Theoretical Physics

"Quantum electrodynamics is studied analytically in a quenched, planar approximation in four dimensions. At sufficiently strong coupling, chiral symmetry is broken spontaneously and the corresponding pseudoscalar Goldstone boson is observed. This phase of the theory is governed by a novel ultraviolet fixed point which requires the mixing of four-fermion interactions with the electrodynamic interactions."

"The observed is assumed to be due to the spontaneous symmetry-breaking mechanism; the Lagrangian is CP invariant but its particular solution is not. The general classification of such theories when coupled with different unified gauge models of the weak and electromagnetic interactions is given. All such theories lead naturally to a basically milliweak CP noninvariant solution. The possibility that for most weak transitions the result may resemble a superweak theory is analysed, and possible experiments to distinguish these two different types of theories are discussed. Detailed calculations for various CP violating amplitudes are carried out for a generalized Georgi-Glashow model."

"The secret of nature is symmetry, but much of the texture of the world is due to mechanisms of symmetry breaking. The are a variety of mechanisms wherein the symmetry of nature can be hidden or broken. The first is explicit symmetry breaking where the is only approximately symmetric, but the magnitude of the symmetry breaking forces is small, so that one can treat the symmetry violation as a small correction. Such approximate symmetries lead to approximate conservation laws. Many of the symmetries observed in nature are of this sort, not really symmetries of the laws of physics at all, but—for what appears sometimes to be accidental reasons—approximate symmetries for a certain class of phenomena. The isotopic symmetry of the is an example of an approximate symmetry; good due to the small values of the and masses and the weakness of the electromagnetic force. A more profound way of hiding symmetry is the phenomenon of spontaneous symmetry breaking. Here the laws of physics are symmetric but the state of the system is not. This situation is common in . The is an example of a solution of Newton’s equations that is not rotationally invariant, although the equations are. Consequently, for an observer of the solar system, the rotational invariance of the law of gravitation is not manifest."

"Will we ever derive the necessary mathematical tools to analytically demonstrate from first principles that confinement is indeed a mathematical property of quantum chromodynamics? This is the million-dollar question, literally. The Clay Mathematics Institute has announced a million-dollar prize for a rigorous mathematical proof that quantum chromodynamics does not allow free quarks or gluons to be produced. While no claimants to the prize have yet come forward, we nevertheless have strong indirect support of this idea, coming not only from experimental observations, but also from numerical simulations that closely approximate the complicated interactions in quantum chromodynamics. This is heartening, if not definitive. We still have to confirm that it is some property of the theory and not of the computer simulation. However, for physicists, if not mathematicians, this seems pretty convincing."

"We investigate the possibility that radiative corrections may produce spontaneous symmetry breakdown in theories for which the semiclassical (tree) approximation does not indicate such breakdown. The simplest model in which this phenomenon occurs is the electrodynamics of massless s. We find (for small coupling constants) that this theory more closely resembles the theory with an imaginary mass (the Abelian Higgs model) than one with a positive mass; spontaneous symmetry breaking occurs, and the theory becomes a theory of a massive vector meson and a massive scalar meson."

"In parallel with the changes it brought in our attitude toward symmetries, the birth of the Standard Model marked changes also in our attitude toward quantum field theory. After 't Hooft's breakthrough in 1971, it became clear that the old problem of infinities in the weak interactions had been solved by the use of spontaneous symmetry breaking to give masses to the W and Z particles. Then the asymptotic freedom of quantum chromodynamics gave us a framework in which we could actually calculate something about the strong interactions - not everything, but at least something. But in scoring these victories, quantum field theory was preparing the way for a further change in our attitude, in which quantum field theory would lose its central position."

"Among the many significant ideas and developments that connect Mathematics with contemporary Physics one of the most intriguing is the role that Quantum Field Theory (QFT) plays in Geometry and Topology. We can argue back and forth on the relevance of such a role, but the perspective QFT offers is often surprising and far reaching. Examples abound, and a fine selection is provided by the revealing insights offered by Yang–Mills theory into the topology of 4-manifolds, by the relation between Knot Theory and topological QFT, and most recently by the interaction between Strings, Riemann moduli space, and enumerative geometry. Doubtless many of the most striking connections suggested by physicists failed to pass the censorship of the Department of Mathematics, and so do not appear in the above official list."

"Starting in the late 1960s and then accelerating in the 1970s, our optimism about quantum field theory once again returned with the advent of the Standard Model — first of electromagnetic and weak interactions and then chromodynamics for the strong interactions. And quantum field theory became the tool of choice for doing calculations in particle physics."

