Theoretical Physics

"Ten years ago, it was perfectly respectable to speculate that there was no such thing as dark matter, just a modification of gravity. (It couldn’t have been MOND alone, which was ruled out by clusters, but it could have been some more elaborate modification.) That’s no longer true. The Bullet Cluster and the CMB both provide straightforward evidence that there is gravity pointing in the direction of something other than the ordinary matter. The source for that gravity is “dark matter.” It could be simple, like an axion or a thermal relic, or it could be quite baroque, like TeVeS + sprinkles of other dark matter as required, but it’s definitely there."

"Mild failures aside, it is clear that there is a broad range of masses, 106 — 1011 M☉, in which systems adhere to MOND in their systematics. This must be telling us something; logically there are the following possibilities. a) MOND is merely an efficient summary of the way DM is distributed in the said systems? b) MOND reveals the dependence of inertia on acceleration for small accelerations? c) MOND betrays hitherto unknown forces particularly effective at astronomical scales? d) MOND encapsulates departures from standard Newtonian-Einsteinian gravity theory at the mentioned scales?"

"Today we can probe regions of physics where space-time curvature is extremely small finding that, again, Newtonian mechanics fails. This may mean that the Theory of General Relativity needs an extension or that we do not yet understand what “space-time” and “mass” are nor how they are fundamentally related. Perhaps it just boils down to the problem of us not understanding the vacuum."

"In matter of fact, whether MOND is a fundamental theory or not, the very special and central role the constant a0 plays in galaxy dynamics is well established and is here to stay. For instance, you will find it everywhere in the data itself. All round systems, from giant molecular clouds, through globular clusters and elliptical galaxies, to clusters of galaxies lie, in the mass-radius plane, near the line with constant M/R2. The value of this ratio when multiplied by G gives a0. Another example: the baryonic Tully-Fisher relation agrees well (over many orders of magnitude in mass) with a relation of the form α M = V–4. The proportionality constant α has dimensions of G times acceleration and when divided by G gives a0 (this is independent of the previous appearance as it refers to asymptotic regions in the galaxies)."

"I've had conversations about MOND with several of the most imaginative theorists I know. Often it went like this: We would be talking about some sober mainstream problem and one of us would mention galaxies. We would look at each other with a glint of recognition and one of would say, "So you worry about MOND, too," as if admitting a secret vice. Then we would share our crazy ideas — because all ideas about MOND that are not immediately wrong turn out to be crazy."

"MOND is so successful that, as a minimum, it is telling us the exact functional form of the force in galaxies. Any theory of galaxy and structure formation must therefore be able to reproduce the MOND phenomenology."

"... Milgrom and those few who work on it, are quite aware of the pressing need to have a fully consistent theory that goes beyond the Newtonian non-relativistic limit to a theory that can be applied to cosmology. They don't have one. They fully admit it and they agree that this is a big gap, big lack in the theory. There it is. They do insist that on the scales of galaxies and smaller where it is intended to apply it works remarkably well, and they're right. There are just a few people working on this theory. The most active of the young people is Stacy McGaugh at the University of Maryland. If you ever get a chance you might be amused to talk to him."

"Fundamental to the idea of MOND is that it is an 'effective' theory, playing a role similar to Kepler's laws (as stressed by Felten 1984). The proponents of MOND have yet to develop the analogue of Newtonian mechanics to explain the effective theory. The absence of a full theory seriously limits the predictive power of MOND, and leads various authors to disagree as to what the observational consequences of this revision will be."

"The observed Higgs mass is compatible with supersymmetry only if the superpartners are quite heavy (tens of TeV) or under special circumstances."

"Without SUSY, there is nothing like a chiral symmetry to protect scalar masses from heavy mass scales. But with SUSY, the chiral symmetry in the fermionic sector protects the scalars too."

"... the discovery of supersymmetry ... would be the first extension of our notions of spacetime since Einstein."

"Edward Witten as quoted by Hirosi Ooguri in (p. 483)"

"... I knew quantum field theory well enough to know that saying that the potential energy for the scalar field is zero is not a meaningful statement quantum mechanically. If it were, we would not have a gauge hierarchy problem in particle physics. ... with supersymmetry the mass renormalization (and even the full effective potential) of a scalar can be zero."

