum mechanics,” or rather he says, “I understand now that there isn’t anything to be understood.” The difficulties which seemed so formidable have mysteriously vanished. What has happened is that he has learned to think directly and unconsciously in quantum-mechanical language. He is no longer trying to explain everything in terms of prequantum conceptions. The duration and severity of the second stage are decreasing as the years go by. Each new generation of students learns quantum mechanics more easily than their teachers learned it. The students are growing more detached from prequantum pictures. There is less resistance to be broken down before they feel at home with quantum ideas. Ultimately, the second stage will disappear entirely. Quantum mechanics will be accepted by students from the beginning as a simple and natural way of thinking, because we shall all have grown used to it. By that time, if science progresses as we hope, we shall be ready for the next big jump into the unknown."

"For me, the important thing about quantum mechanics is the equations, the mathematics. If you want to understand quantum mechanics, just do the math. All the words that are spun around it don’t mean very much. It’s like playing the violin. If violinists were judged on how they spoke, it wouldn’t make much sense."

"Die Quantenmechanik ist sehr achtung-gebietend. Aber eine innere Stimme sagt mir, daß das doch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, daß der nicht würfelt."

"What quantum mechanics tells us, I believe, is surprising to say the least. It tells us that the basic components of objects – the particles, electrons, quarks etc. – cannot be thought of as "self-existent". The reality that they, and hence all objects, are components of is merely "empirical reality"."

"However unfamiliar this direct interparticle treatment compared to the electrodynamics of Maxwell and Lorentz, it deals with the same problems, talks about the same charges, considers the interactions of the same current elements, obtains the same capacitances, predicts the same inductances and yields the same physical conclusions. Consequently action-at-a-distance must have a close connection with field theory."

"...the "paradox" is only a conflict between reality and your feeling of what reality "ought to be.""

"It will be difficult. But the difficulty really is psychological and exists in the perpetual torment that results from your saying to yourself, 'But how can it be like that?' which is a reflection of uncontrolled but utterly vain desire to see it in terms of something familiar. I will not describe it in terms of an analogy with something familiar; I will simply describe it. There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics. So do not take the lecture too seriously, feeling that you really have to understand in terms of some model what I am going to describe, but just relax and enjoy it. I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, 'But how can it be like that?' because you will get 'down the drain', into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that."

"We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me. Okay, I still get nervous with it.... You know how it always is, every new idea, it takes a generation or two until it becomes obvious that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem."

"We choose to examine a phenomenon Double-slit experiment] which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by "explaining" how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics."

"Quantum theory was split up into dialects. Different people describe the same experiences in remarkably different languages. This is confusing even to physicists."

"Many educators, and even politicians, have been firmly convinced that "free will" is not compatible with Newtonian physics, but very much so with quantum theory. They have been convinced also that it is desirable that the citizen should believe in free will, and they have exerted a certain influence in favor of the indeterministic formulation of subatomic physics. What they have in mind is certainly a sociological purpose of science, whatever the technological purposes may be."

"Quantum mechanics, that mysterious, confusing discipline, which none of us really understands but which we know how to use. It works perfectly, as far as we can tell, in describing physical reality, but it is a ‘counter-intuitive discipline’, as social scientists would say. Quantum mechanics is not a theory, but rather a framework, within which we believe any correct theory must fit."

"Just a few months after de Broglie's suggestion, Schrödinger took the decisive step... by determining an equation that governs the shape and the evolution of probability waves, or as they became known, s. It was not long before Schrödinger's equation and the probabilistic interpretation were being used to make wonderfully accurate predictions. By 1927, therefore, classical innocence had been lost. Gone were the days of a whose individual constituents were set in motion at some moment in the past and obediently fulfilled their inescapable, uniquely determined destiny. According to quantum mechanics, the universe evolves according to a rigorous and precise mathematical formalism, but this framework determines only the probability that any particular function will happen—not which future actually ensues."

"Unlike Newton's mechanics, or Maxwell's electrodynamics, or Einstein's relativity, quantum theory was not created—or even definitively packaged—by one individual, and it retains to this day some of the scars of its exhilarating but traumatic youth. There is no general consensus as to what its fundamental principles are, how it should be taught, or what it really "means." Every competent physicist can "do" quantum mechanics, but the stories we tell ourselves about what we are doing are as various as the tales of Scheherazade, and almost as implausible."

"Quantum mechanics is clearly superior to classical mechanics for the description of microscopic phenomena, and in principle works equally well for macroscopic phenomena. Hence it is at least plausible that the mathematical and logical structure of quantum mechanics better reflect physical reality than do their classical counter parts. If this reasoning is accepted, quantum theory requires various changes in our view of physical reality relative to what was widely accepted before the quantum era, among them the following:1. Physical objects never possess a completely precise position or momentum. 2. The fundamental dynamical laws of physics are stochastic and not deterministic, so from the present state of the world one cannot infer a unique future (or past) course of events. 3. The principle of unicity does not hold: there is not a unique exhaustive description of a physical system or a physical process. Instead, reality is such that it can be described in various alternative, incompatible ways, using descriptions which cannot be combined or compared."

"Quantum physics, as our new subject is called, answers such questions as: Why do the stars shine? Why do the elements exhibit the order that is so apparent in the periodic table? How do transistors and other microelectronic devices work? Why does copper conduct electricity but glass does not? In fact, scientists and engineers have applied quantum physics in almost every aspect of everyday life, from medical instrumentation to transportation systems to entertainment industries. Indeed, because quantum physics accounts for all of chemistry, including biochemistry, we need to understand it if we are to understand life itself. Some of the predictions of quantum physics seem strange even to the physicists and philosophers who study its foundations. Still, experiment after experiment has proved the theory correct, and many have exposed even stranger aspects of the theory.The quantum world is an amusement park full of wonderful rides that are guaranteed to shake up the commonsense world view you have developed since childhood."

"Only around the end of the nineteenth century did scientists come across a few observations that did not fit well with Newton's laws, and these led to the net revolution in physics - the theory of relativity and quantum mechanics."

"Einstein was confused, not the quantum theory."

"My attitude — I would paraphrase Goering—is that when I hear of Schrödinger's cat, I reach for my gun."

"After these conversations with Tagore some of the ideas that had seemed so crazy suddenly made much more sense. That was a great help for me."

"Physicists do not believe quantum mechanics because it explains the world, but because it predicts the outcome of experiments with almost miraculous accuracy. Theorists kept predicting new particles and other phenomena, and experiments kept bearing out those predictions."

"Of course, the apparent disarray could have stemmed entirely from my own ignorance. But when I revealed my impression of confusion and dissonance to one of the attendees, he reassured me that my perception was accurate. “It’s a mess,” he said of the conference (and, by implication, the whole business of interpreting quantum mechanics). The problem, he noted, arose because, for the most part, the different interpretations of quantum mechanics cannot be empirically distinguished from one another; philosophers and physicists favor one interpretation over another for aesthetic and philosophical—that is, subjective—reasons."

"Erwin with his psi can do Calculations quite a few. But one thing has not been seen: Just what does psi really mean?"

"It is often stated that of all the theories proposed in this century, the silliest is quantum theory. In fact, some say that the only thing that quantum theory has going for it is that it is unquestionably correct."

"In this connection the "classical object" is usually called apparatus, and its interaction with the electron is spoken of as measurement. However, it must be emphasized that we are here not discussing a process of measurement in which the physicist-observer takes part. By measurement, in quantum mechanics, we understand any process of interaction between classical and quantum objects, occurring apart from and independently of any observer. The importance of the concept of measurement in quantum mechanics was elucidated by N. Bohr. We have defined "apparatus" as a physical object which is governed, with sufficient accuracy, by classical mechanics. Such, for instance, is a body of large enough mass. However, it must not be supposed that apparatus is necessarily macroscopic. Under certain conditions, the part of apparatus may also be taken by an object which is microscopic, since the idea of "with sufficient accuracy" depends on the actual problem proposed. Thus, the motion of an electron in a Wilson chamber is observed by means of the cloudy track which it leaves, and the thickness of this is large compared with atomic dimensions; when the path is determined with such low accuracy, the electron is an entirely classical object. Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation."

"I would like to describe an attitude toward quantum mechanics which, whether or not it clarifies the interpretational problems that continue to plague the subject, at least sets them in a rather different perspective. This point of view alters somewhat the language used to address these issues—a glossary is provided in Appendix C—and it may offer a less perplexing basis for teaching quantum mechanics or explaining it to nonspecialists. It is based on one fundamental in sight, perhaps best introduced by an analogy. My complete answer to the late 19th century question "what is electrodynamics trying to tell us" would simply be this:Fields in empty space have physical reality; the medium that supports them does not.Having thus removed the mystery from electrodynamics, let me immediately do the same for quantum mechanics:Correlations have physical reality; that which they correlate does not."

"Quantum mechanics is a more general model than classical mechanics, in the same way that Einsteinian relativity is a more general model than Galilean relativity. One picture subsumes the other. Quantum mechanics, pushed to the limit of the large, goes over smoothly into classical mechanics, whereas classical mechanics remains resolutely classical even when pushed to the limit of the small. ...Heisenberg, Schrödinger, Dirac, and the other early quantum mechanicians ...needed to peek at the classical equations in order to set the quantum equations on the right track. They needed... an idea of where the broader theory must eventually lead."

"Quantum mechanics fascinates me. It describes a wide variety of phenomena based on very few assumptions. It starts with a framework so unlike the differential equations of classical physics, yet it contains classical physics within it. It provides quantitative predictions for many physical situations, and these predictions agree with experiments. In short, quantum mechanics is the ultimate basis, today, by which we understand the physical world."

"In his standoff with Dr. Ramsay of Harvard last fall, Dr. Leggett suggested that his colleagues should consider the merits of the latter theory. "Why should we think of an electron as being in two states at once but not a cat, when the theory is ostensibly the same in both cases?" Dr. Leggett asked. Dr. Ramsay said that Dr. Leggett had missed the point. How the wave function mutates is not what you calculate. "What you calculate is the prediction of a measurement," he said. "If it's a cat, I can guarantee you will get that it's alive or dead," Dr. Ramsay said. David Gross, a recent Nobel winner and director of the Kavli Institute for Theoretical Physics in Santa Barbara, leapt into the free-for-all, saying that 80 years had not been enough time for the new concepts to sink in. "We're just too young. We should wait until 2200 when quantum mechanics is taught in kindergarten.""

"The general notions about human understanding ... which are illustrated by discoveries in atomic physics are not in the nature of things wholly unfamiliar, wholly unheard of, or new. Even in our own culture, they have a history, and in Buddhist and Hindu thought a more considerable and central place. What we shall find is an exemplification, an encouragement, and a refinement of old wisdom."

"I argue that what breathes fire into the QM equations is field-theoretic what-it's-likeness: "microqualia" to use a philosopher's term of art. The different values of the solutions to the ultimate physical equations exhaustively yield the abundance of different values of subjectivity. There is no room for dualism; "nomological danglers"; causally inert epiphenomena; classical, porridge-like lumps of otherwise insentient but magically mind-secreting matter, etc. There is no "explanatory gap" because there aren't any material objects - not even brains or nerve cells as commonly (mis)perceived. Instead, over millions of years, non-equilibrium thermodynamics and universal, (neo-)Darwinian principles of natural selection have contrived to organise a minimal and self-intimating subjective sludge of microqualia into complex functional living units. Initially, these units have taken the form of self-replicating, information-bearing biomolecular patterns. Eventually, selection-pressure has given rise to complex minds as well, albeit as just one part of the throwaway host vehicles by which our genes leave copies of themselves. Conscious mind, on this proposal, is a triumph of organisation: our egocentric virtual worlds are warm and gappy QM-coherent states of consciousness. Contra materialist metaphysics, sentience of any kind is not the daily re-enactment of an ontological miracle. Moreover the idea that what-it's-like-ness is the fire in the equations is (at least) consistent with orthodox relativistic quantum field theory - because the theorists' key notions (e.g. that of a field, string, brane, etc) are defined purely mathematically. In other cases, they readily lend themselves to such a reconstruction. Using the word "physical" doesn't add anything of substance."

"I should begin by expressing my general attitude to present-day quantum theory, by which I mean standard non-relativistic quantum mechanics. The theory has, indeed, two powerful bodies of fact in its favour, and only one thing against it. First, in its favour are all the marvellous agreements that the theory has had with every experimental result to date. Second, and to me almost as important, it is a theory of astonishing and profound mathematical beauty. The one thing that can be said against it is that it makes absolutely no sense!"

"I also knew the formula that expresses the energy distribution in the normal spectrum. A theoretical interpretation therefore had to be found at any cost, no matter how high. It was clear to me that classical physics could offer no solution to this problem, and would have meant that all energy would eventually transfer from matter to radiation. ...This approach was opened to me by maintaining the two . The two laws, it seems to me, must be upheld under all circumstances. For the rest, I was ready to sacrifice every one of my previous convictions about physical laws. ...[One] finds that the continuous loss of energy into radiation can be prevented by assuming that energy is forced at the outset to remain together in certain quanta. This was purely a formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result."

"Planck ...devised his quanta theory, according to which the exchange of energy between the matter and the ether—or rather between ordinary matter and the small resonators whose vibrations furnish the light of incandescent matter—can take place only intermittently. A resonator can not gain energy or lose it in a continuous manner. It can not gain a fraction of a quantum; it must acquire a whole quantum or none at all."

"Ask anyone today working on foundational questions in quantum theory and you are likely to hear that there is still no consensus on many of these questions—all the while, of course, everybody seems to be in perfect agreement on how to apply the quantum formalism when it comes to making experimental predictions."

"For Mendeleev the rare earths were a complete nightmare because he didn't know where to put them. He couldn't fit them in the table..! Five of them had been found by the time he was building the table, and so he... stuck them in somewhere where things went 3+, and then went "Uh?" and... left it at that. ...[T]his was a real problem, because no one knew where these building blocks went into the periodic table. ...[I]t wasn't ...until Moseley had established what was, that things began to fit together... and suddenly they realized that there couldn't be more than 14... [T]hen as the quantum mechanics rules came through it became clear... that... you'd found the hole. There was the gap... in , and so that became a target."

"If we really want to understand quantum mechanics, the goal should be more about letting go of our biases and embracing what the Universe tells us about itself. Instead, Carroll regressively campaigns for the opposite in teasing his upcoming new book. Unsurprisingly, most physicists are underwhelmed."

"Quantum mechanics is science’s equivalent of political polarization.Voters either take sides and argue with each other endlessly, or stay home and accept politics as it is. Physicists either just accept quantum mechanics and do their calculations, or take sides in the never-ending debate over what quantum mechanics is actually saying about reality."

"The rules of quantum mechanics work, but only if all natural phenomena in the world of the small are subjected to the same rules. This includes viruses, bacteria, even people. However, the bigger and heavier an object is, the harder it becomes to observed the quantum mechanical deviations from the ordinary, 'classical' laws of movement."

"The inner mysteries of quantum mechanics require a willingness to extend one’s mental processes into a strange world of phantom possibilities, endlessly branching into more and more abstruse chains of coupled logical networks, endlessly extending themselves forward and even backwards in time."

"(\left|x\right\rang \left|y\right\rang- \left|y\right\rang \left|x\right\rang) ... was my first lesson in quantum mechanics, and in a very real sense my last, since the rest is mere technique, which can be learnt from books."

"Respectable scientists like de Broglie himself accept wave mechanics because it confers coherence and unity upon the experimental findings of contemporary science, and in spite of the astonishing changes it implies in connection with ideas of causality, time, and space, but it is because of these changes that it wins favor with the public. The great popular success of Einstein was the same thing. The public drinks in and swallows eagerly everything that tends to dispossess the intelligence in favor of some technique; it can hardly wait to abdicate from intelligence and reason and from everything that makes man responsible for his destiny."

"This theoretical failure to find a plausible alternative to quantum mechanics, even more than the precise experimental verification of linearity, suggests to me that quantum mechanics is the way it is because any small change in quantum mechanics would lead to logical absurdities. If this is true, quantum mechanics may be a permanent part of physics. Indeed, quantum mechanics may survive not merely as an approximation to a deeper truth, in the way that Newton's theory of gravitation survives as an approximation to Einstein's general theory of relativity, but as a precisely valid feature of the final theory."

"It is truly surprising how little difference all this makes. Most physicists use quantum mechanics every day in their working lives without needing to worry about the fundamental problem of its interpretation. Being sensible people with very little time to follow up all the ideas and data in their own specialties and not having to worry about this fundamental problem, they do not worry about it. A year or so ago, while Philip Candelas (of the physics department at Texas) and I were waiting for an elevator, our conversation turned to a young theorist who had been quite promising as a graduate student and who had then dropped out of sight. I asked Phil what had interfered with the ex-student’s research. Phil shook his head sadly and said, “He tried to understand quantum mechanics.” So irrelevant is the philosophy of quantum mechanics to its use, that one begins to suspect that all the deep questions about the meaning of measurement are really empty, forced on us by our language, a language that evolved in a world governed very nearly by classical physics. But I admit to some discomfort in working all my life in a theoretical framework that no one fully understands. And we really do need to understand quantum mechanics better in quantum cosmology, the application of quantum mechanics to the whole universe, where no outside observer is even imaginable. The universe is much too large now for quantum mechanics to make much difference, but according to the big-bang theory there was a time in the past when the particles were so close together that quantum effects must have been important. No one today knows even the rules for applying quantum mechanics in this context."

"My own conclusion is that today there is no interpretation of quantum mechanics that does not have serious flaws. This view is not universally shared. Indeed, many physicists are satisfied with their own interpretation of quantum mechanics. But different physicists are satisfied with different interpretations. In my view, we ought to take seriously the possibility of finding some more satisfactory other theory, to which quantum mechanics is only a good approximation."

"Quantum theory does not trouble me at all. It is just the way the world works. What eats me, gets me, drives me, pushes me, is to understand how it got that way. What is the deeper foundation underneath it? Where does it come from? So that we won’t see it as something that is unwelcome by friends that we admire—John Bell and many others—it will be something that will make you say, ‘It couldn’t have been otherwise.’ We haven’t gotten to that stage yet, and until we do, we have not met the challenge that is right there. I continue to say that the quantum is the crack in the armor that covers the secret of existence. To me it’s a marvelous stimulus, hope, and driving force. And yet I am afraid that just the word—‘hope’—is what does not eat, or possess, or drive so many of our colleagues in the field. They’re content to take the theory for granted, rather than to find out where it comes from. But you would hardly feel the drive to find out where from if you don’t feel that the theory is utterly right. I have been brought up from ‘childhood’ to feel that it is utterly right. Here I was, reading that book of Weyl’s at the age of eighteen and just crazy about it."

"I had the feeling that the stuff was beautiful. I learned it from Weyl, and Weyl had the art of putting things in a lovely perspective. More so than anybody else I have ever read. That book was just a treat. So the feeling of ‘rotten’ would be the absolutely last feeling I would ever have about it. ‘Beautiful’ is what I would call it. To me it’s the magic way to do it. I think that having started early and having used it in lots of different contexts, all the way from my doctor’s thesis on the dispersion and absorption of light in a helium atom, to nuclear physics, to the decay of elementary particles, I feel absolutely at home with it. But John Bell’s question I certainly sympathize with. An ‘irreversible act of amplification’? As Eugene Wigner always says, ‘What means it "irreversible"?’ [...] I think it is just wonderful to have puzzles like that staring us in the face. You’d be amused. Every day I try to write down something in my notebook, although I don’t always succeed, pushing things ahead just a little bit. I only got in two or three sentences this morning. ‘Nada. The photon doesn’t exist in the atom. It doesn’t exist in the photodetector after the act of emission, and you have no right to talk of what it’s doing in between. Nada—it’s nothing.’ Then there’s the irreversible act of amplification where you’ve got a whole lot of things. It’s nada to nada."

"For a zeroth slogan about quantum mechanics, I’ve chosenWhat’s hard to understand is classical mechanics, not quantum mechanics."

"The world is not as real as we think.… My personal opinion is that the world is even weirder than what quantum physics tells us."

"I’ve had experts in quantum field theory – people who’ve spent years calculating path integrals of mind-boggling complexity – ask me to explain the Bell inequality to them, or other simple conceptual things like Grover’s algorithm. I felt as if Andrew Wiles had asked me to explain the Pythagorean Theorem."

"The purpose of the first part is to convince the reader that the formalism leading to Bell's inequalities is very general and reasonable. What is surprising is that such a reasonable formalism conflicts with quantum mechanics. In fact, situations exhibiting a conflict are very rare, and quantum optics is the domain where the most significant tests of this conflict have been carried out"

"One of these articles, written by N. David Mermin, gave me a tremendous shock. Mermin described the results of experiments that had been carried out as recently as 1982 to test something called Bell's theorem using two-photon 'cascade' emission from excited calcium atoms. Put simply, Bell's theorem says that my idea of naive realism is in conflict with the predictions of quantum theory in a way that can be tested in the laboratory in special experiments on pairs of quantum particles. These experiments had been done: quantum theory had been proved right and naive realism wrong! There in a montage was a pictorial history of the debate about reality and the experiments that had been done to test it (reproduced opposite). This work struck me as desperately important to my understanding of physical reality, something that as a scientist I felt I ought to know about. This discovery also made me feel rather embarrassed. Here I was, proud of my scientific qualifications and with almost 10 years' experience in chemical physics research at various prestigious institutions around the world, and I had been going around with a conception of physical reality that was completely wrong! Why hadn't somebody told me about this before?"