"The disillusionment with QFT as a basis for the theory of elementary particles was also premature. What was missing was many ingredients, including the identification of the underlying gauge symmetry of the weak interactions, the concept of spontaneous symmetry breaking that could explain how this symmetry was hidden, the identification of the fundamental constituents of the nucleons as colored quarks, the discovery of asymptotic freedom which explained how the elementary colored constituents of hadrons could be seen at short distances yet evade detection through confinement, and the identification of the underlying gauge symmetry of the strong interactions. Once these were discovered, it was but a short step to the construction of the standard model, a gauge theory modeled on QED, which opened the door to the understanding of mesons and nucleons."

"The real world as we know it happens at energies well below the Planck scale, so it is very well described by effective field theory. There is a continuous infinity of consistent effective field theories. Remarkably, only a measure zero fraction of those seem to be obtainable from string theory. These effective field theories arise as low energy descriptions of certain "vacua" of string theory, which in some approximation schemes can be thought of as solutions to the equations of motion for the compactification space."

"Unlike a Feynman graph, which is divided into different lines, which can represent particles of different types with different masses and spins, any part of a string world sheet is equivalent to any other so "there is only one string." Whatever particles there are going to be represent different states of vibration of one basic string. Also there are not any vertices in the string world sheet so we do not have the freedom to tell the string how to interact."

"String theory is not like anything else ever discovered. It is an incredible panoply of ideas about math and physics, so vast, so rich you could say almost anything about it."

"I feel that we are so close with string theory that—in my moments of greatest optimism—I imagine that any day, the final form of the theory may drop out of the sky and land in someone's lap. But more realistically, I feel that we are now in the process of constructing a much deeper theory than anything we have had before and that well into the twenty-first century, when I am too old to have any useful thoughts on the subject, younger physicists will have to decide whether we have in fact found the final theory."

"Generally speaking, all the really great ideas of physics are really spin-offs of string theory... Some of them were discovered first, but I consider that a mere accident of the development on planet earth. On planet earth, they were discovered in this order [general relativity, quantum field theory, superstrings, and supersymmetry]... But I don't believe, if there are many civilizations in the universe, that those four ideas were discovered in that order in each civilization."

"It's been said that string theory is part of the physics of the twenty-first century that fell by chance into the twentieth century. That's a remark that was made by a leading physicist about fifteen years ago. ...String theory was invented essentially by accident in a long series of events, starting with the Veneziano model... No one invented it on purpose, it was invented in a lucky accident. ...By rights, string theory shouldn't have been invented until our knowledge of some of the areas that are prerequisite... had developed to the point that it was possible for us to have the right concept of what it is all about."

"I would expect that a proper elucidation of what string theory really is all about would involve a revolution in our concepts of the basic laws of physics - similar in scope to any that occurred in the past."

"String theory at its finest is, or should be, a new branch of geometry. ...I, myself, believe rather strongly that the proper setting for string theory will prove to be a suitable elaboration of the geometrical ideas upon which Einstein based general relativity."

"It's possible that the true structures in the underlying theory are something quite different — and the obvious candidate is strings. That direction, I think, is certainly worth working on. We don't whether that's the correct direction."

"From the beginning it was clear that, despite its successes, the Standard Model of elementary particles would have to be embedded in a broader theory that would incorporate gravitation as well as the strong and electroweak interactions. There is at present only one plausible candidate for such a theory: it is the theory of strings, which started in the 1960s as a not-very-successful model of hadrons, and only later emerged as a possible theory of all forces."

"... all of these caveats really work only against the idea that the final theory of nature is a quantum field theory. They leave open the view, which is in fact the point of view of my book, that although you can not argue that relativity plus quantum mechanics plus cluster decomposition necessarily leads only to quantum field theory, it is very likely that any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory. Picking up a phrase from Arthur Wightman, I’ll call this a folk theorem. At any rate, this folk theorem is satisfied by string theory, and we don’t know of any counterexamples."

"The most recent chapter in our new understanding of nonperturbative effects in string theory has been the incorporation of unstable branes and open string tachyons into the overall framework of the theory. It has turned out that an understanding of unstable D-branes is necessary to properly describe all D-branes. This is natural from the point of view of K-theory, where brane configurations which are equivalent under the annihilation of unstable branes are identified ... The long-mysterious tachyon instability of open string theory has finally been given a physical interpretation: it is the instability of the D-brane that supports the existence of open strings. The instability disappears in the tachyon vacuum, in which the D-brane decays. Moreover, the belief that D-branes are solitonic solutions of string theory has been confirmed: starting with the appropriate tachyonic field theory of unstable space-filling branes, one can describe lower dimensional D-branes as solitonic solutions. Lower dimensional D-branes are thereby essentially obtained as solitons of the tachyon field theory, so, in some sense, lower-dimensional D-branes can be thought of as being made of tachyons! It has also been shown that the physics of unstable D-branes is captured by string field theory, thus making it a candidate for a non-perturbative formulation of string theory capable of describing changes of the string background."