"SUSY is a unique new symmetry that relates s to s, in a sense explaining why fermions exist. Relating bosons to fermions also makes it possible to explain the smallness of the Higgs mass, since we do know why the smallness of fermion masses can be natural. So that is at least the germ of how SUSY solves the fine-tuning problem."

"… Supersymmetry … the virtues: * SUSY can make a “small” Higgs mass natural; • SUSY is part of a larger vision of physics, not just a technical solution; • the measured value of favors SUSY ’s; • SUSY survives tests; and • the mass has turned out to be heavy, as needed for electroweak symmetry breaking in the context of SUSY."

"... as a very rough analogy, supersymmetric quantum theory is to ordinary quantum theory as differential forms on a manifold are to functions on a manifold. A very large fraction of geometrical applications of quantum field theory found in the eighties and nineties depend on supersymmetry. (Examples include the supersymmetric proofs of the positive energy theorem, the Atiyah-Singer index theorem, and the Morse inequalities, and the quantum field approaches to elliptic cohomology and to Donaldson theory.) ... Surely, if supersymmetry is confirmed in accelerators, mathematical attention will be focussed on this fruitful branch of quantum field theory roughly as the discovery of general relativity focussed attention on Riemannian geometry."

"Supersymmetry is an updating of special relativity to include fermionic as well as bosonic symmetries of spacetime. In developing relativity, Einstein assumed that the spacetime coordinates were bosonic; fermions had not yet been discovered! In supersymmetry the structure of spacetime is enriched by the presence of fermionic as well as bosonic coordinates."

"The unification of forces, even if it were perfected, would leave us with two great kingdoms of particles, still not unified. Technically, these are the fermion and boson kingdoms. More poetically, we may call them the kingdoms of substance (fermions) and force (bosons). By postulating that the fundamental equations enjoy the property of supersymmetry, we heal the division of particles into separate kingdoms. Supersymmetry can be approached from several different angles, but perhaps the most appealing is to consider it as an expansion of space-time, to include quantum dimensions. The defining characteristic of quantum dimensions is that they are represented by coordinates that are Grassmann numbers (i.e., anticommuting numbers) rather than real numbers. Supersymmetry posits that the fundamental laws of physics remain invariant transformations that correspond to uniform motion in the quantum dimensions. Thus supersymmetry extends Galileo/Lorentz invariance."

"In the absence of a canonical model for why and how supersymmetry breaking occurs, the predicted consequences of supersymmetry are not sharply defined."

"Julius Wess and Jonathan Bagger: (p. 3)"

"Frank Wilczek,"

"Arkani-Hamed and Dimopoulos ... have even shown how it is possible to keep the good features of supersymmetry, such as a more accurate convergence of the SU(3) × SU(2) × U(1) couplings to a single value, and the presence of candidates for dark matter WIMPs. The idea of this “split supersymmetry” is that, although supersymmetry is broken at some very high energy, the gauginos and higgsinos are kept light by a chiral symmetry. [An additional discrete symmetry is needed to prevent lepton-number violation in higgsino-lepton mixing, and to keep the lightest supersymmetric particle stable.]"

"Steven Weinberg,"

"There is an infinite number of Lie groups that can be used to combine particles of the same spin in ordinary symmetry multiplets, but there are only eight kinds of supersymmetry in four spacetime dimensions, of which only one, the simplest, could be directly relevant to observed particles."

"Shortly after the development of four-dimensional globally supersymmetric field theories, Zumino (1975) pointed out that supersymmetry in these theories would, if unbroken, imply a vanishing vacuum energy."

"The concept of naturalness is usually cited as the underlying motivation for supersymmetry. We will challenge that concept, and in any case need to point out that there is nothing natural about the development of the theory itself. Its main success is its agility in dodging the facts. The dubious explanation of the convergence of the three scaling coupling constants into a single point can not be taken seriously. It is just another fit, using some of the many free parameters."

"Of the proposed extensions to the Standard Model, supersymmetry (SUSY) has remained among the most popular for decades. It provides exactly the needed compensation to stabilize the Higgs mass, while additionally providing an ideal candidate for dark matter with a stable weakly interacting lightest supersymmetric particle (LSP)."

"Supersymmetry is a subject of considerable interest among physicists and mathematicians. Not only is it fascinating in its own right, but there is a growing belief that it may play a fundamental role in particle physicis. This belief is based on an important result of Haag, Sohnius, and Lopuszanski, who proved that the supersymmetry algebra is the only graded Lie algebra of symmetries of the S-matrix consistent with relativistic quantum field theory."