"Again, part of that psychohistorical study I would like to see is why it did not impress the Copenhagen people, especially Bohr. But in the end it turns out that these other people were, in a way, right, because what I am notorious for, the so-called Bell's theorem, is just for showing that Einstein's explanation doesn't work. Einstein's explanation works so long as you have perfect correlations, which means measuring the same component of spin on the two sides [spin is a measure of a property similar but not identical to the rotation of a particle on its axis]. But as soon as you are measuring in a nonparallel direction, you get results that cannot be explained by Einstein's idea that the answers existed before the experiment."

"The theorem tells you that maybe there must be something happening faster than light, although it pains me even to say that much. The theorem certainly implies that Einstein's concept of space and time, neatly divided up into separate regions by light velocity, is not tenable. But then, to say that there's something going faster than light is to say more than I know."

"The main impact of Bell's theorem is from a philosophical-historical perspective: it reinforces, outside the physics of quantum entanglement, the incompatibility of hidden variable theories with quantum mechanics."

"That's all. That's the difficulty. That's why quantum mechanics can't seem to be imitable by a local classical computer. I've entertained myself always by squeezing the difficulty of quantum mechanics into a smaller and smaller place, so as to get more and more worried about this particular item. It seems to be almost ridiculous that you can squeeze it to a numerical question that one thing is bigger than another. But there you are—it is bigger than any logical argument can produce, if you have this kind of logic."

"Bell’s theorem is the most profound discovery of science."

"The gist of Bell's theorem is this: no local model of reality can explain the results of a particular experiment."

"Bell himself managed to devise such a proof which rejects all models of reality possessing the property of "locality". This proof has since become known as Bells theorem. It asserts that no local model of reality can underlie the quantum facts. Bell's theorem says that reality must be non-local."

"Physicists continue to debate whether Bell's theorem is airtight or not. However, the real question is not whether Bell can prove beyond doubt that reality is non-local, but whether the world is in fact non-local."

"There's an interesting scientific principle that a wrong answer can be much more stimulating to the field than just sort of finding the answer that's in the back of the book. A wrong result gets people excited. Worried. Obviously, you don't really want that to be happening—it's OK for a theorist to come up with a speculative new theory that gets shot down, but experimentalists are supposed to be very careful and their error limits are supposed to be realistic. Unfortunately, with this experiment, whenever you're looking for a stronger correlation, any kind of systematic error you can imagine typically weakens it and moves it toward the hidden-variable range. It was a hard experiment. In those days, at any rate, with the kind of equipment I had, and … well, what can I say? I screwed up."

"The experimental verification of violations of Bell’s inequality for randomly set measurements at space-like separation is the most astonishing result in the history of physics. Theoretical physics has yet to come to terms with what these results mean for our fundamental account of the world. Experimentalists, from Freedman and Clauser and Aspect forward, deserve their share of the credit for producing the necessary experimental conditions and for steadily closing the experimental loopholes available to the persistent skeptic. But the great achievement was Bell’s. It was he who understood the profound significance of these phenomena, the prediction of which can be derived easily even by a freshman physics student. Unfortunately, many physicists have not properly appreciated what Bell proved: they take the target of his theorem— what the theorem rules out as impossible—to be much narrower and more parochial than it is. Early on, Bell’s result was often reported as ruling out determinism, or hidden variables. Nowadays, it is sometimes reported as ruling out, or at least calling in question, realism. But these are all mistakes. What Bell’s theorem, together with the experimental results, proves to be impossible (subject to a few caveats we will attend to) is not determinism or hidden variables or realism but locality, in a perfectly clear sense. What Bell proved, and what theoretical physics has not yet properly absorbed, is that the physical world itself is non-local."

"No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."

"Bell's theorem, for which he is most famous, was more a triumph of character than of intellect. The difficult thing about it was the realization of what was understood and what was not understood in the discussion of hidden variables. Bell's honesty about his own understanding provided the impetus for his formulation and proof of the theorem."

"In the same paper, Bell also discussed two rather unwelcome properties of hidden-variables theories. The first was contextuality. This tells us that, except in trivial cases, any hidden-variable theory must be such that the result of measuring a particular observable will depend on which other observable) are measured simultaneously. The second was nonlocality. All me hidden-variable models that Bell examined, including Bohm's, had the unpleasant feature that the behaviour of a particular particle depended on the properties of all others, however far away they were. In the EPR case, the measurement result obtained on one particle would depend on what measurement is performed on the second. As Bell said, this was the resolution of the EPR problem that Einstein would have liked least, and it is in this sense that it may be said that Bell proved Einstein wrong."

"At the very least, Bell's Theorem prevents us from interpreting quantum amplitudes as probability in the obvious way. You cannot point at a single configuration, with probability proportional to the squared modulus, and say, "This is what the universe looked like all along.""

"As stated repeatedly in this book, John von Neumann's Mathematical Foundations of Quantum Mechanics was an extraordinarily influential work. It is important to recall that the language most commonly used to describe and discuss the measurement process in terms of a collapse or projection of the wave function essentially originates with this classic work. It was von Neumann who so clearly distinguished (in the mathematical sense) between the continuous time-symmetric quantum mechanical equations of motion and the discontinuous, time-asymmetric measurement process. Although much of his contribution to the development of the theory was made broadly within the boundaries of the Copenhagen view, he stepped beyond those boundaries in his interpretation of quantum measurement."

"Thus the formal proof of von Neumann does not justify his informal conclusion: 'It is therefore not, as is often assumed, a question of reinterpretation of quantum mechanics - the present system of quantum mechanics would have to be objectively false in order that another description of the elementary process than the statistical one be possible.' It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made."

"The physicists didn't want to be bothered with the idea that maybe quantum theory is only provisional. A horn of plenty had been spilled before them, and every physicist could find something to apply quantum mechanics to. They were pleased to think that this great mathematician had shown it was so. Yet the Von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It's not just flawed, it's silly. If you look at the assumptions made, it does not hold up for a moment. It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they're nonsense. You may quote me on that: The proof of Von Neumann is not merely false but foolish!"

"There are (at least) two possible ways to formulate precisely (i.e. mathematically) elementary QM. The eldest one, historically speaking, is due to von Neumann in essence, and is formulated using the language of Hilbert spaces and the spectral theory of unbounded operators. A more recent and mature formulation was developed by several authors in the attempt to solve quantum field theory problems in mathematical physics. … The newer formulation can be considered an extension of the former one, in a very precise sense that we shall not go into here, also by virtue of the novel physical context it introduces and by the possibility of treating physical systems with infinitely many degrees of freedom, i.e. quantum fields. In particular, this second formulation makes precise sense of the demand for locality and covariance of relativistic quantum field theories, and allows to extend quantum field theories to a curved spacetime."

"By now it is clear that consistent theories of quantum gravity can be constructed in the context of string theory. This can also be done in diverse dimensions by considering suitable compactifications. This diversity, impressive as it may be for a consistent theory to possess, poses a dilemma: The theory appears to be more permissive than desired!"

"... our most fundamental concept, that of spacetime, is itself being threatened as we probe the quantum nature of dynamical spacetime of quantum gravity."

"Gravity's weakness makes it very difficult to measure its quantum effects; as a result, we have no experimental data to guide theoretical physicists in the development of the missing theory. Detecting a “graviton" – the hypothetical particle making up part of a gravitational field – would require a particle collider the size of the Milky Way or a detector with a mass of the planet Jupiter. These experiments are so detached from our technological capabilities that physicists have focused on trying to remove the mathematical contradictions first, developing approaches like string theory, loop quantum gravity, and asymptotically safe gravity. But to know which theory describes physical reality, experimental tests must eventually be developed."

"Hawking's intitial foray into quantum gravity was more modest than Wheeler's and other[s]... a sneak approach. He first wanted to know what the effect was of an ordinary, classic, curved-space gravitational field on a quantum system. He called this the semiclassical approach. Until that day, most quantum calculations had been done as if gravity didn't exist—they were hard enough without it in normal flat space-time... [Hawking accomplished this by] envisioning an "atom" whose nucleus was a catastrophically powerful black hole... Starobinsky ventured the opinion that rotating black holes would spray elementary particles. ...It was known from Penrose's work, among others, that you could extract energy from the spin of a black hole just like any other dynamo... in particles and radiation just like it did from a particle generator. ...But Hawking ...resolved to redo the calculation for himself ...he decided to warm up first, by calculating the rate of emission from a nonrotating quantum hole. He knew the answer should be no emission. ...his results were embarrassing. His imaginary black hole was spewing matter and radiation... he was reluctant to tell anybody but his closest friends; he was afraid Bekenstein would hear about it. ...It meant that holes had temperatures, just as Bekenstein's work implied."

"Unfortunately, the formulation of general rules for the calculation of in the quantum theory of gravitation has only confirmed the presence of another difficulty: The theory contains infinities, arising from integrals over large virtual momenta. Quantum electrodynamics contains similar infinities, but only in three or four special cases, where they can be dealt with by a renormalization of , , and wave functions. ... In contrast, the quantum theory of gravitation contains an infinite variety of infinities, as can be seen by an elementary dimensional argument: The gravitational constant has dimension ћ/m2, so a term in a dimensionless probability amplitude of order G'm will diverge like a integral ʃ p'2n–1'dp. In this respect, the theory of gravitation is more like other nonrenormizable theories, such as the of , than it is like quantum electrodynamics."

"... it would be a bracing achievement, and major progress, to identify any concrete, observable phenomenon that brings in truly characteristic features of quantum gravity beyond the semiclassical approximation in common use. Actual observation would bring the subject to another level."

"How can quantum gravity help explain the ?"

"Since its inception in Richard Feynman’s 1942 doctoral thesis, the path integral has been a physicist’s dream and a mathematician’s nightmare. To a physicist, the path integral provides a powerful and intuitive way to understand quantum mechanics, building on the simple idea that quantum physics is fundamentally a theory of superposition and interference of probability amplitudes. The “sum over histories” offers a framework for tackling problems ranging from Feynman diagrams to lattice chromodynamics, from quantum cosmology to superfluid vortices to stock-option pricing. To a mathematician, the path integral is at best an ill-defined formal expression. It is some sort of vaguely integral-like object involving a “sum” over a badly specified collection of functions, having an undefined measure, and whose value is apparently determined by a group of unclear and perhaps incompatible limits that may or may not yield finite answers."

"The Feynman method has the virtue that it provides us with a vivid picture of nature’s quantum trickery at work. The idea is that the path of a particle through space is not generally well defined in quantum mechanics. … So when an electron arrives at a point in space—say a target screen—many different histories must be integrated together to create this one event. Feynman’s so-called path-integral, or sum-over-histories approach to quantum mechanics, set this remarkable concept out as a mathematical procedure. It remained more or less a curiosity for many years, but as physicists pushed quantum mechanics to its limits— applying it to gravitation and even cosmology—so the Feynman approach turned out to offer the best calculational tool for describing a quantum universe. History may well judge that, among his many outstanding contributions to physics, the path-integral formulation of quantum mechanics is the most significant."

"You could not imagine the sum-over-histories picture being true for a part of nature and untrue for another part. You could not imagine it being true for electrons and untrue for gravity. It was a unifying principle that would either explain everything or explain nothing. And this made me profoundly skeptical. I knew how many great scientists had chased this will-o’-the-wisp of a unified theory. The ground of science was littered with the corpses of dead unified theories. Even Einstein had spent twenty years searching for a unified theory and had found nothing that satisfied him. I admired Dick tremendously, but I did not believe he could beat Einstein at his own game."

"The sum-over-paths formulation is particularly convenient for integrating out one set of coordinates to concentrate on the remaining set. Thus the photon propagator in quantum electrodynamics is obtained ... by "integrating out" the photon variables, leaving electrons and positrons, both real and virtual, to interact by means of the covariant function \delta (x^2) + (\pi i x^2)^-1."

"The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuitive than the usual approaches. Applications of path integrals are as vast as those of quantum mechanics itself, including the quantum mechanics of a single particle, statistical mechanics, condensed matter physics and quantum field theory. ... It is in quantum field theory, both relativistic and nonrelativistic, that path integrals (functional integrals is a more accurate term) play a much more important role, for several reasons. They provide a relatively easy road to quantization and to expressions for Green’s functions, which are closely related to amplitudes for physical processes such as scattering and decays of particles. The path integral treatment of gauge field theories (non-abelian ones, in particular) is very elegant: gauge fixing and ghosts appear quite effortlessly. Also, there are a whole host of nonperturbative phenomena such as solitons and instantons that are most easily viewed via path integrals. Furthermore, the close relation between statistical mechanics and quantum mechanics, or statistical field theory and quantum field theory, is plainly visible via path integrals."

"The idea behind the Feynman path integral goes back to a paper by P. A. M. Dirac published in 1933 in Physikalische Zeitschrift der Sowjetunion. It formed the core of Richard Feynman’s space–time approach to quantum mechanics and quantum electrodynamics. Although the path integral was not mathematically well defined, it was widely used in quantum field theory, statistical mechanics, and string theory. Recently, path integrals have been the heuristic guide to spectacular developments in pure mathematics."

"In the sixties, a new field of applications of functional integrals appeared — the quantization of gauge fields. The electromagnetic field, the Einstein gravitational field, the Yang-Mills field and the chiral field can serve as examples of gauge fields. The action functionals of those fields are invariant under gauge transformations which depend on one or several arbitrary functions. From a mathematical point of view, gauge fields are fields of geometrical origin which are connected with fibrations over four-dimensional space-time. The specificity of geometrical fields has to be taken into account when quantizing them; otherwise incorrect results may be obtained."

"Path integrals (the terminology for functional integrals, derived from their quantum mechanical origin) had been introduced by Dirac in the 1930s. However, their mathematical fuzziness (in contrast to the precise Euclidean Wiener integrals) had discouraged their serious application in quantum theory. Nonetheless, despite the absence of a mathematical definition it became apparent that path integrals in field theory were ideally suited to (i) implement the symmetries of the theory directly, (ii) incorporate constraints simply, (iii) explore field topology, (iv) isolate relevant dynamical variables, (v) describe non-zero temperature. They were key ingredients in model-making for unified theories, and by the late 1970s a working knowledge of functional integrals had become extremely useful for most field theorists."

"Molecular dynamics is not primarily about making movies of molecules. More often it is about developing quantitative predictions of molecular size and shape, flexibilities, interactions with other molecules, behavior under pressure, and the relative frequency of one state or conformation compared to another. The complex nature of the force fields involved and the large size of typical molecular systems mean that molecular dynamics is almost always chaotic. The changes in a molecule that occur over time are important, but these should be understood in terms of changes in averaged quantities, structural forms, or families of “nearby” structures. Molecular dynamics relies on time-stepping to compute successive snapshots, but these are often used for sampling a probability distribution, or else a number of evolving paths are averaged to describe the likely sequence of changes that would be observed in a typical evolution of the molecule."

"When after some weeks I had a chance to talk to Oppenheimer, I was astonished to discover that his reasons for being uninterested in my work were quite the opposite of what I had imagined. I had expected that he would disparage my program as merely unoriginal, a minor adumbration of Schwinger and Feynman. On the contrary, On the contrary, he considered it to be fundamentally on the wrong track. He thought adumbrating Schwinger and Feynman to be a wasted effort, because he did not believe that the ideas of Schwinger and Feynman had much to do with reality. I had known that he had never appreciated Feynman, but it came as a shock to hear him now violently opposing Schwinger, his own student, whose work he had acclaimed so enthusiastically six months earlier. He had somehow become convinced during his stay in Europe that physics was in need of radically new ideas, that this quantum electrodynamics of Schwinger and Feynman was just another misguided attempt to patch up old ideas with fancy mathematics."

"The theory of quantum electrodynamics describes nature as absurd from the point of view of common sense. And it fully agrees with experiment. So I hope you can accept nature as She is—absurd."

"We physicists are always checking to see if there is something the matter with the theory. That’s the game, because if there is something the matter, it’s interesting! But so far, we have found nothing wrong with the theory of quantum electrodynamics. It is, therefore, I would say, the jewel of physics—our proudest possession. The theory of quantum electrodynamics is also the prototype for new theories that attempt to explain nuclear phenomena, the things that go on inside the nuclei of atoms. If one were to think of the physical world as a stage, then the actors would be not only electrons, which are outside the nucleus in atoms, but also quarks and gluons and so forth—dozens of kinds of particles—inside the nucleus. And though these “actors” appear quite different from one another, they all act in a certain style—a strange and peculiar style—the “quantum” style. At the end, I’ll tell you a little bit about the nuclear particles. In the meantime, I’m only going to tell you about photons—particles of light—and electrons, to keep it simple. Because it’s the way they act that is important, and the way they act is very interesting."

"With the help of W Furry, Weisskopf ... showed that the inclusion of positrons in the electron self-energy calculation reduced the degree of divergence to ln(1/r0), where r0 again represents a minimum cut-off distance. Thus positrons were a partial, but not sufficient, help. Another infinite quantity occurring in quantum electrodynamics arises as a result of the production of virtual electron-positron pairs by a photon. Like the electron self-energy, this vacuum polarization divergence depends on the logarithm of a cut-off parameter. Despite the infinite nature of vacuum polarization, it was possible to calculate its effect on the Coulomb interaction, for instance in a hydrogen atom, by comparing the interaction at large and shorter distances."

"The history of QED in the period from 1946 to 1949 has many similarities with the history of the developments of quantum mechanics from 1925 to 1927, when Schrödinger and Heisenberg had propounded two different approaches to quantum mechanics. The correspondence can be taken to be: Feynman is to Heisenberg what Tomonaga-Schwinger is to Schrödinger, with Dyson initially playing the role of Pauli, Schrödinger, and Eckart in proving the equivalence of the two approaches."

"Once we got past the obscurities produced by spontaneous symmetry breaking in the weak interactions and color trapping in the strong interactions, the Standard Model was revealed to us as a theory that was really not very different from quantum electrodynamics. We had more gauge fields, not just the electromagnetic field but gluon and W and Z fields. There were more fermions, not just the electron but a whole host of charged leptons and neutrinos and quarks. But the Standard Model seemed to be quantum electrodynamics writ large."

"We know something about Einstein's genius we didn't know before."

"While Einstein's belief in an objective reality is similar to that of Weinberg and Sokal, his arguments for his conception of reality are not. In fact, Einstein was no "naive realist," despite such caricaturing of his stand by the Copenhagen orthodoxy. He ridiculed the "correspondence" view of reality that many scientists accept uncritically. Einstein fully realized that the world is not presented to us twice-first as it is, and second, as it is theoretically described-so we can compare our theoretical "copy" with the "real thing." The world is given to us only once - through our best scientific theories. So Einstein deemed it necessary to ground his concept of objective reality in the invariant characteristics of our best scientific theories."

"It deals with an imaginary experiment, but like so many other works of the imagination this one has surpassed the intentions of its creators. In the first place, it is unlikely that they thought that the experiment they were proposing, or any facsimile, would ever be carried out. This, thanks largely to the work inspired by Bell, has now happened. Indeed, our physics journals are now resplendent with new and ever more ingenious versions of the Einstein-Podolsky-Rosen experiment, along with increasingly accurate experimental results. In the second place, and this is also an aftermath of Bell’s work, the Einstein-Podolsky-Rosen experiment has made its way into much of the popular folklore about the quantum theory. (It is usually referred to in the literature, familiarly, as the EPR experiment.)"

"Due to the lucidity and apparently incontestable character of the argument, the paper of Einstein, Podolsky and Rosen created a stir among physicists and has played a large role in general philosophical discussion. Certainly the issue is of a very subtle character and suited to emphasize how far, in quantum theory, we are beyond the reach of pictorial visualization."

"The two photons are entangled and according to local realism, their polarization planes should become independent... a typical EPR situation. Already in 1948, observations... agreed with quantum mechanics, not with local realism."