"Consistent, relativistic string theories had already been written down in two, ten or twenty-six dimensions (the last being relevant only to bosonic strings) in the 1970s. A closed string is a loop which replaces a spacetime point. Its quantum oscillations correspond to particles of higher spins and higher masses, which may be arranged in a linear trajectory in a spin-versus-mass ... (Regge) plot. If the slope parameter of this trajectory — the only parameter in the theory — is adjusted to equal the Newtonian constant, one can show, quite miraculously, that in the zeroth order of the closed bosonic string there emerges from the string theory Einstein's gravity in its fullness! (The higher orders give modifications to Einstein's theory, with corrections which have a range of length = 10–33 cm.)"

"One can ask whether the situation today in string theory is really as favorable as it was for field theory in the early 60's. It is difficult to know. Then, of course we had many more experiments to tell us how quantum field theories actually behave. To offset that, we have today more experience and greater mathematical sophistication."

"String theory is, in fact, a textbook case of a Deceitful Turkey, a beautiful set of ideas that will always remain just barely out of reach. Far from a wonderful technological hope for a greater tomorrow, it is instead the tragic consequence of an obsolete belief system—in which emergence plays no role and dark law does not exist."

"I have no idea whether the properties of the universe as we know it are fundamental or emergent, but I believe that the mere possibility of the latter should give the string theorists pause, for it would imply that more than one set of microscopic equations is consistent with experiment — so that we are blind to these equations until better experiments are designed — and also that the true nature of the microscopic equations is irrelevant to our world."

"We actually have a candidate for the mind of God. The mind of God we believe is cosmic music, the music of strings resonating through 11 dimensional hyperspace. That is the mind of God."

"The number 24 appearing in Ramanujan's function is also the origin of the miraculous cancellations occurring in string theory. ...each of the 24 modes in the Ramanujan function corresponds to a physical vibration of a string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highly sophisticated mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan. ...The string vibrates in ten dimensions because it requires... generalized Ramanujan functions in order to remain self-consistent."

"Question 5. String Phenomenology Here there are many questions that can all be summarized by asking whether one can construct a totally realistic four-dimensional model which is consistent with string theory and agrees with observation?"

"To build matter itself from geometry — that in a sense is what string theory does. It can be thought of that way, especially in a theory like the heterotic string which is inherently a theory of gravity in which the particles of matter as well as the other forces of nature emerge in the same way that gravity emerges from geometry. Einstein would have been pleased with this, at least with the goal, if not the realization. … He would have liked the fact that there is an underlying geometrical principle — which, unfortunately, we don’t really yet understand."

"Now, this change from point-particles to strings that are so small they look like points might not sound like it would accomplish much. But it does. Superstring theory successfully merges general relativity and quantum mechanics."

"Currently, string theorists are in a position analogous to an Einstein bereft of the equivalence principle. ...[A] central organizing principle that embraces ...all ...features of the theory within one overarching and systematic framework ...is still missing."

"String theory... resolves the central dilemma confronting contemporary physics—the incompatibility between quantum mechanics and general relativity—and that unifies our understanding of all of nature's fundamental material constituents and forces. But to accomplish these feats, ...string theory requires that the universe have extra space dimensions. ... Physicists have found that a key signal that a quantum mechanical theory has gone haywire is that particular calculations yield "probabilities" that are not within... acceptable range. For instance... infinite probabilities. ...string theory cures these infinities. ...a residual ...problem remains. In the early days ...calculations yielded negative probabilities ...so string theory appeared to be awash in its own quantum-mechanical hot water. ... Physicists found that the troublesome calculations were highly sensitive to the number of independent directions to which a string can vibrate. ...if strings could vibrate in nine independent spatial directions, all of the negative probabilities would cancel out. ... Kaluza and Klein provide a loophole... in addition to our familiar three... there are six other curled-up... rather than just postulating the existence of extra dimensions, as had been done by Kaluza, Klein, and their followers, string theory requires them."

"String theorists, of course, continue to do whatever it is that string theorists do."

"... one of the main beefs with the string theory is that it is so flexible you can get almost anything out of it. ... String theorists themselves are not too happy about it."