"If dark matter is truly made of the lightest SUSY particle, then experiments designed to see it such as CDMS, XENON, Edelweiss and more should have detected it. Furthermore, SUSY dark matter should annihilate in a very particular way which hasn't been seen. Constraints on WIMP dark matter are quite severe, experimentally. The lowest curve rules out WIMP (weakly interacting massive particle) cross-sections and dark matter masses for anything located above it. This means that most models for SUSY dark matter are no longer viable."

"The mathematical consistency of string theory depends crucially on supersymmetry, and it is very hard to find consistent solutions (quantum vacua) that do not preserve at least a portion of this supersymmetry. This prediction of string theory differs from the other two (general relativity and gauge theories) in that it really is a prediction. It is a generic feature of string theory that has not yet been discovered experimentally."

"Natural SUSY expects s and s to be light. This motivated and to carry out extensive in various final states using full Run-2 data."

"It would not be an exaggeration to say that today supersymmetry dominates theoretical high energy physics. Many believe it will play the same revolutionary role in the physics of the 21st as special and general relativity did in the physics of the 20th century. This belief is based on aesthetical appeal, on indirect evidence, and the fact that no theoretical alternative is in sight."

"If the Standard Model describes the world successfully, how can there be physics beyond it, such as supersymmetry? There are two reasons. First, the Standard Model does not explain aspects of the study of the large-scale universe, cosmology. For example, the Standard Model cannot explain why the universe is made of matter and not antimatter, nor can it explain what constitutes the dark matter of the universe. Supersymmetry suggests explanations for both of these mysteries. Second, the boundaries of physics have been changing. Now scientists ask not only how the world works (which the Standard Model answers) but why it works that way (which the Standard Model cannot answer). Einstein asked "why" earlier in the twentieth century, but only in the past decade or so have the "why" questions become normal scientific research in particle physics rather than philosophical afterthoughts."

"Some theorists are led to supersymmetry because it emerges as part of the low energy theory from a superstring theory of everything. Others are particularly confident that nature will be supersymmetric because the Higgs mechanism is not an extra mechanism added ad hoc to the rest of the gauge theory, but emerges as a derived result (see the chapter by Ibáñez and Ross) if Mtop ≳ MW (which seems to be true). That is, supersymmetry can explain the ratio of the weak scale to the unification scale."

"The popular explanation of the Equilibrium-theory is very simple. If we conceive the earth to be wholly or in a great degree with water, and consider that the attraction of the moon upon different particles (according to the law of gravitation) is inversely as the square of their distance, and is therefore greatest for those particles which are nearest to it; then it will be obvious that the moon attracts the water on that side which is next to her, more than she attracts the great mass of the earth, and therefore tends to raise the water from the earth on the side next to her; but she also attracts the great mass of the earth more than she attracts the water upon the side most distant from her, and therefore tends to draw the earth from the water on the side most distant from her; which will produce exactly the same effect as if a force tended to draw the water away from the earth on that side. Thus the moon’s action tends to raise the water on two opposite sides of the earth; and similarly the sun’s action tends to raise the water on two opposite sides. The close relation, however, which the times of high water bear to the times of the moon’s passage, shows that the moon’s influence in raising the tides must be much greater than the sun's. If the sun and moon are together, as seen from the earth, the elevations produced by these two bodies will coincide in place, and will therefore be added together. Thus Spring Tides will be produced. In other relative positions of the sun and moon, it may happen that the elevation produced by the sun will occur at a place where the moon causes depression: the action of the sun there tends to counteract that of the moon, and Neap Tides will be produced."

"A Frenchman who arrives in London, finds a great alteration in philosophy, as in other things. He left the world full, he finds it empty. At Paris you see the universe composed of vortices of subtile matter, at London we see nothing of the kind. With you it is the pressure of the moon which causes the tides of the sea, in England it is the sea which gravitates towards the moon; so that when you think the moon ought to give us high water, these gentlemen believe that you ought to have low water; which unfortunately we cannot test by experience; for in order to do that, we should have examined the moon and the tides at the moment of the creation. You will observe also that the sun, which in France has nothing to do with the business, here comes in for a quarter of it. Among you Cartesians, all is done by an impulsion which one does not well understand; with the Newtonians, it is done by an attraction of which we know the cause no better. At Paris you fancy the earth shaped like a melon, at London it is flattened on the two sides."