"If Monod and Weinberg are truly speaking for the twentieth century, then I prefer the eighteenth. But in fact Monod and Weinberg, both of them first-rate scientists and leaders of research in their specialties, are expressing a point of view which does not take into account the subtleties and ambiguities of twentieth-century physics. The roots of their philosophical attitudes lie in the nineteenth century, not in the twentieth. The taboo against mixing knowledge with values arose during the nineteenth century out of the great battle between the evolutionary biologists led by Thomas Huxley and the churchmen led by Bishop Wilberforce. Huxley won the battle, but a hundred years later Monod and Weinberg were still fighting the ghost of Bishop Wilberforce.… For the biologists, every step down in size was a step toward increasingly simple and mechanical behavior. A bacterium is more mechanical than a frog, and a DNA molecule is more mechanical than a bacterium. But twentieth-century physics has shown that further reductions in size have an opposite effect. If we divide a DNA molecule into its component atoms, the atoms behave less mechanically than the molecule... ... If we divide an atom into nucleus and electrons, the electrons are less mechanical than the atom. There is a famous experiment, originally suggested by Einstein, Podolsky and Rosen in 1935 as a thought experiment to illustrate the difficulties of quantum theory, which demonstrates that the notion of an electron existing in an objective state independent of the experimenter is untenable. The experiment has been done in various ways with various kinds of particles, and the results show clearly that the state of a particle has a meaning only when a precise procedure for observing the state is prescribed. Among physicists there are many different philosophical viewpoints, and many different ways of interpreting the role of the observer in the description of subatomic processes. But all physicists agree with the experimental facts which make it hopeless to look for a description independent of the mode of observation. When we are dealing with things as small as atoms and electrons, the observer or experimenter cannot be excluded from the description of nature. In this domain, Monod's dogma, "The cornerstone of the scientific method is the postulate that nature is objective," turns out to be untrue. … We are saying only that if as physicists we try to observe in the finest detail the behavior of a single molecule, the meaning of the words "chance" and "mechanical" will depend upon the way we make our observations. The laws of subatomic physics cannot even be formulated without some reference to the observer. "Chance" cannot be defined except as a measure of the observer's ignorance of the future. The laws leave a place for mind in the description of every molecule."

"Quantum key distribution as described above was the product of my graduate infatuation with quantum theory. It all happened back in Oxford in the late 1980s and early 1990s. I do not remember exactly what prompted me to visit the Clarendon Laboratory library one rainy day, browse the dusty shelves and pick up the original Einstein, Podolsky and Rosen paper for casual reading. However, I do remember this one sentence in the paper that drew my attention: "...If, without in any way disturbing a system, we can predict with certainty ... the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." This is a definition of perfect eavesdropping! I guess I was lucky to read it in this particular way. The rest was just about rephrasing the subject in cryptographic terms."

"Many years ago David Bohm proposed replacing the hypothetical "completeness" experiment of Einstein, Podolsky, and Rosen...by a modified and more practical version. Bohm's experiment (called the EPRB...) involves the decay of a particle into two photons... the photons travel in opposite directions, have equal energy, and have identical polarizations. ...John Bell revealed that the EPRB... could be used to distinguish quantum mechanics from hypothetical hidden variable theories. Bell's theorem (also called Bell's inequalities) concerns a particular quantity that specifies the correlation between the polarizations of the two photons."

"The principal distortion disseminated ... is the implication, or even the explicit claim, that measuring the polarization, circular or plane, of one of the photons somehow affects the other photon. In fact, the measurement does not cause any physical effect to propagate from one photon to the other. ...If on one branch of history, the plane polarization of one photon is measured and thereby specified with certainty, then on the same branch of history the circular polarization of the other photon is also specified with certainty. On a different branch of history the circular polarization of one of the photons may be measured, in which case the circular polarization of both photons is specified with certainty. On each branch, the situation is like that of Bertlmann's socks, described by John Bell... Bertlmann... always wears one pink and one green sock. If you see just one... you know immediately the other... Yet no signal is propogated... Likewise no signal passes from one photon to the other in the experiment that confirms quantum mechanics. No action at a distance takes place."

"The false report that measuring one of the photons immediately affects the other leads to all sorts of unfortunate conclusions. ...the alleged effect... would violate the requirement of relativity theory that no signal... can travel faster than the speed of light. If it were to do so, it would appear to observers in some states of motion that the signal were traveling backward in time."

"The intent of the Einstein-Podolsky-Rosen paper was to show that quantum mechanics... could not be the final word regarding the physics of the microcosmos. ...they wanted to show that every particle does possess a definite position and a definite velocity at any given instant of time, and thus they wanted to conclude that the uncertainty principle reveals a fundamental limitation of the quantum mechanical approach. ...quantum mechanics provides only a partial description of the universe. ...an incomplete theory of physical reality and, perhaps, merely a stepping-stone toward a deeper framework waiting to be discovered."

"Einstein had drawn attention to nonlocality in 1935 in an effort to show that quantum mechanics must be flawed. ...Einstein proposed a thought experiment—now called the EPR experiment—involving two particles that spring from a common source and fly in opposite directions. According to the standard model of quantum mechanics, neither particle has a definite position or momentum before it is measured; but by measuring the momentum of one particle, the physicist instantaneously forces the other particle to assume a fixed position... Deriding this effect as "spooky action at a distance," Einstein argued that it violated both common sense and his own theory of special relativity, which prohibits the propagation of effects faster than the speed of light; quantum mechanics must therefore be an incomplete theory. In 1980, however, a group of French physicists carried out a version of the EPR experiment and showed that it did indeed give rise to spooky action. (The reason that the experiment does not violate special relativity is that one cannot exploit nonlocality to transmit information.)"

"The only part of this article that will ultimately survive, I believe, is this last phrase, which so poignantly summarizes Einstein's views on quantum mechanics in his later years. The content of this paper has been referred to on occasion as the Einstein-Podolsky-Rosen paradox. It should be stressed that this paper contains neither a paradox nor any flaw of logic. It simply concludes that objective reality is incompatible with the assumption that quantum mechanics is complete. This conclusion has not affected subsequent developments in physics, and it is doubtful that it ever will. [...] Experimentalists have actively participated, as well. A number of experimental tests of quantum mechanics in general and also of the predictions of specific alternative schemes have been made. This has not led to any surprises."

"Finally, in that period the 'EPR paper' appeared, a collaboration with Nathan Rosen and Boris Podolsky which deals with the foundations of quantum mechanics. There are a number of physicists who consider this a fundamental contribution to that subject. I am not one of those."

"This onslaught came down upon us as a bolt from the blue. Its effect on Bohr was remarkable... as soon as Bohr had heard my report of Einstein’s argument, everything else was abandoned: we have to clear up such a misunderstanding at once. We should reply by taking up the same example and showing the right way to speak about it. In great excitement, Bohr immediately started dictating to me the outline of such a reply. Very soon, however, he became hesitant. ‘No, this won’t do, we must try all over again... we must make it quite clear...’ So it went on for a while, with growing wonder at the unexpected subtlety of the argument."

"When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or ψ-functions) have become entangled."

"Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separate and therefore virtually capable of being "best possibly known," i.e., of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known—at least not in the way that it could possibly be known more completely—it is due to the interaction itself. Attention has recently been called * to the obvious but very disconcerting fact that even though we restrict the disentangling measurements to one system, the representative obtained for the other system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are entirely arbitrary. It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter's mercy in spite of his having no access to it."

"Aage Bohr expressed the same point to me the last time I talked to him. He just couldn’t understand why there were all these conferences—conference after conference—on EPR, when it is just the way it works. It is just exactly the wrong thing to be asking about. There is a conference coming up in Finland in August with some people I’d love to talk to, but I’ve written them to say that I’m not going. If you keep trying to pull apples off the apple tree, after a while it doesn’t do. I hope that I am not being too propagandistic in speaking of the idea that when we see it all it will be so simple we’ll all say, ‘How stupid we’ve been all this time!’ We’ve got to look for the right word, the right image. So you try one word for a day, for a week, for a month, or for a year, and then you give it up and try another one."

"In conclusion, although Einstein, Podolsky, and Rosen presented a picture of the world that was disproven experimentally, I would still regard them as having won a moral victory: The then-common interpretation of quantum mechanics did indeed have a one person measuring at A, seeing a single outcome, and then making a certain prediction about a unique outcome at B; and this is indeed incompatible with relativity, and wrong. Though people are still arguing about that."

"In a complete theory there is an element corresponding to each element in reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by the wave function is not complete."

"The elements of the physical reality cannot be determined by a priori philosophical considerations, but must be found by an appeal to results of experiments and measurements. ...We shall be satisfied with the following criterion... If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, there exists an element of physical reality corresponding to this physical quantity. It seems to us that this criterion, while far from exhausting all possible ways of recognizing a physical reality, at least provides us with one such way..."

"The usual conclusion... in quantum mechanics is that when the momentum of a particle is known, its coordinate has no physical reality. More generally, it is shown in quantum mechanics that, if the operators corresponding to two physical quantities... do not commute... then the precise knowledge of one of them precludes such knowledge of the other. Furthermore, any attempt to determine the latter experimentally will alter the state of the system in such a way as to destroy the knowledge of the first. From this follows that either (1) the quantum mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to the two physical quantities do not commute the two quantities cannot have simultaneous reality."

"In quantum mechanics it is usually assumed that the wave function does contain a complete description of the physical reality of the system in the state to which it corresponds. ...We shall show, however, that this assumption, together with the criterion of reality given above, leads to a contradiction."

"As a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. ...Thus, it is possible to assign two different wave functions... to the same reality."

"Starting then with the assumption that the wave function does give a complete description of reality, we arrive to the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."

"One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical realities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously... can be predicted, they are not simultaneously real. ...No reasonable definition of reality could be expected to permit this."

"While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible."

"The theory of the light quantum, which Einstein initiated in 1905 and which is with us still, is certainly the most revolutionary development in the history of physics and arguably in the history of science. The creators of the theory, men such as Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli, and Paul Dirac, were often struck by the apparent "absurdity"—the utterly noncommonsensical aspects—of the world depicted by the quantum theory."

"Light is something like raindrops—each little lump of light is called a photon—and if the light is all one color, all the "raindrops" are the same."

"My favorite deep, elegant and beautiful explanation is Albert Einstein's 1905 proposal that light consists of energy quanta, today called photons. Actually, it is little known, even among physicists, but extremely interesting how Einstein came to this position. It is often said that Einstein invented the concept to explain the photoelectric effect. Certainly, that is part of Einstein's 1905 publication, but only towards its end. The idea itself is much deeper, more elegant and, yes, more beautiful."

"At the first of the 1960's Rochester Coherence Conferences, I suggested that a license be required for use of the word photon, and offered to give such license to properly qualified people."

"All the indistinguishable photons illuminate the array of N slits, or grating, simultaneously. If only one photon propagates, at any given time, then that individual photon illuminates the whole array of N slits simultaneously."

"If quantum mechanics is right, there is no way to get around the uncertainty principle. The reason that the electron’s probability wave spread so much after we confined it, Heisenberg would argue, is that its momentum became almost completely indeterminate. In a manner of speaking, it headed off in all directions."

"At the heart of the quantum revolution is Heisenberg's uncertainty principle... roughly... all physical quantities... are subject to unpredictable fluctuations, so that their values are not precisely defined. ...[e.g., we are] free to measure [position x and the momentum p of a quantum particle] to arbitrary precision, but they cannot possess precise values simultaneously. The spread, or uncertainty, in their values, denoted by Ax and Ap... are such that [their] product... cannot be less than... [after Max Planck], numerically very small... so that quantum effects are generally only important in the atomic domain... not... in daily life. ...[T]his uncertainty is inherent in nature and not merely the result of technological limitations in measurement."

"The uncertainty has deep implications. For example, it means that a quantum particle does not move along a well-defined path through space. ...The smearing of position and momentum leads to an inherent indeterminism in the behaviour of quantum systems. ...[T]he experimenter may fire an electron at a target and find that it scatters to the left, then, on repeating... the next electron scatters to the right. Quantum mechanics still enables the relative probabilities of the alternatives to be specified precisely. ...It can make definite predictions about ensembles of identical systems, but it can generally tell us nothing definite about an individual system."

"Relativity principles require us to associate mass with the energy of radiation, and it is reasonable to suppose... an exchange of ... [T]he exchange of momentum between free electrons and radiation is very similar to the exchange... when two particles collide. ...[A] beam of light should be considered as an assembly of "units", each or which [using light (\nu), (h), speed of light (c)] possesses energy (W), momentum (p), and mass (m), given byW = h\nu; \; p = \frac{h\nu}{c}; \; m = \frac{h\nu}{c^2}. \quad 17(14)...This general picture was first suggested by Einstein... The units are now called photons... [T]he spreading of light by diffraction cannot be permanently concentrated in a small volume like the energy of a material particle. ...The pressure p, exerted by a parallel beam incident normally on a body which completely absorbs it, is...p = \rho_p, \quad ...17(15) where \rho_p is the energy per unit volume of the incident radiation. ...[C]onsider the radiation pressure of a parallel beam of light, incident on an absorbing body... the light is of frequency \nu and... there are N quanta per unit volume. Then...\rho_p = Nh\nu. \quad ...17(18)[A]ll the quanta in a cylinder of volume c [speed of light multiplied by unit area] cubic centimetres are incident upon unit area of the surface in one second, the pressure...p = NcP, \quad ...17(19)where P is the momentum of one photon. Combining...P = \frac{h\nu}{c} = \frac{h}{\lambda}.[Experimental] results... for isotropic radiation are in agreement..."

"Suppose... motion of an electron in the absence of a field of force, is to be investigated... by testing the validity of [no force implies zero acceleration]...\frac{d^2q}{dt^2} = 0, \quad ...18(3)...q ...the position of the particle at time t. The... procedure is to measure the position and momentum of the electron at... time t = t_0... to obtain two "initial conditions" which can be inserted in the solution of 18(3)... then calculate the position and momentum at some later time... and see if the calculation agrees with... observation... Suppose we observe... with light of wavelength \lambda. ...[D]iffraction of the wave sets the limit to the accuracy of a position measurement...\vartriangle q \sim \frac{\lambda}{2sin\theta}, \quad ...18(4)where \vartriangle q is the probable error in... q, and \theta is the semi-angle of the cone of rays accepted by the microscope... [and] \sim means "at least of the order of magnitude of". The experiment of Compton... shows that the interaction... involves an exchange of momentum. We may assume that the momenta... were exactly known before their interaction, but... [those] after the interaction depends on the accuracy [of the] momentum exchanged during the interaction. [T]he photon enters the microscope, and... we know its direction... within an angle 2\theta. Any attempt [to reduce] the effective aperture... increases \vartriangle q. Thus... the momentum of the photon in the plane [in which q is measured] perpendicular to the axis of the microscope... is uncertain by an amount\vartriangle p \sim \frac{2h\nu}{c}sin\theta \quad. ...18(5)The momentum of the particle after the interaction is uncertain by \vartriangle p. Combining... we have\vartriangle p \vartriangle q \sim \frac{\lambda}{2sin\theta} \frac{2h\nu}{c} sin\theta,i.e.,\vartriangle p \vartriangle q \sim h \quad. ...18(6)"

"Une loi de Physique possède une certitude beaucoup moins immédiate et beaucoup plus difficile à apprécier qu'une loi de sens commun; mais elle surpasse cette dernière par la précision minutieuse et détaillée de ses prédictions. ...Cette minutie dans le détail, les lois de la Physique ne la peuvent acquérir qu'en sacrifiant quelque chose de la certitude fixe et absolue des lois de sens commun. Entre la précision et la certitude il y a une sorte de compensation; l'une ne peut croître qu'au détriment de l'autre. A law of physics possesses a certainty much less immediate and much more difficult to estimate than a law of common sense, but it surpasses the latter by the minute and detailed precision of its predictions. ...The laws of physics can acquire this minuteness of detail only by sacrificing something of the fixed and absolute certainty of common-sense laws. There is a sort of balance between precision and certainty; one cannot be increased except to the detriment of the other."

"In September 1973, while I was visiting Moscow, I discussed s with two leading Soviet experts, Yakov Zeldovich and Alexander Starobinsky. They convince me that, according to the uncertainty principle, s should create and emit particles. I believed their arguments on physical grounds but did not like the mathematical way in which they calculated the emission. I therefore set about devising a better mathematical treatment... when I did the calculation I found to my surprise and annoyance, that even nonrotating black holes should create and emit particles at a steady rate. ...[What finally convinced me that the emission was real was that the spectrum of the emitted particles was exactly that which would be emitted by a hot body... at exactly... the correct rate to prevent violations of the second law."

"[T]he laws of physics need not break down at the origin of the universe. The state of the universe and its contents, like ourselves, are completely determined by the laws of physics, up to the limit set by the uncertainty principle. So much for free will."

"Ultimately... one would hope to find a complete, consistent, unified theory that would include all... partial theories as approximations... "the unification of physics." Einstein spent most of his later years unsuccessfully searching... Einstein refused to believe in the reality of quantum mechanics, despite the important role he played in its development. Yet it seems that the uncertainty principle is a fundamental feature of the universe... A successful unified theory must... incorporate this principle."

"It must have been one evening after midnight when I suddenly remembered my conversation with Einstein and particularly his statement, "It is the theory which decides what we can observe." I was immediately convinced that the key to the gate that had been closed for so long must be sought right here. I decided to go on a nocturnal walk through Faelled Park and to think further about the matter. We had always said so glibly that the path of the electron in the cloud chamber could be observed. But perhaps what we really observed was something much less. Perhaps we merely saw a series of discrete and ill-defined spots through which the electron had passed. In fact, all we do see in the cloud chamber are individual water droplets which must certainly be much larger than the electron. The right question should therefore be: Can quantum mechanics represent the fact that an electron finds itself approximately in a given place and that it moves approximately with a given velocity, and can we make these approximations so close that they do not cause experimental difficulties?"

"Instead of Newtonian certainty and determinism, quantum theory answers our questions with probability and statistics. Classical physics told us precisely where Mars was to be found. Quantum theory sends us to the gambling table to locate an electron in an atom. Then there's Heisenberg's uncertainty principle, which places an ultimate limit on our knowledge of the microworld and tells us that we can make no measurement without affecting the result."

"Although Heisenberg began thinking about an electron track in a , he came to realize that the problem of locating an electron is not one of instrumentation... [H]e found that it was never possible... to measure the precise path of an electron: for example, the droplet size is too large in a cloud chamber. Or if you try to "see" an electron with light (or x-rays), then the light photons strike the electron like billiard balls and randomly change the electron's position. ...It was ...a limit in principle on the accuracy ...a theoretical limit—an irreducible degree of uncertainty or Indeterminacy—in measuring the simultaneous position and motion of an electron. ...[T]he two measurements—of position and motion—counterbalance one another. The better you measure the position, the worse you will be able to measure the motion, and vice versa. If you could measure the position (or motion) perfectly, you would know nothing about the motion (position). The uncertainty principle thus seems to reconcile the particle-wave duality. The better you know the position... the more localized... the more it seems to act like a particle. Alternatively, if you know the motion or speed very well... [position] is diffuse, like a wave. ...[A] different point of view on duality but still ...a paradoxical fact of life."

"Heisenberg considered the observation through a microscope of an electron struck by light of an appropriate frequency. The position of the electron could be determined more precisely by increasing the frequency of the light, but the higher the frequency, the larger the jolt to the electron, and, hence, the greater the indeterminacy in the measurement of the electron's velocity. Conversely, the velocity could be determined more precisely by using light of a lower frequency, but the lower the frequency, the greater the indeterminacy in the measurement of the position. On the basis of these considerations, Heisenbery boldly affirmed the acausality of quantum mechanics: For to predict the future, you had to know everything about the present, and according to quantum mechanics, Heisenberg asserted, "We cannot, as a matter of principle, know the present in all its details.""

"Werner Heisenberg not only discovered but proved that in certain subatomic situations neither classical objectivity nor mechanical causality applied: that the act of the physicists observation (more exactly: his attempts at measurement) interfered either with the movement or with the situation of the object—which meant, among other things, a big crack in the fundament of Descartes's and Newton's objectivism and determinism. In other words: the study of the "reality" of matter was inseparable from the interference (and from the mind and purpose) of the scientist."

"... Heisenberg's main work — his wonderful work — on the uncertainty principle ... Some very important people — like Einstein — did not like this. ... His remark was, "I can believe that God governs the world with any set of laws, but I cannot believe that He is playing at dice. ..." ... Einstein's statement — his remark — was really something to the effect: "I cannot imagine any science without causality ...""

"Une hypothèse qui permet de prévoir certains effets qui se reproduisent toujours ressemble absolument à une vérité démontrée. Le système de Newton ne repose guère sur un autre fondement. Si en réalité et de l aveu du. [A hypothesis which permits the prediction of certain effects that always reoccur under certain conditions does, in its way amount to a demonstrable certainty. Even the Newtonian system had no more than such a foundation.]"

"To recap, I had four numbers to add together. If the total came to under 2, then Einstein’s version of quantum reality was correct and the world is deterministic, rather than probabilistic, with quantum entities existing prior to being observed. But if the total came to over 2, then Niels Bohr was right and there is no objective reality out there in the absence of measurement and the subatomic world is ruled by chance and probability. [...] So, sorry Einstein, victory goes to Bohr instead."

"The discomfort that I feel is associated with the fact that the observed perfect quantum correlations seem to demand something like the ‘genetic’ hypothesis [identical twins, carrying with them identical genes]. For me, it is so reasonable to assume that the photons in those experiments carry with them programs, which have been correlated in advance, telling them how to behave. This is so rational that I think that when Einstein saw that, and the others refused to see it, he was the rational man. The other people, although history has justified them, were burying their heads in the sand. I feel that Einstein’s intellectual superiority over Bohr, in this instance, was enormous; a vast gulf between the man who saw clearly what was needed, and the obscurantist. So for me, it is a pity that Einstein’s idea doesn’t work. The reasonable thing just doesn’t work."