"... within , string theory is a complex subject. There are few large particle theory groups within major universities that do not have at least one person doing string theory. In total, there are probably a couple of thousand people at universities around the world whose mortgage is paid either by doing string theory, using string theory, working with tools made within string theory, solving problems using methods developed in string theory, or simply having their mental map of the world at the set by string theory."

"Is string theory a futile exercise as physics, as I believe it to be? It is an interesting mathematical specialty and has produced and will produce mathematics useful in other contexts, but it seems no more vital as mathematics than other areas of very abstract or specialized math, and doesn't on that basis justify the incredible amount of effort expended on it."

"Superstring theory starts off by proposing a new answer to an old question: what are the smallest, indivisible constituents of matter? For many decades, the conventional answer has been that matter is composed of particles... that can be modeled as dots that are indivisible and that have no size and no internal structure. Conventional theory claims, and experiments confirm, that these particles combine in various ways to produce protons, neutrons, and a wide variety of atoms and molecules... Superstring theory tells a different story. ...it does claim that these particles are not dots. Instead... every particle is composed of a tiny filament of energy, some hundred billion billion times smaller than a single atomic nucleus, which is shaped like a string. And just as a violin string can vibrate in different patterns, each of which produces a different musical tone, the filaments of superstring theory can also vibrate in different patterns. But these vibrations... produce different particle properties. ...All species of particles are unified in superstring theory since each arises from a different vibrational pattern executed by the same underlying entity."

"The notion of a smooth spatial geometry, the central principle of general relativity, is destroyed by the violent fluctuations of the quantum world on short distance scales. ...The equations of general relativity cannot handle the rolling frenzy of the quantum foam. ... Physicists have made numerous attempts at modifying either general relativity or quantum mechanics in some manner so as to avoid the conflict, but the attempts... have been met with failure after failure. That is, until the discovery of superstring theory."

"Of all the fields in fundamental physical theory, the gravitational field is picked out as controlling, in Einsteinian fashion, the structure of space-time. This is true even in a unified description of all the fields and all the particles of nature. Today, in superstring theory, we have the first respectable candidate for such a theory, apparently finite in perturbation theory and describing, roughly speaking, an infinite set of local fields, one of which is the gravitational field linked to the metric of space-time. If all the other fields are dropped, the theory becomes an Einsteinian theory of gravitation."

"I don’t like that they’re not calculating anything. I don’t like that for anything that disagrees with an experiment, they cook up an explanation – a fix-up to say 'Well. it still might be true'. For example, the theory requires ten dimensions. Well, maybe there's a way of wrapping up six of the dimensions. Yes, that's possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there's no reason whatsoever in superstring theory that it isn't eight of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn't produce anything; it has to be excused most of the time. It doesn't look right."

"I have noticed when I was younger, that lots of old men in the field couldn't understand new ideas very well, and resisted them with one method or another, and that they were very foolish in saying these ideas were wrong — such as Einstein not being able to take quantum mechanics. I’m an old man now, and these are new ideas, and they look crazy to me, and they look like they’re on the wrong track. Now I know that other old men have been very foolish in saying things like this, and, therefore, I would be very foolish to say this is nonsense. I am going to be very foolish, because I do feel strongly that this is nonsense! I can’t help it, even though I know the danger in such a point of view. So perhaps I could entertain future historians by saying I think all this superstring stuff is crazy and is in the wrong direction."

"What philosophical conclusions should we draw from the abstract style of the superstring theory? We might conclude, as Sir James Jeans concluded long ago, that the Great Architect of the Universe now begins to appear as a Pure Mathematician, and that if we work hard enough at mathematics we shall be able to read his mind. Or we might conclude that our pursuit of abstractions is leading us far away from those parts of the creation which are most interesting from a human point of view. It is too early yet to come to conclusions."

"Imagine, if you can, four things that have very different sizes. First, the entire universe. Second, the planet Earth. Third, the nucleus of an atom. Fourth, a superstring. The step in size from each of these things to the next is roughly the same... twenty powers of ten...."

"Scientifically speaking, a butterfly is at least as mysterious as a superstring. When something ceases to be mysterious it ceases to be of absorbing interest to scientists. Almost all things scientists think and dream about are mysterious."

"Superstring theories demand our spacetime dimension to be 10, which means we should reduce them to an effectively 4-dimensional theory. The standard solution of string compactification, as a generalization of Kaluza-Klein compactification, renders the extra six dimensions Calabi-Yau (CY). Thus, the study of Calabi-Yau and algebraic geometry has entered the field of theoretical physics."

"Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space."

"If our world-line and that of the body which is being observed are parallel, the body is said to be at rest. But if the two world-lines are not parallel, then, when interpreting things in terms of space and time, the body will be said to be in relative motion."

"Consider two observers, one... moving uniformly along a straight line, the other [stationary] on the embankment. At the precise instant these two observers pass each other at a point P, a flash of light is produced at the point P. The light wave produced by the instantaneous flash will present the shape of an expanding sphere. Since the invariant velocity of light holds equally for either observer, we must assume that either observer will find himself at all times situated at the centre of the expanding sphere. Our first reaction might be to say: "What nonsense! How can different people, travelling apart, all be at the centre of the same sphere?" Our objection, however, would be unjustified."

"The most precise experiments have proved the correctness of the Einsteinian laws of mechanics and...Bucherer's experiment proving the increase in mass of an electron in rapid motion is a case in point. Very important differences distinguish the theory of Einstein from that of Lorentz. Lorentz also had deduced from his theory that the mass of the electron should increase and grow infinite when its speed neared that of light; but the speed in question was the speed of the electron through the stagnant ether; whereas in Einstein's theory it is merely the speed with respect to the observer. According to Lorentz, the increase in mass of the moving electron was due to its deformation or Fitzgerald contraction. The contraction modified the lay of the electromagnetic field round the electron; and it was from this modification that the increase in mass observed by Bucherer was assumed to arise. In Einstein's theory, however, the increase in mass is absolutely general and need not be ascribed to the electromagnetic field of the electron in motion. An ordinary unelectrified lump of matter like a grain of sand would have increased in mass in exactly the same proportion; and no knowledge of the microscopic constitution of matter is necessary in order to predict these effects, which result directly from the space and time transformations themselves. Furthermore, the fact that this increase in mass of matter in motion is now due to relative motion and not to motion through the stagnant ether, as in Lorentz's theory, changes the entire outlook considerably. According to Lorentz, the electron really increased in mass, since its motion through the ether remained a reality. According to Einstein, the electron increases in mass only in so far as it is in relative motion with respect to the observer. Were the observer to be attached to the flying electron no increase in mass would exist; it would be the electron left behind which would now appear to have suffered the increase. Thus mass follows distance, duration and electromagnetic field in being a relative and having no definite magnitude of itself and being essentially dependent on the conditions of observation. Owing to the general validity of the Lorentz-Einstein transformations, it becomes permissible to apply them to all manner of phenomena.. ...temperature, pressure and many other physical magnitudes turned out to be relatives. ...entropy, electric charge and the velocity of light in vacuo were absolutes transcending the observer's motion. ...a number of other entities are found to be absolutes, the most important of which is that abstract mathematical quantity called the Einsteinian interval, which plays so important a part in the fabric of the new objective world of science, the world of four-dimensional space-time."

"Einstein's definition... does not differ in spirit from the definitions in classical science; its sole advantage is that it entails a minimum of assumptions, and is susceptible of being realised in a concrete way permitting a high degree of accuracy in our measurements. Einstein's definition, is then, as follows: If we consider a ray of light passing through a Galilean frame, its velocity in the frame will be the same regardless of the relative motion of the luminous source and frame, and regardless of the direction of the ray. ...when it was found that contrary to the anticipations of classical science not the slightest trace of anisotropy could be detected even by ultra-precise experiment, the objections which classical science may might have presented... lost all force. ...ether drift appeared to exert no influence one way or the other. ... Isotropy signifies that the velocity of light is the same in all directions. And how can we ascertain the equality of a velocity in all directions when we do not yet know how to measure time? Experimenters solved the difficulty by appealing to the observation of coincidences. ... Waves of light leaving the centre of a sphere simultaneously are found to return to the centre also in concidence, after having suffered a reflection against a highly polished inner surface of the sphere. ...the light waves have thus covered equal distances in the same time; whence we conclude that their speed is the same in all directions. Inasmuch as this experiment has been performed, yielding the results we have just described, even though the ether drift caused by the earth's motion should have varied in direction and intensity, the isotropy of space to luminous propogations was thus established. (The experiment described constitutes but a schematic form of Michelson's.) It is to be noted that in this experiment the observation of coincidences is alone appealed to (even spatial measurements can be eliminated). This is because in Michelson's experiment it is not necessary to consider a sphere. The two arms of the apparatus may be of different lengths; and all that is observed is the continued coincidence of the interference-bands with markings on the instrument. When it is realised that coincidences constitute the most exact form of observation, we understand why it is that Einstein's definition is justified"