"The theory of the tides has been reduced [in this work] into an extremely simple form, which appears to agree better with all the phenomena, than the more intricate calculations which they have commonly been supposed to require."

"Here lies Isaac Newton, Knight, Who, by a Vigour of Mind almost supernatural, First demonstrated The Motions and Figures of the Planets, The Paths of the Comets, and the Tides of the Ocean. He diligently investigated The different Refrangibilities of the Rays of Light, And the Properties of the Colours to which they give rise. An assiduous, sagacious, and faithful Interpreter Of Nature, Antiquity, and the Holy Scriptures, He asserted his Philosophy of the Majesty of God, And exhibited in his conduct the Simplicity of the Gospel. Let mortals rejoice That there has existed such and so great An Ornament of Human Nature."

"The application of the general doctrines of mechanics to fluids was a natural and inevitable step, when the principles of the science had been generalised. It was easily seen that a fluid is, for this purpose, nothing more than a body of which the parts are moveable amongst each other with entire facility; and that the mathematician must trace the consequences of this condition upon his equations. This accordingly was done, by the founders of mechanics, both for the cases of the equilibrium and of motion. ... The explanation of the Tides, in the way in which Newton attempted it in the third book of the Principia, is another example of a hydrostatical investigation: for he considered only the form that the ocean would have if it were at rest. The memoirs of Maclaurin, Daniel Bernoulli, and Euler, on the question of the tides, which shared among them the prize of the Academy of Sciences in 1740, went upon the same views. The Treatise of the Figure of the Earth by Clairaut, in 1743, extended Newton's solution of the same problem, by supposing a solid nucleus covered with a fluid of different density. No peculiar novelty has been introduced into this subject, except a method employed by Laplace for determining the attractions of s of small eccentricity, which is, as Professor Airy has said, "a calculus the most singular in its nature, and the most powerful in its effects, of any which has yet appeared.""

"Laplace... took up the subject of waves propagated along the surface of water; and deduced a very celebrated theory of the tides, in which he considered the ocean to be, not in equilibrium, as preceding writers had supposed, but agitated by a constant series of undulations, produced by the solar and lunar forces. The difficulty of such an investigation may be judged of from this, that Laplace, in order to carry it on, is obliged to assume a mechanical proposition, unproved, and only conjectured to be true; namely, that "in a system of bodies acted upon by forces which are periodical, the state of the system is periodical like the forces." Even with this assumption, various other arbitrary processes are requisite; and it appears still very doubtful whether Laplace's theory is either a better mechanical solution of the problem, or a nearer approximation to the laws of the phenomena, than that obtained by D. Bernoulli, following the views of Newton."

"In most cases, the solutions of problems of hydrodynamics are not satisfactorily confirmed by the results of observation. Poisson and Cauchy have prosecuted the subject of waves, and have deduced very curious conclusions by a very recondite and profound analysis. The assumptions of the mathematician here do not represent the conditions of nature; the rules of theory, therefore, are not a good standard to which we may refer the aberrations of particular cases; and the laws which we obtain from experiment are very imperfectly illustrated by à priori calculation. The case of this department of knowledge, hydrodynamics, is very peculiar... we want, in addition to what we have, true and useful principles, intermediate between the highest and the lowest;—between the extreme and almost barren generality of the laws of motion, and the endless varieties and inextricable complexity of fluid motions in special cases. The reason of this peculiarity in the science of hydrodynamics appears to be, that its general principles were not discovered with reference to the science itself, but by extension from the sister science of the mechanics of solids...by a perception that the parts of fluids are included in that range of generality which we are entitled to give to the supreme laws of motion of solids. ...[S]olid and fluid dynamics resemble two edifices which have their highest apartment in common, and though we can explore every part of the former building, we have not yet succeeded in traversing the staircase of the latter, either from the top or from the bottom. If we had lived in a world in which there were no solid bodies, we should probably not yet have discovered the laws of motion; if we had lived in a world in which there were no fluids, we should have no idea how insufficient a complete possession of the laws of motion may be, to give us a true knowledge of particular results."