"Bohr was inconsistent, unclear, willfully obscure and right. Einstein was consistent, clear, down-to-earth and wrong."

"The EPR paper came out in 1935 and for at least two decades no one paid much attention to it. However, in 1964, the late John Bell published a paper that changed everything. He showed that Einstein’s idea that the results of quantum mechanics could be reproduced by a theory in which Einstein’s notions of realism were included could be tested in the laboratory. Having spent a good deal of time talking to Bell I can tell you that his heart was with Einstein. He often referred to Bohr as an “obscurantist.” But the experiments were carried out by Alain Aspect and others and showed that Einstein was wrong and Bohr was right. I cannot believe that anyone familiar with this would still agree with Einstein."

"His thought experiment with photon and film had not challenged Heisenberg's principle, but now Einstein did turn his attention there. He began looking for an experiment that would allow a more complete collection of data than the Heisenberg team thought possible. If he could find a technique that allowed the simultaneous discovery of position and momentum or time and energy, he would prove the quantum mechanics had indeed not yet brought us to the the secret of the Old One. This effort led the most famous set-pieces of the Einstein "debate" with Bohr over quantum mechanics. Einstein, Bohr, and Ehrenfest would meet in the hotel dining room for breakfast. Einstein would propose a thought experiment. Bohr would think about it. ... During the day's program at Solvay, Heisenberg and Pauli would analyze the experiment that Einstein had proposed. They would find some point where the uncertainty principle fought back, and over dinner, Bohr wold refute the experimental effort while Ehrenfest looked on."

"The mid-twentieth century “Bohr-Einstein debate” about quantum theory is often misinterpreted as a personal clash between wizards. So counter-intuitive are quantum theory’s predictions that, under the leadership of one of its pioneers, Neils Bohr, a myth grew that there is no underlying reality that explains them. Particles get from A to B without passing through the intervening space, where they have insufficient energy to exist; they briefly “borrow” the energy, because we are “uncertain” about what their energy is. Information gets from A to B without anything passing in between – what Einstein called “spooky action at a distance.” And so on....So, while most accounts say that Bohr won the debate, my view is that Einstein, as usual, was seeking an explanation of reality, while his rivals were advocating nonsense. Everett’s interpretation doesn’t make Einstein a demigod. But it does make him right."

"Einstein was not prepared to let us do what, to him, amounted to pulling the ground from under his feet. Later in life, also, when quantum theory had long since become an integral part of modern physics, Einstein was unable to change his attitude—at best, he was prepared to accept the existence of quantum theory as a temporary expedient. "God does not throw dice" was his unshakable principle, one that he would not allow anybody to challenge. To which Bohr could only counter with: "Nor is it our business to prescribe to God how He should run the world.""

"Their dispute went to the fundamental heart of the design of the cosmos. Was there an objective reality that existed whether or not we could ever observe it? Were there laws that restored strict causality to phenomena that seemed inherently random? Was everything in the universe predetermined?"

"Einstein's thinking is always on the ontological level traditional in physics; trying to describe the realities of Nature. Bohr's thinking is always on the epistemological level, describing not reality but only our information about reality."

"The famous debate between Einstein and Bohr began at the Solvay Council in 1927. The debate was about the interpretation of quantum mechanics, but also addressed the fundamental question of what the purpose and aim of a physical theory should be. Their conflicting positions were based on two diametrically opposed philosophical approaches to the fundamental problems of physics. The many books popularising quantum mechanics quite rightly place the emphasis on the problem of interpretation: they discuss the opposing positions of Einstein’s “realism” and the “Copenhagen interpretation” of which Bohr is seen as the leading protagonist."

"We, of course, were sure that on that particular debate Bohr was right and Einstein was wrong."

"The refutation of Einstein’s criticism does not add any new element to the conception of complementarity, but it is of great importance in laying bare a very deep-lying opposition between Bohr’s general philosophical attitude and the still widespread habits of thought belonging to a glorious but irrevocably bygone age in the evolution of science."

"Albert Einstein, who was in many ways the father of quantum mechanics, had a notorious love-hate relation with the subject. His debates with Niels Bohr—Bohr completely accepting of quantum mechanics and Einstein deeply skeptical— are famous in the history of science. It was generally accepted by most physicists that Bohr won and Einstein lost. My own feeling, I think shared by a growing number of physicists, is that this attitude does not do justice to Einstein’s views. Both Bohr and Einstein were subtle men. Einstein tried very hard to show that quantum mechanics was inconsistent; Bohr, however, was always able to counter his arguments. But in his final attack Einstein pointed to something so deep, so counterintuitive, so troubling, and yet so exciting, that at the beginning of the twenty-first century it has returned to fascinate theoretical physicists. Bohr’s only answer to Einstein’s last great discovery—the discovery of entanglement—was to ignore it."

"To this day, many researchers agree with Bohr's pragmatic attitude. The history books say that Bohr has proved Einstein wrong. But others, including myself, suspect that, in the long run, the Einsteinian view might return: that there is something missing in the Copenhagen interpretation. Einstein's original objections could be overturned, but problems still arise if one tries to formulate the quantum mechanics of the entire universe (where measurements can never be repeated), and if one tries to reconcile the laws of quantum mechanics with those of gravitation. But I am running far ahead in my story (I will return to this point in chapter 28). For a correct description of atoms and molecules, quantum mechanics is a perfect theory."

"The other mistake that is widely attributed to Einstein is that he was on the wrong side in his famous debate with Niels Bohr over quantum mechanics, starting at the Solvay Congress of 1927 and continuing into the 1930s. In brief, Bohr had presided over the formulation of a “Copenhagen interpretation” of quantum mechanics, in which it is only possible to calculate the probabilities of the various possible outcomes of experiments. Einstein rejected the notion that the laws of physics could deal with probabilities, famously decreeing that God does not play dice with the cosmos. But history gave its verdict against Einstein—quantum mechanics went on from success to success, leaving Einstein on the sidelines. All this familiar story is true, but it leaves out an irony. Bohr’s version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe."

"In the famous Einstein–Bohr debates, Bohr defended quantum mechanics against Einstein's yearning for a more classical theory; but some of us are coming to feel in defending his hard-won ground he compromised too much. Quantum mechanics should be pushed as hard as possible, to the point where it can describe within itself a recognizable caricature of the world as it is experienced, and thus begin to provide its own self-consisted interpretation — or else there should be some definite change in its equations. As yet this task has not been accomplished."

"Einstein’s great friend and intellectual sparring partner Niels Bohr had a nuanced view of truth. Whereas according to Bohr, the opposite of a simple truth is a falsehood, the opposite of a deep truth is another deep truth. In that spirit, let us introduce the concept of a deep falsehood, whose opposite is likewise a deep falsehood. It seems fitting to conclude this essay with an epigram that, paired with the one we started with, gives a nice example:“Naïveté is doing the same thing over and over, and always expecting the same result.”"

"When the quantum system contains more than one particle, the superposition principle gives rise to the phenomenon of entanglement. It is now not just a particle interfering with itself—it is a system interfering with itself: an entangled system. Amazingly enough, Erwin Schrödinger himself realized that particles or photons produced in a process that links them together will be entangled, and he actually coined the term entanglement, both in his native German and in English. Schrödinger discovered the possibility of entanglement in 1926, when he did his pioneering work on the new quantum mechanics, but he first used the term entanglement in 1935, in his discussion of the Einstein, Podolsky, and Rosen (EPR) paper."

"The two photons are entangled and according to local realism, their polarization planes should become independent... a typical EPR situation. Already in 1948, observations... agreed with quantum mechanics, not with local realism."

"Quantum theory is now discussing instantaneous connections between two entangled quantum objects such as electrons. This phenomenon has been observed in laboratory experiments and scientists believe they have proven it takes place. They’re not talking about faster than the speed of light. Speed has nothing to do with it. The entangled objects somehow communicate instantaneously at a distance. If that is true, distance has no meaning. Light-years have no meaning. Space has no meaning. In a sense, the entangled objects are not even communicating. They are the same thing. At the “quantum level” (and I don’t know what that means), everything may be actually or theoretically linked. All is one. Sun, moon, stars, rain, you, me, everything. All one."

"I'd like to think that the moon still exists, even if I'm not looking at it."

"... you consider my attitude towards to be strange and archaic. ... the theory cannot be reconciled with the idea that physics should represent a reality in time and space free from spooky actions at a distance."

"He [Albert Einstein] didn’t think the spooky action at a distance would be verified, but it was, He thought that was somehow unphysical. He presented this as an example of why quantum mechanics is probably wrong, but in fact it’s right."

"When two systems, of which we know the states by their respective representation, enter into a temporary physical interaction due to known forces between them and when after a time of mutual influence the systems separate again, then they can no longer be described as before, viz., by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics."

"The phenomenon of entanglement is the essential fact of quantum mechanics, the fact that makes it so different from classical physics. It brings into question our entire understanding about what is real in the physical world. Our ordinary intuition about physical systems is that if we know everything about a system, that is, everything that can in principle be known, then we know everything about its parts. If we have complete knowledge of the condition of an automobile, then we know everything about its wheels, its engine, its transmission, right down to the screws that hold the upholstery in place. It would not make sense for a mechanic to say, “I know everything about your car but unfortunately I can’t tell you anything about any of its parts.” But that’s exactly what Einstein explained to Bohr — in quantum mechanics, one can know everything about a system and nothing about its individual parts — but Bohr failed to appreciate this fact. I might add that generations of quantum textbooks blithely ignored it."

"There is a troubling weirdness about quantum mechanics. Perhaps its weirdest feature is entanglement, the need to describe even systems that extend over macroscopic distances in ways that are inconsistent with classical ideas."

"In contemplating the papers Einstein wrote in 1905, I often find myself wondering which of them is the most beautiful. It is a little like asking which of Beethoven’s symphonies is the most beautiful. My favorite, after years of studying them, is Einstein’s paper on the blackbody radiation. [...] Part of being a great scientist is to know—have an instinct for—the questions not to ask. Einstein did not try to derive the Wien law. He simply accepted it as an empirical fact and asked what it meant. By a virtuoso bit of reasoning involving statistical mechanics (of which he was a master, having independently invented the subject over a three-year period beginning in 1902), he was able to show that the statistical mechanics of the radiation in the cavity was mathematically the same as that of a dilute gas of particles. As far as Einstein was concerned, this meant that this radiation was a dilute gas of particles—light quanta. But, and this was also characteristic, he took the argument a step further. He realized that if the energetic light quanta were to bombard, say, a metal surface, they would give up their energies in lump sums and thereby liberate electrons from the surface in a predictable way, something that is called the photoelectric effect. [...] In the first place, not many physicists were even interested in the subject of blackbody radiation for at least another decade. Kuhn has done a study that shows that until 1914 less than twenty authors a year published papers on the subject; in most years there were less than ten. Planck, who was interested, decided that Einstein’s paper was simply wrong."

"Bohr’s principle of complementarity – the heart of the Copenhagen philosophy – implies that quantum phenomena can only be described by pairs of partial, mutually exclusive, or ‘complementary’ perspectives. Though simultaneously inapplicable, both perspectives are necessary for the exhaustive description of phenomena. Bohr aspired to generalize complementarity into all fields of knowledge, maintaining that new epistemological insights are obtained by adjoining contrary, seemingly incompatible, viewpoints. [...] The value of Bohr’s philosophy for the advancement of physics is controversial. His followers consider complementarity a profound insight into the nature of the quantum realm. Others consider complementarity an illuminating but superfluous addendum to quantum theory. More severe is the opinion that Bohr’s philosophy is an obscure ‘web of words’ and mute on crucial foundational issues."

"Unfortunately, people writing about quantum mechanics often use the phrase "collapse of the wave-function" to describe what happens when an object is observed. This phrase gives a misleading idea that the wave-function itself is a physical object. A physical object can collapse when it bumps into an obstacle. But a wave-function cannot be a physical object. A wave-function is a description of a probability, and a probability is a statement of ignorance. Ignorance is not a physical object, and neither is a wave-function. When new knowledge displaces ignorance, the wave-function does not collapse; it merely becomes irrelevant."

"If the packet is to be reduced, the interaction must have produced knowledge in the brain of the observer. If the observer forgets the result of his observation, or loses his notebook, the packet is not reduced. We are again led to emphasize the fact that the wave function of a pure-state assemblage is merely a mathematical tool for computing from all previous observations what the relative probabilities are for different results when we make our next observation."

"To Einstein the collapse postulate was an even more pernicious retreat from realism than that described above: it implied that physical quantities usually have no values until they are observed, and therefore that the observer must be intrinsically involved in the physics being observed. This suggests that there might be no real world in the absence of an observer!"

"If collapse actually worked the way its adherents say it does, it would be:1. The only non-linear evolution in all of quantum mechanics. 2. The only non-unitary evolution in all of quantum mechanics. 3. The only non-differentiable (in fact, discontinuous) phenomenon in all of quantum mechanics. 4. The only phenomenon in all of quantum mechanics that is non-local in the configuration space. 5. The only phenomenon in all of physics that violates CPT symmetry. 6. The only phenomenon in all of physics that violates Liouville's Theorem (has a many-to-one mapping from initial conditions to outcomes). 7. The only phenomenon in all of physics that is acausal / non-deterministic / inherently random. 8. The only phenomenon in all of physics that is non-local in spacetime and propagates an influence faster than light.WHAT DOES THE GOD-DAMNED COLLAPSE POSTULATE HAVE TO DO FOR PHYSICISTS TO REJECT IT? KILL A GOD-DAMNED PUPPY?"

"Quantum computational approaches improve upon classical methods for a number of specialized tasks. The extent of quantum computing’s applicability is still being determined. It does not provide efficient solutions to all problems; neither does it provide a universal way of circumventing the slowing of Moore’s law. Strong limitations on the power of quantum computation are known; for many problems, it has been proven that quantum computation provides no significant advantage over classical computation. Grover’s algorithm, the other major algorithm of the mid- 1990s, provides a small speedup for unstructured search algorithms. But it is also known that this small speedup is the most that quantum algorithms can attain. Grover’s search algorithm applies to unstructured search. For other search problems, such as searching an ordered list, quantum computation provides no significant advantage over classical computation. Simulation of quantum systems is the other significant application of quantum computation known in the mid-1990s. Of interest in its own right, the simulation of increasingly larger quantum systems may provide a bootstrap that will ultimately lead to the building of a scalable quantum computer. After Grover’s algorithm, there was a hiatus of more than five years before a significantly new algorithm was discovered. During that time, other areas of quantum information processing, such as quantum error correction, advanced significantly. In the early 2000s, several new algorithms were discovered. Like Shor’s algorithm, these algorithms solve specific problems with narrow, if important, applications. Novel approaches to constructing quantum algorithms also developed. Investigations of quantum simulation from a quantum-information-processing point of view have led to improved classical techniques for simulating quantum systems, as well as novel quantum approaches. Similarly, the quantum-information-processing point of view has led to novel insights into classical computing, including new classical algorithms. Furthermore, alternatives to the standard circuit model of quantum computation have been developed that have led to new quantum algorithms, breakthroughs in building quantum computers, new approaches to robustness, and significant insights into the key elements of quantum computation."

"The problem of measurement and the observer is the problem of where the measurement begins and ends, and where the observer begins and ends. Consider my spectacles, for example: if I take them off now, how far away must I put them before they are part of the object rather than part of the observer? There are problems like this all the way from the retina through the optic nerve to the brain and so on. I think, that—when you analyze this language that the physicists have fallen into, that physics is about the results of observations—you find that on analysis it evaporates, and nothing very clear is being said."

"... quantum mechanics, as it is enshrined in s, seems to require separate rules for how quantum objects behave when we’re not looking at them, and how they behave when they are being observed. When we’re not looking, they exist in “s” of different possibilities, such as being at any one of various locations in space. But when we look, they suddenly snap into just a single location, and that’s where we see them. We can’t predict exactly what that location will be; the best we can do is calculate the probability of different outcomes. The whole thing is preposterous. Why are observations special? What counts as an “observation,” anyway? When exactly does it happen? Does it need to be performed by a person? Is consciousness somehow involved in the basic rules of reality? Together these questions are known as the “measurement problem” of quantum theory."

"We show that Shor's algorithm, the most complex quantum algorithm known to date, is realizable in a way where, yes, all you have to do is go in the lab, apply more technology, and you should be able to make a bigger quantum computer … It might still cost an enormous amount of money to build — you won’t be building a quantum computer and putting it on your desktop anytime soon — but now it’s much more an engineering effort, and not a basic physics question."

"Quantum computation is … nothing less than a distinctly new way of harnessing nature … It will be the first technology that allows useful tasks to be performed in collaboration between parallel universes, and then sharing the results."

"Turning to quantum mechanics, we know immediately that here we get only the ability, apparently, to predict probabilities. Might I say immediately, so that you know where I really intend to go, that we always have had (secret, secret, close the doors!) we always have had a great deal of difficulty in understanding the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me. Okay, I still get nervous with it. And therefore, some of the younger students ... you know how it always is, every new idea, it takes a generation or two until it becomes obvious that there's no real problem. It has not yet become obvious to me that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem. So that's why I like to investigate things. Can I learn anything from asking this question about computers—about this may or may not be mystery as to what the world view of quantum mechanics is? So I know that quantum mechanics seem to involve probability—and I therefore want to talk about simulating probability."

"Back in the 1940s, researchers were just discovering how to use vacuum tubes as simple switches. … These switches could then form logic gates, which could be linked together to form the first logic circuits. That’s where we’re at now with quantum processors. We have verified that all the components work. The next step is to engineer the smallest, yet most interesting circuit possible."

"In less than ten years quantum computers will begin to outperform everyday computers, leading to breakthroughs in artificial intelligence, the discovery of new pharmaceuticals and beyond. The very fast computing power given by quantum computers has the potential to disrupt traditional businesses and challenge our cyber-security. Businesses need to be ready for a quantum future because it's coming."

"This book tells the story of the researches that are traditionally lumped together under the label "radiation theory" and revolving, loosely speaking around the familiar heat-and-light exchange (hot bodies emit light and radiate; the absorption of light, especially sunlight, is warming)."

"These days it is common knowledge that short waves are more powerful than long ones, as the very short ones, known as x-rays, damage living tissues. It took half-a-century to learn this fact: it was one of the great discoveries of young Albert Einstein of 1905. When he announced it leading researchers found it most incredible..."

"The development of quantum mechanics in the beginning of the twentieth century was a unique intellectual adventure, which obliged scientists and philosophers to change radically the concepts they used to describe the world. After these heroic efforts, it became possible to understand the stability of matter, the mechanical and thermal properties of materials, the interaction of radiation and matter, and many other properties of the microscopic world that had been impossible to understand with classical physics. A few decades later, that conceptual revolution enabled a technological revolution, at the root of our information-based society. It is indeed with the quantum mechanical understanding of the structure and properties of matter that physicists and engineers were able to invent and develop the transistor and the laser—two key technologies that permit the high-bandwidth circulation of information, as well as many other scientific and commercial applications."

"The realization of the importance of entanglement and the clarification of the quantum description of single objects have been at the root of a second quantum revolution, and the John Bell was its prophet."

"The blackbody oven embodied an... instance of radiation interacting with matter. ...Planck first... derived an empirical equation to fit the data. ...His more ambitious aim now was to find a theoretical entropy-energy connection applicable to the blackbody problem. ...Ludwig Boltzmann interpreted the second law of thermodynamics as a "probability law." If the relative probability or disorder for the state of the system was W, he concluded, then the entropy S of the system in that state was proportional to the logarithm of W,S ∝ lnW ...Plank applied this to the blackbody problem by writingS = k lnW (1)for the total entropy of the vibrating molecules... "resonators"—in the blackbody oven's walls... k is now called Boltzmann's constant. ...Boltzmann's theory taught the lesson that conceivably—but against astronomically unfavorable odds—any macroscopic process can reverse... contradicting the second law of thermodynamics. Boltzmann's conclusions seemed fantastic to Planck, but by 1900 he was becoming increasingly desparate, even reckless... The counting procedure Planck used to calculate the disorder W... was borrowed from... Boltzmann's theoretical techniques. He considered... that the total energy of the resonators was made up of small indivisible "elements," each one of magnitude ε. It was then possible to evaluate W as a count of the number of ways a certain number of energy elements could be distributed to a certain number of resonators... His argument would not succeed unless he assumed that the energy ε of the elements was proportional to the frequency with which the resonators vibrated, ε ∝ v, or ε = hv, with h the proportionality constant."

"The second quantum revolution... was the term coined by... Alain Aspect to describe the changes in physics, the beginnings of which date back to the 1960s. ...he brought together two different threads. The first one embraced the emergence of the awareness of the importance of... entanglement. ...It started a conceptual revolution, including the perspective of building quantum computers... The second thread derives from physicists' ability to isolate, control, and observe single quantum systems such as electrons, neutrons and atoms. Finally these threads merged into a new field of research entitled quantum information. In Aspect's formulation... he posited two quantum revolutions taking place in the twentieth century. The first one, in the first half of the century, created the scientific theory that describes the behavior of atoms, radiation, and their interactions. The second one occurred in the second half and is still evolving... intellectual aspects... arose from the renewal of research on the foundations of quantum physics."

"Intuitively, Planck knew that his work was as important as Newton's in paving a new physics, as he privately confided to his son in 1901. Because of his own conservative beliefs, Planck felt stymied by the revolutionary nature of his own ideas... In truth, Planck was only underestimating the importance of his work, if anything. In retrospect... it started a revolution in science—the quantum revolution—compared to which the Copernican revolution pales."

"The development of the quantum ideas themselves occurred in discrete jumps, quantum leaps of the creative insights of a few people."

"The start of the revolution that produced the old quantum theory is moved from the end of 1900 to 1906... The preceding crisis...resulted from the difficulties in reconciling Planck's derivation with the tenets of classical physics. Planck's change in vocabulary—from "resonator" to "oscillator" and from "element" to "quantum"—is the central symptom of incommensurability. It signals the changed meaning of the quantity hv from a mental subdivision of the energy continuum to a physically separable atom of energy. That my critics continue to apply the term "energy quantum" to pre-1906 papers and lectures in which Planck consistently used "energy element" reveals something of the difficulty of reversing the gestalt switch that took place during that year and those which followed. ...Boltzmann's probabilistic derivation of the entropy of a gas ...illustrates the problem to which the concept of paradigm was a response. The derivation was not reduced to rules but instead served as a model to be applied by means of analogy. As a result, when its application was transferred from gases to radiation, Planck and Lorentz could invent different analogies with which to effect change."

"Quantum mechanics is revolutionary because it overturned scientific concepts that seemed to be so obvious and so well confirmed by experience that they were beyond reasonable question, but it is an incomplete because we still do not know precisely where quantum mechanics will lead us—nor even why it must be true!"

"In the brief period between 1900 and 1935 there occurred one of the most astonishing outbursts of scientific creativity in all of history. ...no other historical era has crammed so much scientific creativity, so many discoveries of new ideas and techniques, into so few years. Although a few outstanding individuals dominate... they were assisted in their work by an army of talented scientists and technicians."

"In the years 1925 and 1926 modern quantum theory came into being fully fledged. These anni mirabiles remain an episode of great significance in the folk memory of the theoretical physics community... Werner Heisenberg had been struggling to understand the details of atomic spectra. ...Heisenberg's discovery came to be known as matrix mechanics. ...In 1925 matrices were... mathematically exotic to the average theoretical physicist... Prince Louis de Broglie... made the bold suggestion that if undulating light also showed particle-like properties... one should expect particles such as electrons to manifest wavelike properties... by generalizing the Planck formula. The latter had made the particlelike property of energy proportional to the wavelike property of frequency. De Broglie suggested that... particlelike... momentum... should analogously be related to... wavelength, with Planck's universal constant again... These equivalences provided... for translating from particles to waves, and vice versa. In 1924, de Broglie laid out these ideas in his doctoral thesis. ...To attain a full dynamical theory, a further generalization... allowed the incorporation of interactions... This is the problem that Schrödinger succeeded in solving. Early in 1926 he published the famous equation... led to its discovery by exploiting an analogy drawn from optics."

"The quantum revolution describes our deepest insight, so far, into the physical structure of nature. It is comparable only with the Copernican revolution, switching from a finally oriented anthropomorphic description of physical phenomena to one using general laws with initial or boundary conditions, connected with the names Kepler, Galileo, and Newton, or with the change from tangible mass points as basic structures to Faraday's and Maxwell's field concepts and, shortly before quantum theory, with the relativization of space and time by the lonely genius Einstein."

"Quantum mechanics, created early this century in response to certain experimental facts which were inexplicable according to previously held ideas (...'classical physics'), caused three great revolutions. In the first place it opened up a completely new set range of phenomena to which the methods of physics could be applied. ...The second revolution was the apparent breakdown of determinism, which had always been an unquestioned ingredient and an inescapable prediction of classical physics. ...The outcome of any particular experiment is not, even in principle, predictable, but is chosen at random from a set of possibilities; all that can be predicted is the probability of particular results when the experiment is repeated many times. ...even if we had complete knowledge of the initial state... The third revolution ...challenged the basic belief, implicit in all science and indeed in almost the whole of human thinking, that there exists an objective reality ...that does not depend for its existence on its being observed."

"Above all, the ominous clouds of those phenomena that we are with varying success seeking to explain by means of the quantum of action, are throwing their shadows over the sphere of physical knowledge, threatening no one knows what new revolution."

"A few years after writing the preface of that book, Popper fell into an opposite, and equally serious error about an "EPR situation," On this occasion, contrary to the preceding one, there is an over- rather than an underevaluation of the EPR analysis. On p. 27 of the same book, Popper proposes an experiment that constitutes a variant of the EPR argument, asserting that if the the Copenhagen interpretation is correct, the experiment just analyzed would allow for sending signals faster than the speed of light. This work is one of a lengthy series we will discuss later, in which it is maintained that quantum formalism would permit us to use the reduction of the wave packet to violate one of the postulates at the basis of relativity (i.e., that the speed of light cannot be exceeded). Now, despite the peculiarity of the situation addressed by EPR, this conclusion is fundamentally erroneous and arise from an incorrect use of quantum formalism. I recall a spirited discussion I once had with Popper at the International Center for Theoretical Physics at Miramare in 1983. Professor Abdus Salam informed me that on the occasion of Popper's visit (for delivering a lecture on the foundations of quantum mechanics), he would be very pleased if the Center would have on hand some competent person in the field, and asked me to take part in the discussion. I knew Popper's work well and told Professor Salam that my intervention could be critical. Salam's reply was simple: "I have full confidence in you, and if you think you are right, you should explain your position without any fear." Popper presented his thought experiment (a variant of the EPR argument), which, according to him, left us with only two alternatives: either the orthodox interpretation was correct, and it would then be possible to send signals faster than the speed of light, or there would not be any action at a distance and the experiment would constitute a falsification of quantum theory. At the end of the conference I explained to him in simple, but mathematically precise terms, the reasons why his point of departure was erroneous: he had not correctly applied the rules of the theory and in fact, the impossibility of sending superluminal signals would confirm the theory rather than falsify it—the exact opposite of what he maintained. At the end of my intervention he only said that he could not answer my objection since he did not have a mastery of the mathematics of the formalism, but was still convinced that the theory implied the possibility of superluminal signals. This strange, and, as we shall see, fundamentally erroneous idea has been supported by various researchers in various scientific works, and published in prestigious journals."

"If one translates this result into terms of particles, only one interpretation is possible: \Phi_{n,m} (\alpha, \beta, \gamma)[* addition in proof: More careful consideration shows that the probability is proportional to the square of the quantity \Phi_{n,m}] gives the probability for the electron, arriving from the z-direction, …"

"One of the most profound and mysterious principles in all of physics is the Born Rule, named after Max Born. In quantum mechanics, particles don’t have classical properties like “position” or “momentum”; rather, there is a wave function that assigns a (complex) number, called the “amplitude,” to each possible measurement outcome. The Born Rule is then very simple: it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. (The wave function is just the set of all the amplitudes.) … The Born Rule is certainly correct, as far as all of our experimental efforts have been able to discern. But why? … The status of the Born Rule depends greatly on one’s preferred formulation of quantum mechanics. … Everett, … is claiming that all the weird stuff about “measurement” and “wave function collapse” in the conventional way of thinking about quantum mechanics isn’t something we need to add on; it comes out automatically from the formalism. The trickiest thing to extract from the formalism is the Born Rule."

"... the Born rule just comes out of the blue without being derived from the time-dependent Schrödinger equation — which is supposed to account for everything."

"And Po'mi has just asked: "Why should the subjective probability of finding ourselves in a side of the split world, be exactly proportional to the square of the thickness of that side?"When the initial hubbub quiets down, the respected Nharglane of Ebbore asks: "Po'mi, what is it exactly that you found?" "Using instruments of the type we are all familiar with," Po'mi explains, "I determined when a splitting of the world was about to take place, and in what proportions the world would split. I found that I could not predict exactly which world I would find myself in—" "Of course not," interrupts De'da, "you found yourself in both worlds, every time -" "—but I could predict probabilistically which world I would find myself in. Out of all the times the world was about to split 2:1, into a side of two-thirds width and a side of one-third width, I found myself on the thicker side around 4 times out of 5, and on the thinner side around 1 time out of 5. When the world was about to split 3:1, I found myself on the thicker side 9 times out of 10, and on the thinner side 1 time out of 10.""

"One serious mystery of decoherence is where the Born probabilities come from, or even what they are probabilities of. What does the integral over the squared modulus of the amplitude density have to do with anything? … So what could it mean, to associate a "subjective probability" with a component of one factor of a combined amplitude distribution that happens to factorize? … But what does the integral over squared moduli have to do with anything? On a straight reading of the data, you would always find yourself in both blobs, every time. How can you find yourself in one blob with greater probability? What are the Born probabilities, probabilities of? Here's the map—where's the territory? I don't know. It's an open problem. This problem is even worse than it looks, because the squared-modulus business is the only non-linear rule in all of quantum mechanics."

"So, what is quantum mechanics? Even though it was discovered by physicists, it’s not a physical theory in the same sense as electromagnetism or general relativity. In the usual “hierarchy of sciences” – with biology at the top, then chemistry, then physics, then math – quantum mechanics sits at a level between math and physics that I don’t know a good name for. Basically, quantum mechanics is the operating system that other physical theories run on as application software (with the exception of general relativity, which hasn’t yet been successfully ported to this particular OS). There’s even a word for taking a physical theory and porting it to this OS: “to quantize.” But if quantum mechanics isn’t physics in the usual sense – if it’s not about matter, or energy, or waves, or particles – then what is it about? From my perspective, it’s about information and probabilities and observables, and how they relate to each other."

"Information? Whose information? Information about what?"

"I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles."

"In my view the most fundamental statement of quantum mechanics is that the wavefunction, or more generally the density matrix, represents our knowledge of the system we are trying to describe. I shall return later to the question "whose knowledge?". It is well known that we have to use a wavefunction if we have a "pure state" i.e. if our knowledge of the system is complete, in the sense that any further knowledge is barred by the uncertainty principle. Failing such complete knowledge we must use a density matrix, which therefore contains both quantum and classical ignorance. The wavefunction is a special case of a density matrix, and I shall here talk about "density matrix" when I mean "wavefunction or density matrix"."

"It from bit. Otherwise put, every it—every particle, every field of force, even the spacetime continuum itself—derives its function, its meaning, its very existence entirely—even if in some contexts indirectly—from the apparatus-elicited answers to yes or no questions, binary choices, bits. It from bit symbolizes the idea that every item of the physical world has at bottom—at a very deep bottom, in most instances—an immaterial source and explanation; that what we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and this is a participatory universe."

"The light-quantum has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely, the zero state, in which its momentum and therefore also its energy, are zero. When a light-quantum is absorbed it can be considered to jump into this zero state, and when one is emitted it can be considered to jump from the zero state to one in which it is physically in evidence, so that it appears to have been created. Since there is no limit to the number of light-quanta that may be created in this way, we must suppose that there are an infinite number of light quanta in the zero state..."

"One hopes will soon demonstrate the incorrectness of the hypothesis of zero-point energy, the theoretical untenability of which became glaringly obvious to me soon after the publication of the paper I coauthored with Mr. Stern."

"Zero-point energy is now dead as a doornail."

"In his Theorie der Wärmestrahlung, Planck emphasized that the existence of a zero-point energy was completely foreign to classical physics. However, it seemed to be a ghost-like entity which it was difficult to connect to experiments."

"I fear that your hatred of the zero-point energy extends to the electrodynamic emission hypothesis that I introduced and that leads to it. But what’s to be done? For my part, I hate discontinuity of energy even more than discontinuity of emission."

"We here face a fundamental problem of outstanding importance. Its solution may still require a radical change in our theories beyond our present imagining."

"From quantum theory there follows the existence of so called zero-point oscillations; for example each oscillator in its lowest is not completely at rest but always is moving about its equilibrium position. Therefore electromagnetic oscillations also can never cease completely. Thus the quantum nature of the electromagnetic field has as its consequence zero point oscillations of the field strength in the lowest energy state, in which there are no light quanta in space... The zero point oscillations act on an electron in the same way as ordinary electrical oscillations do. They can change the eigenstate of the electron, but only in a transition to a state with the lowest energy, since empty space can only take away energy, and not give it up. In this way spontaneous radiation arises as a consequence of the existence of these unique field strengths corresponding to zero point oscillations. Thus spontaneous radiation is induced radiation of light quanta produced by zero point oscillations of empty space."

"We know what kind of a dance to do experimentally to measure this number very accurately, but we don't what kind of a dance to do on a computer to make this number come out—without putting it in secretly!"

"... an understanding of the numerical value of the fine structure constant may emerge ... charge might be an emergent property generated by a simple interaction mechanism between point-like particles and the electromagnetic vacuum, similar to the process that generates the Lamb shift."

"The theoretical determination of the fine structure constant is certainly the most important of the unsolved problems of modern physics. To reach it, we shall, presumably, have to pay with further revolutionary changes of the fundamental concepts of physics with a still farther digression from the concepts of the classical theories."

"The title of my talk may seem a bit ambitious, but please note the plural “constants”. To calculate the fine structure constant, 1/137, we would need a realistic model of just about everything, and this we do not have. In this talk I want to return to the old question of what it is that determines gauge couplings in general, and try to prepare the ground for a future realistic calculation."

"The hierarchy problem is somewhat unique. ... It's one of a trifecta of problems in the Standard Model that don't come from incontrovertible evidence ... Those three problems are ... the strong CP problem ... the cosmological constant problem ... and the hierarchy problem. ... These .. are ... problems where it's a conflict between our expectations for the size of parameters in quantum field theory and what we see."

"The hierarchy problem is hard to explain. ... Basically, the problem is that there are two main energy (or mass) scales in nature, but that situation shouldn’t be stable. One is the Planck scale, which is defined via fundamental constants: the speed of light, c; Planck’s quantum size, h; and Newton’s gravitational force strength, G. The associated energy scale is about 1019 GeV. The other is the electroweak scale, which is set by the masses of the Higgs and the W and Z bosons, at about 102 GeV. (Protons and atoms have smaller scales, but we understand how to derive those.) It is a conceptual problem, not a conflict with observations."

"Following 't Hooft, we can formulate a technical definition of naturalness: The smallness of a dimensionless parameter η would be considered natural only if a symmetry emerges in the limit η → 0. Thus, fermion masses could be naturally small, since, as you will recall from chapter II.1, a chiral symmetry emerges when a fermion mass is set equal to zero. On the other hand, no particular symmetry emerges when we set either the bare or renormalized mass of a scalar field equal to zero. This represents the essence of the hierarchy problem."

"The discipline of collider physics involves going from the direct collider observables to the underlying lagrangian of the theory. One of the simplest questions one can ask is how to recognize the presence of new particles. In colliders the answer to this question is simple, one collects groups of particles (pairs, for example) and one plots the invariant mass. If a bump is seen in this distribution, one says that there is a new particle. One can also look at the angular distribution of the particles and read off the spin of the new particle."

"Very high energy collisions occur naturally in cosmic ray interactions; they also occurred in the early moments of our universe according to big-bang cosmology. Both these sources provide useful information but they cannot compare with systematic experimentations in accelerator laboratories when this is possible."

"Stephen Hawking and I wrote an essay about future colliders that is relevant to both the CEPC and the FCC. We were encouraged by others to chime in because of discussions that arose in China about physics and economic and cultural issues surrounding building a future collider. The theoretical arguments for building a larger collider with several times the energy of the LHC are thus very strong, particularly in regard to solving the hierarchy problem. What would happen without data? Someone may get or even already have the solution, but no one will be convinced. With data pointing to the solution, we may be able to move on and obtain consensus about a comprehensive theory that incorporates the standard models of particle physics and cosmology and a quantum theory of general relativity, giving us a profound understanding of our universe."

"There is virtually no chance that we will be able to do experiments involving processes at particle energies like 1016 GeV. With current technology the diameter of an accelerator is proportional to the energy given to the accelerated particles. To accelerate particles to an energy of 1016 GeV would require an accelerator a few light-years across."

"Baryon-number-violating processes, including proton decay, and the existence of superpartners are dramatic, make-or-break predictions ... Either would open new worlds of phenomena to investigation. According to our best estimates, neither proton decay nor superpartners lie beyond the reach of a heroic search. They should be found, well within 100 years."

"It is commonly believed that grand unified theories (GUTs) predict proton decay. This is because the exchange of extra GUT gauge bosons gives rise to dimension 6 proton decay operators. We show that there exists a class of GUTs in which these operators are absent. Many string and supergravity models in the literature belong to this class."

"I expect that sooner or later we will be seeing another departure from the renormalizable Standard Model in the discovery of proton decay, or some other example of baryon nonconservation."

"Besides non-zero neutrino masses, the other classic experimental implication of unification is proton instability, with a very long but perhaps not inaccessible lifetime. That prediction has not yet been verified, despite heroic efforts. The existing limits put significant pressure on the framework. Reading it optimistically: There is an excellent chance that further efforts along this line would be rewarded."

"Proton decays into a positron and neutral pion, p → e+π0, are a dominant decay mode in many GUT models. It also has a very clean experimental signature in a water Cherenkov detector with full reconstruction of the event. After decades of search, the sensitivity is still improving with advancement of detector technology and analysis technique. One of examples for such a technique is the background suppression with the neutron tagging. In the proton decay events, the probability of neutron emission is rather small, while in the atmospheric neutrino events, which is the dominant background of proton decay searches, often neutrons are produced. Thus, neutron tagging can provide an additional handle to suppress the background for the proton decay search and improve the sensitivity."

"Masashi Yokoyama and Proto-collaboration:"

"What we have learnt from this chapter is that we cannot have a direct evidence of, i.e. directly measure, a quantum state of a single system. Our experience is only connected with the experimental values of observables, and any time we measure an observable we can only have a partial experience of a system under a certain perspective but we can never have a complete experience that would be represented by an observation of the state vector, which is – in a quantum-mechanical sense – a complete description of the system. In other words, the quantum state is not an observable in the classical sense. However, since this feature of the quantum state is not due to subjective ignorance but rather to an intrinsic characteristic of the microscopic world, there are no definitive reasons to deny the reality of a quantum state."

"By now the reader will have realized that measurement in quantum physics is fundamentally different from that in classical physics. In classical physics, a measurement reveals a pre-existing property of the physical system that is tested. If a car is driving at 180 km h−1 on the highway, the measurement of its speed by radar determines a property that exists prior to the measurement, which gives the police the legitimacy to give a ticket to the driver. On the contrary, the measurement of the Sx component of a spin-1/2 particle in the state |+〉 does not reveal a value of Sx existing before the measurement. The spread in the results of measuring Sx in this case is sometimes attributed to “uncontrollable perturbation of the spin due to the measurement process,” but the value of Sx does not exist before the measurement, and that which does not exist cannot be perturbed."

"When we measure a real dynamical variable ξ, the disturbance involved in the act of measurement causes a jump in the state of the dynamical system. From physical continuity, if we make a second measurement of the same dynamical variable ξ immediately after the first, the result of the second measurement must be the same as that of the first. Thus after the first measurement has been made, there is no indeterminacy in the result of the second. Hence, after the first measurement has been made, the system is in an eigenstate of the dynamical variable ξ, the eigenvalue it belongs to being equal to the result of the first measurement. This conclusion must still hold if the second measurement is not actually made. In this way we see that a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement. We can infer that, with the dynamical system in any state, any result of a measurement of a real dynamical variable is one of its eigenvalues. Conversely, every eigenvalue is a possible result of a measurement of the dynamical variable for some state of the system, since it is certainly the result if the state is an eigenstate belonging to this eigenvalue. This gives us the physical significance of eigenvalues. The set of eigenvalues of a real dynamical variable are just the possible results of measurements of that dynamical variable and the calculation of eigenvalues is for this reason an important problem."

"Hence coherence survives to the extent to which an experiment fails to distinguish between different eigenvalues of the observable being measured, and distinct final states of the object display interference if the final states of the apparatus overlap. […] When the apparatus ends up in one member of a set of orthogonal states, the experiment unambiguously determines whether or not the object is then in a particular eigenstate of the observable of interest, but the result becomes increasingly ambiguous with increasing overlap between the states of the apparatus; furthermore, states of the object with distinct eigenvalues interfere with a visibility that is a measure of the ambiguity with which the states assumed by the apparatus determine the states of the object."

"We see that the measuring process in quantum mechanics has a "two- faced" character: it plays different parts with respect to the past and future of the electron. With respect to the past, it "verifies" the probabilities of the various possible results predicted from the state brought about by the previous measurement. With respect to the future, it brings about a new state. Thus the very nature of the process of measurement involves a far-reaching principle of irreversibility."