"That all the parts of the universe are drawn and held together by love, or harmony, or some affection to which, among other names, that of attraction may have been given, is an assertion which may very possibly have been made at various times, by speculators writing at random, and taking their chance of meaning and truth. The authors of such casual dogmas have generally nothing accurate or substantial, either in their conception of the general proposition, or in their reference to examples of it... But among those who were really the first to think of the mutual attraction of matter, we cannot help noticing Francis Bacon; for his notions were so far from being chargeable with the looseness and indistinctness to which we have alluded, that he proposed an experiment which was to decide whether the facts were so or not;—whether the gravity of bodies to the earth arose from an attraction of the parts of matter towards each other, or was a tendency towards the centre of the earth. And this experiment is, even to this day, one of the best which can be devised, in order to exhibit the universal gravitation of matter: it consists in the comparison of the rate of going of a clock in a deep mine, and on a high place. Huyghens, in his book "De Causâ Gravitatis," published in 1690, showed that the earth would have an oblate form, in consequence of the action of the centrifugal force; but his reasoning does not suppose gravity to arise from the mutual attraction of the parts of the earth. The influence of the moon upon the tides had long been remarked; but no one had made any progress in truly explaining the mechanism of this influence; and all the analogies to which reference had been made, on this and similar subjects, as magnetic and other attractions, were rather delusive than illustrative, since they represented the attraction as something peculiar in particular bodies, depending upon the nature of each body. That all such forces, cosmical and terrestrial, were the same single force, and that this was nothing more than the insensible attraction which subsists between one stone and another, was a conception equally bold and grand; and would have been an incomprehensible thought, if the views which we have already explained had not prepared the mind for it."

"Newton, in the Principia, had inserted a series of propositions, the object of which was to prove, that the machinery of vortices could not be accommodated to one part of the celestial phenomena, without contradicting another part. A more obvious difficulty was the case of gravity of the earth; if this force arose, as Descartes asserted, from the rotation of the earth's vortex about its axis, it ought to tend directly to the axis, and not to the centre. The asserters of vortices often tried their skill in remedying this vice in the hypothesis, but never with much success. ...The mathematical prize-questions proposed by the French Academy, naturally brought the two sets of opinions into conflict. The Cartesian Memoir of John Bernoulli... was the one which gained the prize in 1730. ...The last act of homage of this kind to the Cartesian system was performed in 1740, when the prize on the question of the tides was distributed between Daniel Bernoulli, Euler, Maclaurin, and Cavallieri; the last of whom had tried to amend and patch up the Cartesian hypothesis on this subject."

"We propose... to enter at some length into the mathematical theories, and the experimental observations, applying to the two subjects of Tides and Waves of water. But we do not intend to treat them with the same extension. We shall give the various theories of Tides in detail sufficient to enable the reader to understand the present state of the science... and we shall advert to the principal observations which throw light either on the ordinary phænomena of tides, or on the extraordinary deviations that occur in peculiar circumstances. In thus treating the Tides, it will be necessary for us to enter largely into the theory of Waves. We shall take advantage of this circumstance for the introduction several propositions, not applying to the theory Tides, but elucidating some of the ordinary observations upon small Waves. But these investigations will be limited to that class which is most closely connected with tides, namely, that in which similar waves follow each other in a continuous series, or in which the same mathematical process may be used as when similar waves follow each other. In this class will be included nearly all the phænomena of waves produced by natural causes, and therefore possessing general interest. But it will not include the waves of discontinuous nature produced by the sudden action of arbitrary causes, which have been the subject of several remarkable mathematical memoirs, but which possess no interest for the general reader."

"We shall describe cursorily the ordinary phænomena of tides."

"We shall explain the Equilibrium-Theory of Tides, including the first tidal theory given by Newton, and the more detailed theory of his successors, especially Daniel Bernoulli."

"We shall give a sketch of Laplace's investigations, (founded essentially on the theory of the motion of water,) in the general form in which he first attempted the theory, as well as with the arbitrary limitations which he found it necessary to use for practical application."

"We shall give an extended Theory of Waves on water, applying principally to the motion of water in canals of small breadth, but with some indications of the process to be followed for the investigation of the motion of Waves in extended surfaces of water."

"The results of a few Experiments on Waves will be given, in comparison with the preceding theory."

"We shall investigate the mathematical expressions for the Disturbing Forces of the Sun and Moon which produce the Tides, and shall use them in combination with the theory of Waves to predict some of the laws of Tides."