"As we apply the results obtained in this section, we should remember that common terms like "measurement" and "information" are being used here with a specific technical meaning. In particular, this is not the place for a detailed analysis of real experimental measurements and their relation to the theoretical framework. We merely note that, in the information theoretic view of quantum mechanics, the probabilities and the related density operators and entropies, which are employed to assess the properties of quantum states and the outcomes of measurement, provide a coherent and consistent basis for understanding and interpreting the theory."

"We now consider measurements of A and B when they are compatible observables. Suppose we measure A first and obtain result a'. Subsequently, we may measure B and get result b'. Finally we measure A again. It follows from our measurement formalism that the third measurement always gives a' with certainty; that is, the second (B) measurement does not destroy the previous information obtained in the first (A) measurement. This is rather obvious when the eigenvalues of A are nondegenerate:|\alpha \rangle \xrightarrow{\text{A measurement}} |a', b' \rangle \xrightarrow{\text{B measurement}} |a', b' \rangle \xrightarrow{\text{A measurement}} |a', b' \rangle."

"This strange fact—that the system evolves one way between measurements and another way during a measurement—has been a source of contention and confusion for decades. It raises a question: Shouldn’t the act of measurement itself be described by the laws of quantum mechanics? The answer is yes. The laws of quantum mechanics are not suspended during measurement. However, to examine the measurement process itself as a quantum mechanical evolution, we must consider the entire experimental setup, including the apparatus, as part of a single quantum system."

"A theory of superconductivity is presented, based on the fact that the interaction between electrons resulting from virtual exchange of phonons is attractive when the energy difference between the electrons states involved is less than the phonon energy, ℏω. It is favorable to form a superconducting phase when this attractive interaction dominates the repulsive screened Coulomb interaction. The normal phase is described by the Bloch individual-particle model. The ground state of a superconductor, formed from a linear combination of normal state configurations in which electrons are virtually excited in pairs of opposite spin and momentum, is lower in energy than the normal state by amount proportional to an average (ℏω)2, consistent with the isotope effect. A mutually orthogonal set of excited states in one-to-one correspondence with those of the normal phase is obtained by specifying occupation of certain Bloch states and by using the rest to form a linear combination of virtual pair configurations. The theory yields a second-order phase transition and a Meissner effect in the form suggested by Pippard. Calculated values of specific heats and penetration depths and their temperature variation are in good agreement with experiment."

"Identifying the pairing mediators in the Fe-based superconductors is a necessary but far from sufficient step toward a full understanding of the materials’ superconductivity. Electrons in crystals are confined to bands shaped by the lattice’s constituent atoms and crystal structure. In the superconducting state, which is an intrinsically many-body state, the electrons must also obey Hund’s and other quantum rules. Those restrictions on electrons’ freedom of movement and association dictate perhaps the most eagerly sought characteristic of a new superconductor: its pairing symmetry."

"As you know, very many metals become superconducting below a certain temperature...—the temperature is different for different metals. When you reduce the temperature sufficiently the metals conduct electricity without any resistance. This phenomenon has been observed for a very large number of metals but not for all, and the theory of this phenomenon has caused a great deal of difficulty. It took a very long time to understand what was going on inside of superconductors, and I will only describe enough of it for our present purposes. It turns out that due to the interactions of the electrons with the vibrations of the atoms in the lattice, there is a small net effective attraction between the electrons. The result is that the electrons form together, if I may speak very qualitatively and crudely, bound pairs."

"The high-temperature copper-oxide superconductors, which offer resistance-free current flow at temperatures extending well above 100 K, are formed by doping certain copper-oxide compounds or by adding excess oxygen to them. The parent compounds—all antiferromagnetic insulators—couldn't be more different from their superconducting offspring: Magnetism and superconductivity are generally antithetical."

"I think that the single most important thing accomplished by the theory of John Bardeen, Leon Cooper, and Robert Schrieffer (BCS) was to show that superconductivity is not part of the reductionist frontier (Bardeen et al. 1957). Before BCS this was not so clear. For instance, in 1933 Walter Meissner raised the question of whether electric currents in superconductors are carried by the known charged particles, electrons and ions. The great thing that Bardeen, Cooper, and Schrieffer showed was that no new particles or forces had to be introduced to understand superconductivity. According to a book on superconductivity that Cooper showed me, many physicists were even disappointed that “superconductivity should, on the atomistic scale, be revealed as nothing more than a footling small interaction between electrons and lattice vibrations”. (Mendelssohn 1966)."

"The difference between Type I and Type II superconductivity is of substantial practical importance. To understand why, let us return to the Meissner effect ... What happens when the external magnetic field is increased to the critical value Bcrit? Magnetic flux begins to penetrate the material, initially in the form of flux lines. In the Type I case, the flux lines attract, forming a large region of normal metal, and superconductivity is soon lost altogether. In the case of a Type II superconductor, when one reaches the critical magnetic field, flux lines appear inside the superconductor. But since they repel each other, they can form a stable arrangement, a “lattice” of parallel flux lines ... As a result, a Type II superconductor for B > Bcrit can reach a stable arrangement in which part of an externally applied magnetic field is expelled to the outside world, while part penetrates the superconductor in the form of the flux lattice. The material remains superconducting in such a state, and as a result, Type II superconductors can support considerably higher magnetic fields and currents, making them more useful for many applications. Superconductivity is finally lost at a higher value of the magnetic field called the upper critical field."

"When certain materials are cooled below a certain critical temperature Tc, they suddenly become superconducting. Historically, physicists had long suspected that the superconducting transition, just like the superfluid transition, has something to do with Bose-Einstein condensation. But electrons are fermions, not bosons, and thus they first have to pair into bosons, which then condense. We now know that this general picture is substantially correct: Electrons form Cooper pairs, whose condensation is responsible for superconductivity. With brilliant insight, Landau and Ginzburg realized that without having to know the detailed mechanism driving the pairing of electrons into bosons, they could understand a great deal about superconductivity by studying the field φ(x) associated with these condensing bosons. In analogy with the ferromagnetic transition in which the magnetization M⃗(x) in a ferromagnet suddenly changes from zero to a nonzero value when the temperature drops below some critical temperature, they proposed that φ(x) becomes nonzero for temperatures below Tc. (In this chapter x denotes spatial coordinates only.) In statistical physics, quantities such as M⃗(x) and φ(x) that change through a phase transition are known as order parameters. The field φ(x) carries two units of electric charge and is therefore complex."

"That electron-phonon interactions lead to an effective attractive interaction between electrons by exchange of virtual photons was shown by Fröhlich by use of field-theoretic methods. His analysis was extended by Pines and myself to include Coulomb interactions. In second order, there is an effective interaction between the quasi-particle excitations of the normal state which is the sum of the attractive phonon-induced interaction and a screened Coulomb interaction. In the Handbuch article, I suggested that one should take the complete interaction, not just the self-energy tens, and use it for a theory of superconductivity."

"The phonons are well known as the quanta of lattice vibrations. These lattice vibrations are the most understood from successful theories of condensed matter physics and can be considered as a milestone in the comprehension of the properties of condensed materials. Ranging from the infrared, Raman, neutron and in the recent years, synchrotron spectra to the anharmonic properties as well as the electron phonon interactions and superconductivity, there is little left in which the phonons do not play a vital role. In order to understand the material properties, however, the modelling of microscopic interactions intrinsically involved between electrons and ions, remained an ambitious and affordable tool over decades. One of the remarkable developments in physics of phonons is the concept of Rigid Shell Model (RSM) ... derived from the original crystal lattice theory of Born and Huang ..., which includes the microscopic electron-phonon interactions in several classes of non-metallic solids."

"As the concept of a phonon originates from relative motion of the atoms, rather than the motion of their centre of mass, a phonon in a crystal does not carry a momentum. However, for practical purposes we assign a momentum \hbarq to a phonon in the qth mode. For this reason a phonon is called a quasi-particle."

"The spin S = 3/2 nucleus and the S = 1/2 valence electron spin combine to gives states with a total spin of either S = 2 or S = 1."

"Protons behave like quantum spinning tops with spin ½ in units of Planck's constant \hbar. This spin is responsible for many fundamental properties of nature including the proton's magnetic moment, the different phases of matter in low temperature physics, the properties of neutron stars and the stability of the known Universe. One of the main questions in particle and nuclear physics the last 20 years has been: How is the proton's spin built up from its quark and gluon constituents?"

"In dealing with problems about electrons according to quantum mechanics, one finds one does not get agreement with experiment if one assumes the electrons to be simply point charges repelling one another according to the Coulomb law of forces. It is necessary to make the assumption that each electron is spinning and so has an internal angular momentum, and also that it has a magnetic moment. To make the theory agree with experiment we must assume that the eigenvalues of the Cartesian component of the spin angular momentum in any direction are ½\hbar and –½\hbar, and that the magnetic moment of the electron (with its sign reversed) always lies in the same direction as the spin angular momentum ..."

"... when one has to deal with very many electrons ... one then requires a ... simpler and rougher method. Such a method is provided by Thomas' atomic model, in which the electrons are regarded as forming a perfect gas satisfying the Fermi statistics and occupying the region of phase space of lowest energy. This region of phase space is assumed to be saturated, with two electrons with opposite spins in each volume (2πh)3, and the remainder is assumed to be empty. Although this model hitherto has not been justified theoretically, it seems to be a plausible approximation for the interior of a heavy atom and one may expect it to give with some accuracy the distribution of electric charge there."

"For a spin-one particle only—because it requires three amplitudes—the correspondence with a vector is very close. In each case, there are three numbers that must transform with coordinate changes in a certain definite way. In fact, there is a set of base states which transform just like the three components of a vector."

"The investigations of Dirac (1936) on relativistic wave equations for particles with arbitrary spin have recently been followed up by one of us (Fierz, 1939, ... ) It was there found possible to set up a scheme of second quantization in the absence of an external field, and to derive expressions for the current vector and the energy-momentum tensor. These considerations will be extended in the present paper to the case when there is an external electromagnetic field, but we shall in the first instance disregard the second quantization and confine ourselves to a c-number theory. The difficulty of this problem is illustrated by the fact that the most immediate method of taking into account the effect of the electromagnetic field, proposed by Dirac (1936), leads to inconsistent equations as soon as the spin is greater than 1."

"... Before the year 1928, every physicist knew what we meant by an elementary particle. The electron and the proton were the obvious examples, and at that time we would have liked simply to take them as point charges, infinitely small, defined simply by their charge and their mass. We had to agree reluctantly that they must have a radius, since their electromagnetic energy had to be finite. We did not like the idea that such objects should have properties like a radius, but still we were happy that at least they seemed to be completely symmetrical, like a sphere. But then the discovery of electron spin changed this picture considerably. The electron was not symmetrical. It had an axis, and this result emphasized that perhaps such particles have more than one property, and that they are not simple, not so elementary as we had thought before."

"With the exception of experts on the classification of spectral terms, the physicists found it difficult to understand the exclusion principle, since no meaning in terms of a model was given to the fourth degree of freedom of the electron. The gap was filled by Uhlenbeck and Goudsmit’s idea of electron spin ..., which made it possible to understand the anomalous Zeeman effect simply by assuming that the spin quantum number of one electron is equal to 1/2 and that the quotient of the magnetic moment to the mechanical angular moment has for the spin a value twice as large as for the ordinary orbit of the electron. Since that time, the exclusion principle has been closely connected with the idea of spin."

"In brief, the conditions for BEC in alkali gases are reached by combining two cooling methods. Laser cooling is used to precool the gas. The principle of laser cooling is that scattered photons are on average blue-shifted with respect to the incident laser beam. As a result, the scattered light carries away more energy than has been absorbed by the atoms, resulting in net cooling. Blue-shifts are caused by Doppler shifts or ac Stark shifts. The different laser cooling schemes are described in the 1997 Nobel lectures in physics ... After the precooling, the atoms are cold enough to be confined in a magnetic trap. Wall-free confinement is necessary, otherwise the atoms would stick to the surface of the container. It is noteworthy that similar magnetic confinement is also used for plasmas which are too hot for any material container. After magnetically trapping the atoms, forced evaporative cooling is applied as the second cooling stage ... In this scheme, the trap depth is reduced, allowing the most energetic atoms to escape while the remainder rethermalize at steadily lower temperatures. Most BEC experiments reach quantum degeneracy between 500 nK and 2 μK, at densities between 1014 and 1015 cm-3. The largest condensates are of 100 million atoms for sodium, and a billion for hydrogen; the smallest are just a few hundred atoms."

"Evidence for a BEC has long been seen in superfluids and superconductors, but the constituents of those condensates have strong interactions with one another, so the pristine nature of the BECs is hard to predict and observe. The best hope for clear manifestation of BEC behavior, it seemed, lay with a gas of weakly interacting atoms. ... Possible applications are still on the far horizon. Those that people often mention are ones that exploit the unprecedented control and manipulation of atoms that BECs offer at the quantum level. BECs offer hope of enhanced precision for atomic interferometry, rotation measurements, and atomic clocks. They might find a use in nanofabrication and atom lithography. Or they might play a role in quantum computing."

"Most Bose-Einstein condensates of atomic gases have internal degrees of freedom originating from the spin. ... When a Bose-Einstein condensate is trapped in a magnetic potential, the spin aligns along the direction of a local magnetic field, and the internal degrees of freedom are virtually frozen. The condensate is therefore described by a scalar order parameter. When it is trapped in an optical potential, the internal degrees of freedom are liberated because the optical potential exerts the same force on an atom irrespective of which magnetic sublevel it is in. Such a condensate is called a spinor Bose-Einstein condensate; the order parameter of this condensate has 2ƒ+1 components, where ƒ is the hyperfine spin for alkali atoms and the electronic spin for chromium."

"This paper presents an attempt of explaining the phenomenon of superfluidity on the basis of the theory of degeneracy of a non-perfect Bose-Einstein gas. By using the method of the second quantization together with an approximation procedure we show that in the case of the small interaction between molecules the low excited states of the gas can be described as a perfect Bose-Einstein gas of certain “quasi-particles” representing the elementary excitations, which cannot be identified with the individual molecules. The special form of the energy of a quasi-particle as a function of its momentum is shown to be connected with the superfluidity. The object of this paper is an attempt to construct a consistent molecular theory explaining the phenomenon of superfluidity without assumptions concerning the structure of the energy spectrum."

"Superfluid phenomena in liquid helium-4 have fascinated both experimentalists and theoreticians since the discovery of superfluidity in 1938 simultaneously by Peter Kapitsa at the Soviet Academy of Sciences and by Jack Allen and Donald Misener at the Royal Society Laboratories in Cambridge, England. These phenomena include a vanishingly small viscosity, a very high heat conductivity (30 times greater than copper), and many other bizarre effects, such as the He fountain, film flow and creep, and quantized vortices (see the article by Russell Donnelly in Physics Today, July 1995, page 30)."

"There is no unique relation between BEC and superfluidity. An ideal-gas Bose-Einstein condensate shows no superfluity and a two-dimensional superfluid shows no BEC. However, there are many cases in which BEC and superfluidity do occur simultaneously. Under such circumstances, a generic argument can be made, which offers insight into the interplay between BEC and superfluidity."

"At the heart of the Higgs physics programme is the question of how the Higgs boson couples to Standard Model elementary particles. Within the SM itself, all these couplings are uniquely determined. But new physics beyond the SM (BSM) can modify these couplings in many different ways."

"The discovery ... of the Higgs boson will mark a watershed in particle physics. In the future, the calendar of particle physics will surely be divided into BH (before Higgs) and AH (after Higgs), with 2012 being year 0. The discovery of the Higgs will signpost the direction that both theoretical and experimental physics will take in the decades to come."

"... the famous particle bears the name of Peter Higgs alone. Over the years, all the prestigious prizes in physics have been awarded to different combinations of Brout, Englert, Higgs, Guralnik, Hagen and Kibble. ... Nambu's insight to apply the concept of spontaneous symmetry breaking to empty space was profound. In the strange quantum world, what we think of as empty space is anything but. Instead, it is a seething soup of particles constantly popping in and out of existence, and it is this structure in the vacuum itself that gives rise to particle masses. The thing that fills the vacuum is what has come to be known as the Higgs field. Some particles interact strongly with this field, others not at all, and it is the strength of the interaction with the HIggs field that determines the fundamental particles' masses."

"In regards to the mechanism, Brout and Englert make some comments, but give no quantitative argument. In the Higgs PRL paper, there is only the specific scalar electrodynamics example. In his physics letters paper Higgs observed that an argument given by Gilbert to negate the Goldstone theorem includes a term with a fixed vector such as can be found in radiation gauge electrodynamics. He makes no follow-up or extension of this argument in his PRL paper. The GHK paper does an extensive analysis of the mechanism and shows that in broken gauge theories in the radiation gauge that charge leaks out of any volume and consequently is not conserved since currents continue to exist in the limit of volumes tending toward infinity. This negates the basic assumption of a conserved charge that is required by the Goldstone theorem."

"... string theory seems to require our world to have a property called supersymmetry. And a supersymmetric Standard Model with string theory boundary conditions has Higgs bosons and explains their properties. ... Finding a Higgs boson thus strongly supports the supersymmetric Standard Model, which in turn supports the notion that string theory is indeed the right approach to nature."

"... inclusion of the Higgs boson into the unified electroweak theory guaranteed that one could perform finite calculations of physical quantities in the standard model."

"The key fact about the Higgs boson is that it is the origin of the masses of all known massive elementary particles, including the quarks, leptons, and W and Z bosons. That raises the following question: Why is a new particle needed to give other particles mass? An audience of physicists can understand the answer: Detailed experiments on the weak interactions establish that the Hamiltonian governing elementary particles has exact symmetries that forbid quark, lepton, and boson mass terms. Thus those symmetries must be spontaneously broken. That, in turn, requires a new field to provide the order parameter for the symmetry breaking. Those familiar with contemporary physics will recognize each step in the above argument. But for nonscientists, every step brings in new, technical, and difficult concepts. It is an unsolved problem to present this argument in popular language."

"Higgs, 84, shares the 8m Swedish kronor (£775,000) prize – and no shortage of kudos – with François Englert at the Free University of Brussels for showing how fundamental particles get their masses. Before the theory, the answer to this basic question was unknown. The Royal Swedish Academy awarded the prize for "the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the Atlas and CMS experiments at Cern's Large Hadron Collider.""

"... one of the consequences of the electroweak symmetry is that, if nothing new is added to the theory, all elementary particles, including electrons and quarks, would be massless, which of course they are not. So, something has to be added to the electroweak theory, some new kind of matter or field, not yet observed in nature or in our laboratories. The search for the Higgs particle has been a search for the answer to the question: What is this new stuff we need?"

"There is a puzzle about the Higgs particle — which is what makes it relatively light compared to the natural mass scale of elementary particles."

"Armed with the most advanced mathematics and a good dose of imagination, a generation of theoretical physicists conceived the idea that the universe must have undergone a transformation, after a tenth of a billionth of a second from the Big Bang. At that instant, the entire structure of space-time crystallized into a new form, following a phase transition, just as water turns into ice below zero degrees. [...] The same physicists also realized that this idea so suggestive as to sound like science fiction implied the existence of a new particle, a granule of the substance that permeates all of space-time. Data presented yesterday at Cern show that that particle-the Higgs boson - really exists, corroborating the fantastic story of the cosmic phase transition."

"The discovery of the Higgs boson was a fantastic achievement, but not enough to answer all the questions in particle physics. It was a bit like walking into a three-star restaurant and being served soup. We theoretical physicists confidently await a more tantalizing second course."

"Logarithmic perturbation theory is an alternative way of solving the perturbation equations in the coordinate representation. It was developed many years ago ... and has lately been widely discussed and applied to many problems in quantum mechanics. ... perturbation theory allows us to transform the nonlinear Riccati equation ... into a set of linear differential equations ..."

"The separation of the hamiltonian into an unperturbed part and a perturbation is not unique, but in most problems of interest there is a separation which presents itself in a most natural way. In quantum electrodynamics for example, the unperturbed system consists of the electron-positron field and the photon field without interaction. In the theory of an imperfect gas the unperturbed system will be taken as the ideal gas obtained by neglecting interparticle interactions. In the application of perturbation theory to large quantum systems one encounters problems not met with in the usual perturbation theory of systems with a finite number of degrees of freedom. These problems are related to the following phenomena:"

"The great success of calculations in quantum electrodynamics using the renormalization idea generated a new enthusiasm for quantum electrodynamics. After this change of mood, probably most theorists simply didn’t worry about having to deal with infinite renormalizations. Some theorists thought that these infinities were just a consequence of having expanded in powers of the electric charge of the electron, and that not only observables but even quantities like the “bare” electron charge (the charge appearing in the field equations of quantum electrodynamics) would be found to be finite when we learned how to calculate without perturbation theory. But at least two leading theorists had their doubts about this, and thought that the appearance of infinite renormalizations in perturbation theory was a symptom of a deeper problem, a problem not with perturbation theory but with quantum field theory itself. They were Lev Landau, and Gunnar Källén."

"An electronic semiconductor is typically a valence crystal whose conductivity depends markedly on temperature and on the presence of minute amounts of foreign impurities. The ideal crystal at the absolute zero is an insulator. When the valence bonds are completely occupied and there are no extra electrons in the crystal, there is no possibility for current to flow. Charges can be transferred only when imperfections are present in the electronic structure, and these can be of two types: excess electrons which do not fit into the valence bonds and can move through the crystal, and holes, places from which electrons are missing in the bonds, which also behave as mobile carriers. While the excess electrons have the normal negative electronic charge -e, holes have a positive charge, +e. It is a case of two negatives making a positive ; a missing negative charge is a positive defect in the electron structure. The bulk of a semiconductor is electrically neutral; there are as many positive charges as negative. In an intrinsic semiconductor, in which current carriers are created by thermal excitation, there are approximately equal numbers of excess electrons and holes. Conductivity in an extrinsic semiconductor results from impurity ions in the lattice. In n-type material, the negative charge of the excess electrons is balanced by a net positive space charge of impurity ions. In p-type, the positive charge of the holes is balanced by negatively charged impurities. Foreign atoms which can become positively charged on introduction to the lattice are called donors; atoms which become negatively ionized are called acceptors. Thus donors make a semiconductor n-type, acceptors p-type. When both donors and acceptors are present, the conductivity type depends on which is in excess. Mobile carriers then balance the net space charge of the impurity ions."

"One of the remarkable and dramatic developments in recent years has been the application of solid state science to technical developments in electrical devices such as transistors. The study of semiconductors led to the discovery of their useful properties and to a large number of practical applications. ... The semiconductor substances in most common use today are silicon and germanium. These elements crystallize in the diamond lattice, a kind of cubic structure in which the atoms have tetrahedral bonding with their four nearest neighbors. They are insulators at very low temperatures—near absolute zero—although they do conduct electricity somewhat at room temperature. ... somehow put an extra electron into a crystal of silicon or germanium which is at a low temperature ... The electron will be able to wander around in the crystal jumping from one atomic site to the next. Actually, we have looked only at the behavior of electrons in a rectangular lattice, and the equations would be somewhat different for the real lattice of silicon or germanium. All of the essential points are, however, illustrated by the results for the rectangular lattice."

"A consequence of the discovery of electricity was the observation that metals are good conductors while nonmetals are poor conductors. The latter were called insulators. Metallic onductivity is typically between 106 and 104 (Ω cm)–1, while typical insulators have conductivities of less than 10–10 (Ω cm)–1. Some solids with conductivities between 104 and 10–10 (Ω cm)–1 are classified as semiconductors. ... semiconductors have an energy gap while semimetals and metals have no such gap. However, very impure semiconductors show a more or less metallic behavior and with many substances, the art of purification is not so far advanced that a distinction can easily be made. The transition between semiconductors and insulators is even more gradual and depends on the ratio of the energy gap to the temperature of investigation. Very pure semiconductors may become insulators when the temperature approaches the absolute zero."

"The 1982 Aspect Experiment in France demonstrated, that two once-connected quantum particles separated by vast distances remained somehow connected. If one particle was changed, the other changed - instantly. Scientists don't know the mechanics of how this faster-than-the-speed-of-light travel can happen, though some theorists suggest that this connection takes place via doorways into higher dimensions... We are connected, rather than separate, from all of life... the full spectrum of consciousness encompasses both physical and a multitude of non physical dimensions of reality. At core, this new world view involves seeing yourself, others, and all of life, not through the eyes of our small, earthly self that lives in time and is born in time. But rather through the eyes of the soul, our Being, the True Self. One by one, people are jumping to this higher orbit."

"Just four weeks after the first paper (Q1) the Annalen received on February 23 the second paper (Q2) in the series 'Quantization as an Eigenvalue Problem'. ... It consists of a detailed exploration of the Hamiltonian analogy between mechanics and optics leading to a new derivation of the wave equation, an analysis of the relations between security making geometrical and undulatory mechanics, and applications of the wave equation to the harmonic oscillator and the diatomic molecule."

"With the growing importance of models in statistical mechanics and in field theory, the path integral method of Feynman was soon recognized to offer frequently a more general procedure of enforcing the first quantization instead of the Schrödinger equation. To what extent the two methods are actually equivalent, has not always been understood. ... the Coulomb potential and the harmonic oscillator ... point the way: For scattering problems the path integral seems particularly convenient, whereas for the calculation of discrete security eigenvalues the Schrödinger equation."

"Interactions that look instantaneous are well suited to Schrödinger’s equation, which requires the potential between particles at equal times. It would be quite awkward to explicitly describe finite-velocity forces in the Schrödinger equation because the potential for one particle at a time t would depend on the positions of the others at the retarded times, and one would need the past histories of all the particles to propagate the system forward in time."

"Wick rotation is a very basic procedure for inter-playing Lorentzian and Riemannian geometry. The simplest example applies to \mathbb R^{n+1} endowed with both the standard Minkowski metric {-dx_0}^2+\cdots+dx_{n-1}^2+dx_n^2\, and the Euclidean metric dx_0^2+\cdots+dx_n^2\,. By definition ... these are related via a Wick rotation directed by the vector field \frac{\partial}{\partial x_0}. Sometimes one refers to it as "passing to the imaginary time"."

"The celebrated Schrödinger equation is mathematically close to the ordinary diffusion equation. What is the main difference is that time is imaginary, there is the Wick rotation. This means that classical and quantum are related partly by a rotation of 90 degrees in the complex plane (multiplying by the imaginary unit)."

"The Wick rotation idea is now so commonly used in quantum field theory that it is often taken as an automatic procedure in numerous different kinds of situation, with barely a mention, and its validity is hardly ever questioned. It does, in fact, have a broad applicability, but it is not a universally valid procedure. Most particularly. it is highly questionable in the context of the curved space-times arising in general relativity, when in normal circumstances the procedure cannot even be applied, because there is no natural time coordinate. In string theory, this is a problem both in the 10-dimensional space-time, in general curved-space situations, and also in the string world-sheet ..."

"The Einstein–Podolsky–Rosen (EPR) argument has been enormously influential in the debate on the foundations of quantum mechanics. While EPR argue for the incompleteness of quantum mechanics, Bell's 'no-go' theorem, which is in a sense an extension of the EPR argument, appears to support the opposite conclusion."

"The Kochen–Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models."

"One possibility that comes to mind is that the spin-two graviton might arise as a composite of two spin-one gauge bosons. This interesting idea would seem to be rigorously excluded by a no-go theorem of Weinberg & Witten ... The Weinberg–Witten theorem appears to assume nothing more than the existence of a Lorentz-covariant energy momentum tensor, which indeed holds in gauge theory. The theorem does forbid a wide range of possibilities, but (as with several other beautiful and powerful no-go theorems) it has at least one hidden assumption that seems so trivial as to escape notice, but which later developments show to be unnecessary. The crucial assumption here is that the graviton moves in the same spacetime as the gauge bosons of which it is made!"

"The question of the possibility for a completion of quantum mechanics received its most famous (partial) answer in 1964 by, again, Bell ... He proved what today is known simply as Bell's theorem, to wit, that is such a more complete description exists, it cannot be local, i.e. dependent only on the events in a system's past lightcone, and agree with quantum mechanics in all instances. To this day, this result forms the paradigm example of a 'no-go' theorem."

"... If you start off with switches and gears, or whatever, you can never construct a universe in which you see quantum mechanical phenomena, according to Bell. We call such a thing a 'no-go theorem'. You may already suspect that I still believe in the hidden variables hypothesis. Surely our world must be constructed in such an ingenious way that some of the assumptions that Einstein, Bell and others found quite natural will turn out to be wrong. But how this will come about, I do not know. Anyway, for me, the hidden variables hypothesis is still the best way to ease my conscience about quantum mechanics. And as for 'no-go theorems', we will encounter several of these and discuss their fate."

"It has been recently conjectured that scattering amplitudes in planar N = 4 super Yang-Mills are given by the volume of the (dual) amplituhedron. In this paper we show some interesting connections between the tree-level amplituhedron and a special class of differential equations. In particular we demonstrate how the amplituhedron volume for NMHV amplitudes is determined by these differential equations. The new formulation allows for a straightforward geometric description, without any reference to triangulations. Finally we discuss possible implications for volumes related to generic NkMHV amplitudes."

"In the past decade a new, geometric picture has emerged for scattering amplitudes in planar N = 4 super Yang-Mills (SYM) theory. It originated from the observation that the tree-level amplitudes and loop-level integrands of n-point amplitudes for all helicity sectors can be computed using integrals over the Grassmannian space ... In such formulation, amplitudes can be extracted from a Grassmannian integral over a suitable contour which selects a particular sum of residues. Building upon this idea, novel studies revealed the interrelation between the rich combinatorial structure of positive Grassmannians and the physical properties of amplitudes ... From this point of view, the aforementioned residues are associated with positroid cells, which are particular subvarieties inside the positive Grassmannian."

"Perturbative scattering amplitudes in gauge theories have remarkable simplicity and hidden infinite dimensional symmetries that are completely obscured in the conventional formulation of field theory using Feynman diagrams. This suggests the existence of a new understanding for scattering amplitudes where locality and unitarity do not play a central role but are derived consequences from a different starting point. In this note we provide such an understanding for N = 4 SYM scattering amplitudes in the planar limit, which we identify as “the volume” of a new mathematical object–the Amplituhedron–generalizing the positive Grassmannian. Locality and unitarity emerge hand-in-hand from positive geometry."

"The amplituhedron has now given us a new description of planar N = 4 SYM amplitudes which does not have a usual space-time/quantum mechanical description, but does make all the symmetries manifest. This is not a “duality” in the usual sense, since we are not identifying an equivalence between existing theories with familiar physical interpretations. We are seeing something rather different: new mathematical structures for representing the physics without reference to standard physical ideas, but with all symmetries manifest. Might there be an analogous story for superstring scattering amplitudes?"

"When I recall the days of twenty years ago, when the conception of the physical quantum of 'action' was first beginning to disentangle itself from the surrounding mass of available experimental facts, and when I look back upon the long and tortuous road which finally led to its disclosure, this development strikes me at times as a new illustration of Goethe's saying, that 'man errs, so long as he is striving'."

"The pursuit of a goal, the brightness of which is undimmed by initial failure, is an indispensable condition, though by no means a guarantee, of final success. In my own case, such a goal has been for many years the solution of the question of the distribution of energy in the normal spectrum of radiant heat."

"The discovery by Gustav Kirchhoff that the quality of the heat radiation produced in an enclosure surrounded by any emitting or absorbing bodies whatsoever, all at the same temperature, is entirely independent of the nature of such bodies, established the existence of a universal function, which depends only upon the and the , and is entirely independent of the particular properties of the substance. ...[T]his remarkable function promised a deeper insight into the relation between energy and temperature, which is the principal problem of thermodynamics and therefore also of the entire field of ."

"A most suitable body... seemed H. Hertz's rectilinear oscillator (dipole) whose laws of emission for a given frequency he had just... developed. If a number of such oscillators be distributed in an enclosure surrounded by reflecting walls, there would take place, in analogy with sources and resonators in... sound, an exchange of energy by means of the emission and reception of electro-magnetic waves, and... corresponding to Kirchoff's law should establish itself in the vacuum-enclosure."

"I expected, in a [naive] way... that the laws of classical electrodynamics would suffice, if one adhered sufficiently to generalities and avoided too special hypotheses..."

"I... first developed in... general terms... the laws of the emission and absorption of a linear ... by a... circuitous route which might have been avoided had I used the electron theory which had just been put forward by H. A. Lorentz."

"The outcome of this long series of investigations... was the establishment of a general relation between the energy of a resonator of a definite free frequency and the energy radiation of the corresponding spectral region in the surrounding field in equilibrium with it."

"The remarkable result was obtained that this relation is independent of the nature of the resonator, and in particular of its coefficient of damping—a result which... introduced the simplification that the energy of the radiation could be replaced by the energy of the resonator, so that a simple system of one degree of freedom could be substituted for a complicated system having many degrees of freedom."

"But this result constituted only a preparatory advance towards the attack on the main problem... [M]y original hope [was] that the radiation emitted by the resonator would differ in some characteristic way from the absorbed radiation... The resonator reacted only to those rays which were emitted by itself, and exhibited no trace of resonance to neighbouring spectral regions."

"[M]y suggestion that the resonator might be able to exert a one-sided, i. e. irreversible, action on the energy of the surrounding radiation field called forth the emphatic protest of Ludwig Boltzmann... showing that according to... classical dynamics... the processes I was considering could take place in... the opposite sense. Thus a spherical wave emitted from a resonator when reversed shrinks... continually decreasing... on to the resonator, is absorbed by it, and so permits the resonator to send out again into space the energy formerly absorbed in the direction from which it came."

"[I]t became more and more evident that... an essential link was missing which should lead to... comprehension... The only way out... was to attack the problem from the opposite side, from... thermodynamics, a domain in which I felt more at home."

"[M]y previous studies on the second law of thermodynamics served me here... in that my first impulse was to bring not the but the of the resonator into relation with its energy, more accurately not the entropy itself but its with respect to the energy... [T]his differential coefficient... has a direct physical significance for the irreversibility of the exchange of energy between the resonator and the radiation."

"But as I was... too much devoted to pure phenomenology to inquire more closely into the relation between entropy and probability, I felt compelled to limit myself to the available experimental results. Now, at that time... 1899, interest was centred on the law of the distribution of energy... proposed by W. Wien... On calculating the relation following from this law between the entropy and energy of a resonator the remarkable result is obtained that the reciprocal value of the above differential coeffcient... R, is proportional to the energy. This extremely simple relation can be regarded as an adequate expression of Wien's law..."

"I believed... that the basis of the law of the distribution of energy could be expressed by the theorem that the value of R is proportional to the energy. But in view of the results of new measurements this conception soon proved untenable."

"[T]wo simple limits were established by direct observation for the function R: for small energies proportionality to the energy, for large energies proportionality to the square of the energy. Nothing... seemed simpler than to put in the general case R equal to the sum of a term proportional to the first power and another proportional to the square of the energy... and thus was found a new radiation formula which... has withstood experimental examination fairly satisfactorily."

"But even if this radiation formula should prove to be absolutely accurate it would after all be only an formula found by happy guesswork..."

"I was... occupied with the task of giving it a real physical meaning, and this... led me, along Boltzmann's line... to the consideration of the relation between and probability... after some weeks of the most intense work of my life clearness began to dawn... and an unexpected view revealed itself..."

", according to Boltzmann, is a measure of a physical probability, and the meaning of the second law of thermodynamics is that the more probable a state is, the more frequently will it occur in nature."

"[W]hat one measures are only the differences of entropy, and never entropy itself, and consequently one cannot speak... of the absolute entropy of a state. But nevertheless the introduction of an appropriately defined absolute magnitude of entropy is... recommended... by its help certain general laws can be formulated with great simplicity."

"[T]he case is... the same as with energy. Energy... cannot itself be measured; only its differences can."

"[T]he concept used by our predecessors was not energy but work, and even Ernst Mach, who devoted much attention to the law of but... avoided all speculations exceeding the limits of observation, always abstained from speaking of energy..."

"[I]n the early days of one was content to deal with heats of reaction, that is to say again with differences of energy, until emphasized that... calculations could be... shortened if energies instead of calorimetric numbers were used."

"The additive constant which... remained undetermined for energy was later finally fixed by the relativistic law of the proportionality between energy and inertia."

"As in the case of energy, it is now possible to define an absolute value of entropy, and thus of physical probability, by fixing the additive constant so that together with the energy (or better still, the temperature) the entropy also should vanish."

"Such... led to a comparatively simple method of calculating the physical probability of a given distribution of energy in a system of resonators, which yielded precisely the same expression for entropy as that corresponding to the radiation law; and it gave me particular satisfaction, in compensation for the many disappointments... to learn from Ludwig Boltzmann of his interest and... acquiescence in my... reasoning."

"To work out these probability considerations the knowledge of two universal constants is required, each of... independent meaning, so... evaluation of these... from the radiation law could serve as an a posteriori test whether the... process is merely a mathematical artifice or has a true physical meaning."

"The first constant... is connected with the definition of temperature. If temperature were defined as the mean of a molecule in a , which is a minute energy indeed, this constant would have the value ⅔. But in the conventional scale of temperature the constant ...[instead] assumes an extremely small value... intimately connected with the energy of a single molecule... [I]ts accurate determination would lead to the calculation of the mass of a molecule and... associated magnitudes. This constant is frequently termed Boltzmann's constant, although to the best of my knowledge Boltzmann... never introduced it (...he, as appears from... his statements, never believed it would be possible to determine this constant accurately)..."

"Nothing can better illustrate the rapid progress of experimental physics within the last twenty years than the fact that... [by] a host of methods ...the mass of a single molecule can be measured with almost the same accuracy as that of a planet."

"While at the time when I carried out this calculation on the basis of the radiation law an exact test of the value... was... impossible... it was not long before E. Rutherford and H. Geiger succeeded, by... a direct count of the α-particles, in determining the value of the electrical as 4.65 10-10, the agreement... with my value 4.69 10-10... a decisive confirmation of my theory. ...[M]ethods ...by E. Eegener, R. A. Millikan, and others... have led to a but slightly higher value."

"Much less simple... was the interpretation of the second universal constant of the radiation law... the product of energy and time (...a first calculation to 6.55 10-27 erg. sec.) I called the elementary quantum of action."

"While this constant was absolutely indispensable to... a correct expression for —for only with its aid could be determined the magnitude of the 'elementary region' or 'range' of probability, necessary for the statistical treatment of the problem—it obstinately withstood all attempts at fitting it... into the frame of the classical theory. So long as it could be regarded as infinitely small, that is to say for large values of energy or long periods of time, all went well; but in the general case a difficulty arose... which became the more pronounced the weaker and... more rapid the oscillations."

"The failure placed one before the dilemma: either the quantum of action was only a fictitious magnitude, and... the entire deduction from the radiation law was illusory and a mere juggling with formulae, or there is at the bottom of this method of deriving the radiation law some true physical concept."

"If the latter were the case, the would have to play a fundamental role in physics, heralding the advent of a new state of things, destined, perhaps, to transform completely our physical concepts which since the introduction of the infinitesimal calculus by Leibniz and Newton have been founded upon the assumption of the continuity of all causal chains of events."

"Experience has decided for the second alternative. But that the decision should come so soon... was due not to the examination of the law of distribution of the energy of heat radiation, still less to my special deduction of this law, but to the steady progress of the work of those investigators who have applied the concept of the quantum of action to their researches."

"The first advance... was made by A. Einstein, who... pointed out that the... quanta of energy associated with the quantum of action seemed capable of explaining... a series of remarkable properties of light action discovered experimentally, such as Stokes's rule, the emission of electrons, and the ionization of gases, and on the other hand, by the identification of the expression for the energy of a system of resonators with the energy of a solid body, derived a formula for the specific heat of solid bodies which on the whole represented it correctly as a function of temperature, more especially exhibiting its decrease with falling temperature."

"With regard to specific heat of solid bodies, Einstein's view, which rests on the assumption of a single free period of the atoms, was extended by M. Born and Th. von Karman to the case which corresponds better to reality, viz. that of several free periods; while P. Debye, by a bold simplification of the assumptions as to the nature of the free periods, succeeded in developing a comparatively simple formula for the specific heat of solid bodies which excellently represents its values, especially those for low temperatures obtained by W. Nernst and his pupils, and... compatible with the elastic and optical properties of such bodies."

"But the influence of the quanta asserts itself also in the case of the specific heat of gases. At the very outset it was pointed out by W. Nernst that to the energy quantum of vibration must correspond an energy quantum of rotation, and it was therefore... expected that the rotational energy of gas molecules would also vanish at low temperatures. ...That 'quantized' rotations of gas molecules... do actually occur in nature can no longer be doubted... although a[n] exhaustive explanation of... rotation spectra is still outstanding."

"The inverse of the process of producing light quanta by the impact of electrons is the emission of electrons on exposure to light-rays, or s, and here... energy quanta following from the action quantum and the vibration period play a characteristic role, as was early recognized from the striking fact that the velocity of the emitted electrons depends not upon the intensity but only on the colour of the impinging light. [T]the relations to the light quantum, pointed out by Einstein, have proved successful in every direction... shown especially by R. A. Millikan, by measurements of the velocities of emission of electrons, while the importance of the light quantum in inducing photo-chemical reactions was disclosed by E. Warburg."

"[T]he results... hitherto quoted... taken in their totality, form an overwhelming proof of the existence of the quantum of action, the quantum hypothesis received its strongest support from the theory of the structure of atoms (Quantum Theory of Spectra) proposed and developed by Niels Bohr... the long-sought key to the gates of the wonderland of spectroscopy which since the discovery of spectrum analysis... stubbornly refused to yield. And... once clear, a stream of new knowledge poured in a sudden flood, not only over this... field but into the adjacent territories of physics and chemistry."

"Its first brilliant success was the derivation of Balmer's formula for the spectrum series of and , together with the reduction of the universal constant of Rydberg to known magnitudes; and even the small differences of the Rydberg constant for these two gases appeared as a necessary consequence of the slight wobbling of the massive atomic nucleus (accompanying the motion of electrons around it). As a sequel came the investigation of other series in the visual and especially the X-ray spectrum aided by Ritz's resourceful combination principle, which only now was recognized in its fundamental significance."

"But whoever may have still felt inclined... in the face of this almost overwhelming agreement... to believe it... a coincidence, must... give up... doubt when A. Sommerfeld deduced, by a logical extension of the laws of the distribution of quanta in systems with several degrees of freedom, and by a consideration of the variability of inert mass required by the principle of relativity, that magic formula before which the spectra of both and revealed the mystery of their ... by the most delicate measurements...of F. Paschen..."

"P. Epstein achieved a complete explanation of the of the electrical splitting of spectral lines, P. Debye obtained a simple interpretation of the K-series of the X-ray spectrum investigated by , and then... a long series of further researches... illuminated... the dark secret of atomic structure."

"[T]he quantum of action, which in every one of the many and most diverse processes has always the same value, namely 6.52 10-27 erg. sec., deserves to be... incorporated into the system of the universal s."

"[A]t just the same time as the idea of general relativity arose... nature revealed, precisely... where ...least ...expected, an absolute and strictly unalterable unit, by means of which the amount of action contained in a space-time element can be expressed by a perfectly definite number, and thus is deprived of its former relative character."

"[T]he mere introduction of the quantum of action does not yet mean that a true Quantum Theory has been established. Nay, the path which research has yet to cover... is perhaps not less long than that from the discovery of the velocity of light by Olaf Römer to the foundation of Maxwell's theory of light."

"The difficulties which the introduction of the quantum of action into the well-established classical theory has encountered from the outset... have gradually increased rather than diminished; and although research... has... passed over some of them, the remaining gaps in the theory are the more distressing..."

"[W]hat in Bohr's theory served as the basis of the laws of action consists of certain hypotheses which a generation ago would doubtless have been flatly rejected by every physicist. That with the atom certain quantized orbits [i.e. picked out on the quantum principle] should play a special role could well be granted; somewhat less easy to accept is the further assumption that the electrons moving on these curvilinear orbits, and therefore accelerated, radiate no energy. But that the sharply defined frequency of an emitted light quantum should be different from the frequency of the emitting electron would be regarded... in the classical school as monstrous and almost inconceivable. But numbers decide... the tables have been turned."

"While originally it was a question of fitting in with as little strain as possible a new and strange element into an existing system... generally regarded as settled, the intruder... having won an assured position, now has assumed the offensive; and... is about to blow up the old system... The only question... is, at what point and to what extent this will happen."

"[O]ut of the classical theory the great principles of thermodynamics will not only maintain intact their central position in the quantum theory, but will perhaps even extend their influence."

"The significant part played in the origin of the classical thermodynamics by mental experiments is now taken over in the quantum theory by P. Ehrenfest's hypothesis of the adiabatic invariance; and just as the principle introduced by R. Clausius, that any two states of a material system are mutually interconvertible on suitable treatment by reversible processes, formed the basis for the measurement of , just so do the new ideas of Bohr show a way into the midst of the wonderland he has discovered."

"[O]ne... question... will... lead to an extensive elucidation of the entire problem. What happens to the energy of a light-quantum after its emission? Does it pass outwards in all directions, according to Huygens's wave theory, continually increasing in volume and tending towards infinite dilution? Or does it, as in Newton's emanation theory, fly like a projectile in one direction only? In the former case the quantum would never again be in a position to concentrate its energy at a spot strongly enough to detach an electron from its atom; while in the latter case it would be necessary to sacrifice the chief triumph of Maxwell's theory — the continuity between the static and the dynamic fields — and with it the classical theory of the interference phenomena which accounted for all their details, both alternatives leading to consequences very disagreeable..."

"[S]cience will some day master the dilemma, and what may now appear to us unsatisfactory will appear from a higher standpoint as endowed with a particular harmony and simplicity. But until... [then] the problem of the quantum of action will not cease to stimulate research, and the greater the difficulties encountered in its solution the greater will be its significance for the broadening and deepening of all our physical knowledge."

"The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger's scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles."

"One of the reasons for being interested in the geometric phase is that it connects a number of different areas."

"Whenever a quantum system undergoes a cyclic evolution governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov–Bohm phase and the Pancharatnam and Berry phase, but both earlier and later manifestations exist. Although traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and become increasingly influential in many areas from condensed-matter physics and optics to high-energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review, we first introduce the Aharonov–Bohm effect as an important realization of the geometric phase. Then, we discuss in detail the broader meaning, consequences and realizations of the geometric phase, emphasizing the most important mathematical methods and experimental techniques used in the study of the geometric phase, in particular those related to recent works in optics and condensed-matter physics."

"One of the simplest chemical exchange reactions involves a system of three hydrogen atoms: H+H2→H2+H. Surely, chemists have felt, one should be able to calculate the cross sections for this reaction from first principles. But the computations have not been easy. Only in the last six years or so have theorists, aided by efficient methodologies and access to supercomputers, been able to predict the cross sections in sufficient detail for comparison with experiments, which themselves have evolved in precision. The agreement has been good—well, almost. Small discrepancies, especially at higher total energies, stubbornly refused to yield to adjustments in either the calculations or the experiments. Now Yi‐Shuen Mark Wu and Aron Kuppermann of Caltech have erased these pesky discrepancies by including a topological effect known as the geometric phase. Michael Berry (University of Bristol) has called attention to the presence of this phase, which now bears his name, in a wide variety of physical systems."

"The geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two-form through a surface in the system’s parameter space. We obtain the classical limit of this two-form in a form applicable to systems whose classical dynamics is chaotic. For integrable systems the expression is equivalent to the Hannay two-form. We discuss various properties of the classical two-form, derive semiclassical corrections to it (associated with classical periodic orbits), and consider implications for the semiclassical density of degeneracies."

"Examples of geometric phases abound in many areas of physics. Many familiar problems that we do not ordinary associate with geometric phases may be phrased in terms of them. Often, the result is a clearer understanding of the structure of the problem, and of its solution."

"Gauge symmetry: Whenever the Hamiltonian is such as to conserve the total number of particles of a particular sort—or, more generally, where there is a conserved "charge"-like quantity, such as lepton or baryon number, or electric charge itself—we shall find that the Hamiltonian will exhibit a gauge invariance property."

"In their theory of superconductivity, Bardeen, Cooper, and Schrieffer made use of a reduced Hamiltonian which included only scattering of pairs of particles of opposite momentum and spin. It is shown that the solution they obtained by a variational method is correct to O(\tfrac{1}{n}) for a large system. The single particle Green's function is derived and used to calculate the interaction energy."

"This work is a part of an effort to analyze the physical limitations of computers due to the laws of physics. For example, Bennett ... has made a careful study of the free energy dissipation that must accompany computation. He found it to be virtually zero. He suggested to me the question of the limitations due to quantum mechanics and the uncertainty principle. I have found that, aside from the obvious limitation to size if the working parts are to be made of atoms, there is no fundamental limit from these sources either. We are here considering ideal machines; the effects of small imperfections will be considered later. This study is one of principle; our aim is to exhibit some Hamiltonian for a system which could serve as a computer."

"Some time in 1960 or early 1961, I learned of an idea which had originated earlier in solid-state physics and had been brought into particle physics by Heisenberg, Nambu, and Goldstone, who had worked in both areas. It was the idea of "broken symmetry," that the Hamiltonian and commutation relations of a quantum theory could possess an exact symmetry, and the physical states might nevertheless not provide neat representations of the symmetry. In particular, a symmetry of the Hamiltonian might turn out to be not a symmetry of the vacuum."

"Perhaps the best-known giant resonance in nuclei is the giant dipole resonance (GDR). The GDR is described in classical hydrodynamics as a class of nuclear motion in which the neutrons and protons within a nucleus move collectively against one another, providing a separation between the centers of mass and charge, thus creating a dipole moment."

"Nuclei interact with the external environment through a number of different fields—electromagnetic, weak and hadronic. The collective excitations induced by these interactions are known as giant resonances. The best known example is the giant dipole resonance, which is stimulated when the electric field of an incident gamma ray exerts a force on the positively charged protons in a nucleus, moving them relative to the uncharged neutrons ... Other giant resonances that have been studied are the monopole, quadrupole and spin-isospin modes of oscillation. The spin-isospin mode involves charge-changing processes, in particular beta decay. The quadrupole and monopole giant resonances are most easily seen with fields that act equally on neutrons and protons, because in these modes the neutrons and protons oscillate in the same mode. The giant resonances are collective oscillations and the various modes of oscillation depend on specific aspects on the nuclear force to sustain them. In the monopole mode, the motion is radial and the frequency depends on the compressibility of the nucleus. In the dipole and spin-isospin resonances, the protons and neutrons are excited out of phase, and the proton-neutron interaction provide the restoring force."

"The spectrum of gamma-radiation emitted by a highly excited nucleus can be calculated in two ways. In the first method the transition probability for gamma emission is related to the photon absorption cross-section by detailed balance. The second method relies on the fact that an excited hot nucleus has thermal fluctuations. In particular it has a fluctuating dipole moment which produces thermal radiation. The two methods are closely related and in both cases the spectrum of the radiation emitted is dominated by the giant dipole resonance. The equivalence of the detailed balance and thermal radiation theories can be demonstrated explicitly for a coupled oscillator model of the giant resonance."

"A powerful method to study the properties of a system is to subject it to a weak external perturbation and to examine its response. For the atomic nucleus subjected to the absorption of a photon or to the scattering of a particle (electron, proton, etc.) the response is ... a function of the energy and linear momentum transferred to the system. ... Up to about 10 MeV the nucleus responds through the excitation of relatively simple states often involving only one or a few particles. In the energy range between 10 and 30 MeV the system response exhibits broad resonances. These are the giant resonances ... Giant resonances correspond to a collective motion involving many if not all the particles in the nucleus. The occurrence of such a collective motion is a common feature of many-body quantum systems. In quantum-mechanical terms the resonance corresponds to a transition between the ground state and the collective state and its strength is described by a transition amplitude. Intuitively it is clear that the strength of the transition will depend on the basic properties of the system such as the number of particles participating in the response and the size of the system. This implies that the total transition strength should be limited by a sum rule which depends 'only' on ground-state properties. If the transition strength of an observed resonance exhausts a major part, say greater than 50%, of the corresponding sum rule we call it a giant resonance."

"Maurice Goldhaber has emphasized that the situation with respect to possible nuclear resonances in (γ,n) or (γ,fission) reactions was quite unclear at the time of George C. Baldwin and G. Stanley Klaiber’s papers on these reactions. ... This was because the rapid rise of their yield to a prominent peak with increasing energy, followed by a slower fall off was then thought to have been due to the competition between the rapidly rising density of nuclear states and the eventual domination of other reaction channels at higher energies. Goldhaber realized, however, that there could be an analogy between a possible collective nuclear resonance and the restrahl resonance (essentially the transverse optical phonon mode) in polar crystals. Goldhaber sought out Teller because of his paper with Russell Lyddane and Robert Sachs, ... relating the restrahl frequency to the asymptotic behavior of the crystal’s dielectric function. Goldhaber and Teller, in their paper together, went on to predict universal, giant photo-nuclear resonances. ..."

"... In my view, after more than 25 years, the preferred solution to the strong CP problem still remains the idea that the Standard Model has an additional U(1)PQ symmetry. Such a solution, necessarily, predicts the existence of a concomitant axion."

"… Remember that quark masses arise in the Standard Model because the Higgs field has a non-zero vacuum value. Roberto and I saw that one could add an additional symmetry to the theory in such a way that it is automatic that the vacuum energy is minimized for θeffective = 0. Technically this new global U(1) symmetry is not quite an exact symmetry. Like the strong CP symmetry itself, it is a pseudo-symmetry, broken only by non-perturbative or instanton (tunneling) effects. … The additional pseudo-symmetry has a consequence, as was pointed out by Weinberg … and Wilczek … namely that there is an additional pseudo-Goldstone boson, now known as the axion, associated with it. The fact that Roberto and I did not notice this obvious phenomenological consequence of our model shows that we were focused on the general solution to the strong CP problem. I, at least, was so happy to find a general solution to that that I did not stop to examine other phe- nomenological implications of the model we built to demonstrate the idea before we published it. But the axion implication is common to all such models, for it arises from the symmetry itself."

"It is pointed out that a global U(1) symmetry, that has been introduced in order to preserve the parity and time-reversal invariance of strong interactions despite the effects of instantons, would lead to a neutral pseudoscalar boson, the "axion," with mass roughly of order 100 keV to 1 MeV."

"There seems to be a vast landscape of possible universes. ... We live in one in which life is possible, but if the universe were only slightly different, beings like us could not exist. What are we to make of this fine-tuning? Is it evident that the universe, after all, was designed by a benevolent creator? Or does science offer a different explanation?"

"The old cosmological constant problem is to understand why the is so small; the new problem is to understand why it is comparable to the present mass density. ... does not help with either; anthropic considerations offer a possibility of solving both. In theories with a that takes random initial values, the anthropic principle may apply to the cosmological constant, but probably to nothing else."

"Once one starts to admit anthropic interpretations of fine-tuning problems like the cosmological constant, it is clear that such a proposal might be made for other fine-tuning problems, such as the problem of the Higgs boson mass. Certainly, we would not be here if the Higgs boson mass, and hence also the and and and masses, were greatly bigger. If they were near the , for example, any collection of more than a few elementary particles would collapse into a Black Hole. More generally, if the elementary particle masses were scaled up by a factor N, the number of elementary particles in a star or planet would scale down like N–3, and for very modest N the stars would stop shining."

"I remember a lunch in which Schwinger began by saying to Weisskopf, “Now I will make you a world.” The “world” was written down on a few paper napkins, one of which I saved. In any event, one of the things that he said, which has stuck with me ever since, was that scalar particles were the only ones that could have nonvanishing vacuum expectation values. He then went on to say that if you couple one of these to a fermion \Psi by a of the form \Phi \overline \Psi\Psi, then this vacuum expectation value would act like a mass. This sort of coupling is how mass generation is done in principle for the fermions. All particles in this picture would acquire their masses from the vacuum."

"A new conceptual foundation for Tμν on locally flat —to obtain the so-called Casimir effect—is presented. The Casimir ground state is viewed locally as a (nonvacuum) state on Minkowski space-time and the expectation value of the normal-ordered is taken. The same ideas allow us to treat, for the first time, self-interacting fields for arbitrary mass in —using traditional flat-space-time renormalization theory. First-order results for zero-mass λφ4 theory agree with those recently announced by . We point out the crucial role played by the simple renormalization condition that the vacuum expectation value of Tμν must vanish in Minkowski space-time, and in a critical discussion of other approaches, we clarify the question of renormalization ambiguities for Tμν in curved space-times."

"Vacuum expectation values of products of neutral operators are discussed. The properties of these distributions arising from , the absence of states and the of the scalar product are determined. The vacuum expectation values are shown to be s of s. Local commutativity of the field is shown to be equivalent to a symmetry property of the analytic functions. The problem of determining a theory of a neutral scalar field given its vacuum expectation values is posed and solved."

"The fundamental scientific purpose of the LHC is to explore the inner structure of matter and the forces that govern its behavior, and thereby understand better the present content of the Universe and its evolution since the Big Bang, and possibly into the future. The unparalleled high energy of the LHC, which is designed to be 7 TeV per proton in each colliding beam, and its enormous collision rate, which is planned to attain about a billion collisions per second, will enable the LHC to examine rare processes occurring at very small distances inside matter. It will be a microscope able to explore the inner structure of matter on scales an order of magnitude smaller than any previous collider. The energies involved in the proton-proton collisions will be similar to those in particle collisions in the first trillionth of a second of the history of the Universe. By studying these processes in the laboratory, the LHC experiments will, in a sense, be looking further back into time than is possible with any telescope."

"… The relied not on the detection of photon pairs with a certain energy but on the detection of more of those pairs than expected. That reliance on probabilities is why the L.H.C. and other major collider experiments often have independent teams, working with separate detectors, analyzing the same types of collisions—to avoid biasing each other. It is also the reason for the ."

"With the discovery of the Higgs boson, the next burning question at the LHC is why its mass is so low. Nobody knows the answer to that question, but it is definitely the next hot topic for LHC physicists ..."

"On July 4, scientists working with data from ongoing experiments at the Large Hadron Collider (LHC) announced the discovery of a new particle "consistent with" the Higgs boson — a subatomic particle also colloquially referred to as the "God particle." After years of design and construction, the LHC first sent protons around its 27 kilometer (17 mile) underground tunnel in 2008. Four years later, the LHC's role in the discovery of the Higgs boson provides a final missing piece for the Standard Model of Particle Physics — a piece that may explain how otherwise massless subatomic particles can acquire mass. Gathered here are images from the construction of the massive $4-billion-dollar machine that allowed us peer so closely into the subatomic world."

"The origins of es is one of the biggest mysteries in modern physics since they are beyond the realm of the Standard Model. As massive particles, neutrinos undergo throughout their propagation. In this paper we show that when a neutrino oscillates from a flavor state α to a flavor state β, it follows three possible paths consistent with the Quantum Yang- Baxter Equations. These trajectories define the transition probabilities of the oscillations. Moreover, we define a probability matrix for flavor transitions consistent with the Quantum Yang-Baxter Equations, and estimate the values of the three neutrino mass eigenvalues within the framework of the triangular formulation."

"It has been known for some time that the Yang-Baxter equations can be solved using s. More recently it was discovered … that the YBE for the N state could be solved using special curves of (N – 1)2."

"About 40 years ago, in the study of quantum s … , in particular in the framework of the … , new algebraic structures arose, the generalizations of which were later called quantum groups … The Yang-Baxter equations became a unifying basis of all these investigations. The most important nontrivial examples of quantum groups are quantizations (or deformations) of ordinary classical s and algebras (more precisely, one considers the deformations of the algebra of functions of a Lie group and the universal enveloping of a Lie algebra). The quantization is accompanied by the introduction of an additional parameter q (the deformation parameter), which plays a role analogous to the role of in quantum mechanics. In the limit q → 1, the quantum groups and algebras go over into the classical ones."

"At an early stage the Yang-Baxter equation (YBE) appeared in several different guises in the literature, and sometimes its solutions have preceded the equation. One can trace basically three streams of ideas from which YBE has emerged: the , commuting in statistical mechanics, and factorizable in field theory."

"That the pion-nucleon (πN) coupling constant is fundamental in our understanding of the Cosmos has been adequately emphasised in numerous works. In meson-exchange models of the , a significantly weaker coupling between the pions and the nucleons would have prevented the s from combining fast with s in the ; they would have decayed before they had any chance to be enmeshed first in s, then in other light nuclei. According to the , within half an hour of the Big Bang, all existing matter had assumed the form of free electrons, protons, and helium nuclei (as well as traces of other nuclei up to 7). On the contrary, a significantly stronger coupling would have resulted in the rapid creation of bound diprotons and would have led to a helium-dominated Universe. It is hard to imagine how life could emerge in such a Universe: typical stars burn hydrogen to helium for about 90 % of their lives."

"... Roughly speaking, in renormalizable theories no coupling constants can have the dimensions of negative powers of . But every time we add a field or a space-time derivative to an interaction, we reduce the dimensionality of the associated coupling constant. So only a few simple types of interaction can be renormalizable. ..."

"… what is the wave function? Is it a complete and comprehensive representation of the world? Or do we need additional physical quantities to fully capture reality, as Albert Einstein and others suspected? Or does the wave function have no direct connection with reality at all, merely characterizing our personal ignorance about what we will eventually measure in our experiments? Until s definitively answer these questions, they can’t really be said to understand quantum mechanics — thus Feynman’s lament. Which is bad, because quantum mechanics is the most fundamental theory we have, sitting squarely at the center of every serious attempt to formulate deep laws of nature. If nobody understands quantum mechanics, nobody understands the universe."

"What is the empirical content of quantum mechanics? Or how does the wave function assigned to a relate to the results of measurements on the system? The well-known connection rule is the , which has been precisely tested by experiments. It says that a (projective) measurement of an A on a system with the wave function |ψ⟩ will randomly obtain one of the of A, and the probability of obtaining an eigenvalue ai is given by |⟨ai|ψ⟩|2, where |ai⟩ is the corresponding to the eigenvalue ai."

"... was formalized by a number of authors ... as the requirement that the wave function of a multi-electron system is in the coordinates, and , of all the electrons. This principle was incorporated into statistical mechanics by Enrico Fermi ... and Dirac, ... and for this reason particles obeying the exclusion principle are generally called 's,' just as particles like photons for which the wave function is symmetric and which obey the statistics of Bose and Einstein are called 's.'"