Theoretical Physics

"We shall advert to the methods which been used, or which may advantageously be used, for Observation of Tides, and for the Reduction of the Observations."

"We shall give the results of extensive observations of the Tides, as well with regard to the change of the phænomena of tides at different times in the same place, as with respect to the relation which the time and height of tide at one place bear to the time and height at other places, and shall compare these with the results of the preceding theories, as far as possible."

"And as Conclusion, we shall point out what we consider to be the present Desiderata in the Theory and Observations of Tides."

"Caesar, in his account of the invasion of Britain, (De Bella Gallico, lib. iv.) alludes to the nature of spring tides as perfectly well understood in connection with the moon’s age. Some of the peculiarities of river tides, however, were not published in scientific works till the beginning of the last century; and some of the properties of the tides in the English and other channels were not known till the end of that century. ...In the present century, the elaborate discussions of immense collections of accurate tide-observations by M. Laplace, Sir John W. Lubbock, and Professor Whewell, have brought to light and reduced to law many irregularities which were before that time unknown."

"Before entering upon either of the theories explaining the Tides, we must allude to their inadequacy, perhaps not to the explanation of the facts already observed, but certainly to the prediction of new ones. This inadequacy does not appear to arise from any defect in the principles upon which the theory is based,... but from the extreme difficulty of investigating mathematically the motions of fluids under all the various circumstances in which the waters of the sea and of rivers are found. For the problem of the Tides, it is evident, is essentially one of the motion of fluids. Yet so difficult are the investigations of motion that, till the time of Laplace, no good attempt was made to determine, by theory, the laws of the Tides, except on the supposition that the water was at -rest. Since that time theories of motion have been applied..."

"Indeed, throughout the whole of this subject, the selection of the proper theoretical ground of explanation is a matter of judgment. In some cases we may conceive that we are justified in using the Equilibrium theory; in others the Wave-theory will apply, completely or partially... as a last resource, in almost every case, we shall be driven to the same arbitrary suppositions which Laplace introduced. ...In the instances which it does not master completely, it will show that there are ample grounds for the arbitrary alterations of constants introduced by Laplace in his suppositions..."

"[W]e are precluded from further advance, partly by our almost necessary ignorance of the forms of the bottom in deep seas, and partly by the imperfection of our mathematics. ...the first principles of our explanation are correct."

"Newton pointed out and assigned generally, not only the nature and the magnitude of the periodical forces which are concerned in producing the tides, but likewise indicated their true character as undulations, in one very remarkable proposition, as well as in a special explanation of... the tides of the Port of Batsha. The equilibrium theory of Daniel Bernoulli adopted the first part of Newton's views but altogether neglected the second."

"It had been shown that if the earth was a spherical body covered with water, and if both the earth and moon were at rest, the water would assume the form of a spheroid of equilibrium, of extremely small eccentricity, such as would be due to the disturbing action of the moon's forces. A similar but less eccentric spheroid would be formed beneath the sun. Under such circumstances the joint effect of the elevations or depressions of the two spheroids would produce the elevation or depression of the water, or the tide. The theory further assumes that the same effects would follow if the earth revolved round her axis and the earth and moon in their orbits, and that no effect was produced by the spontaneous oscillations of the sea. Totally false as are the principal assumptions upon which this theory is founded, it is extremely remarkable that it not only sufficiently separates from each other the principal movements of the tides, but represents generally the law and order of succession of the periodical phenomena which they present. "The greatest mathematicians and the most laborious observers of the present day," says Professor Airy, "including Sir John Lubbock and Dr. Whewell... have agreed equally in rejecting the foundation of this theory, and comparing all their observations with its results.""

"The same eminent authority [Professor Airy] has pronounced the theory proposed by La Place in the Mécanique Céleste,—if viewed with reference to the boldness and comprehensive character of its design rather than to the success of its execution—"as one of the most splendid works of the greatest mathematician of the past age." The problem, however, was not considered by him [La Place] in the most general form which it is capable of receiving. He assumed the earth to be entirely covered by water, and its depth to be uniform, at least throughout the same parallel of latitude, and he neglected the resistance both of the particles of the fluid amongst each other, and of that which arises from the irregular surfaces in the channels over which the tide is transmitted. He was consequently obliged to omit the consideration of the tides in canals, rivers, and narrow seas, which constitute some of the most interesting, and by no means the most unmanageable, of the problems which later, and even in some respects more simple, investigations of the oscillations of the sea have brought within the control of analysis. Imperfect, however, as the results of this theory were as it came from the hand of its author, their importance cannot easily be estimated too highly. Dr. Young adopted the general principles which they involved, though he has subjected them to a totally different treatment; and Professor Airy, who has materially simplified the investigations which it contains, by rejecting some conditions which they included, such as the density of the sea, by which they were made needlessly difficult and complicated, has not only verified the more remarkable of the conclusions at which La Place arrived, but has also made important use of his methods in his own theory of waves and tides, which is by far the most complete and comprehensive that has ever yet appeared."

"There is one result of a very unexpected kind, which La Place regarded as one of the happiest of his discoveries,—it is the entire evanescence, if the sea be of uniform depth, of the diurnal tide in elevation, but not in horizontal motion. At the equator, under such circumstances, the water moves north and south, resting for a moment at the change of motion. At the poles the motion is transverse to the meridian passing through the luminary. At all other points on the earth's surface it is perpetually changing. Few persons have attempted to follow the mazes of the difficult analysis by which this great mathematician has arrived at this conclusion, which has been verified by the Astronomer Royal. Its correctness, however, has been disputed by Dr. Young, who contends that the diurnal tide will not disappear, unless the depth of the sea be not merely uniform, but evanescent."

"Though Dr. Young was not disposed to give his assent to the results of an extremely difficult analysis,—which few persons of his age could venture to follow, and which might appear to those who could not trace them through the long train of consequences... to be either paradoxical or contradictory to the first principles of mechanics—he was sufficiently prepared to seize the general purport of other parts of this comprehensive theory; and by divesting it of the unnecessary generalizations by which it was encumbered, not only to bring its principles to bear immediately upon the ordinary phenomena of the tides, but to apply it to cases which it was otherwise incompetent to reach. Such were the tides of narrow seas and rivers, and the modifications which those tides undergo from the effects of the resistance of the particles of water upon each other, or upon the channels through which they are propagated. The same questions have been made the principal subject of the investigations of the Astronomer Royal, in his Article on Tides and Waves, in the Encyclopædia Metropolitana, where they have been treated with that rare combination of mathematical skill and clearness and completeness of exposition for which all his writings are so remarkable. It will be found, however, that there are not many of his results which Young had not already attained, though in a much less definite form, by methods which are, it is true, much less regular and systematic, but which are not less distinguished for the sagacity and philosophical power which they display."

"Afterwards that incomparable Philosopher Sir Isaac Newton, improv'd the hint, and wrote so amply upon this Subject as to make the Theory of the Tides his own, by shewing that the Waters of the Sea rise under the Moon and the Place opposite to it: For Kepler believ'd "that the Impetus occasion'd by the presence of the Moon, by the absence of the Moon, occasions another Impetus; till the Moon returning, stops and moderates the Force of that Impetus, and carries it round with its motion." Therefore this Spheroidical Figure which stands out above the Sphere (like two Mountains, the one under the Moon and the other in the place opposite to it) together with the Moon (which it follows) is carried by the Diurnal Motion, (or rather, according to the truth of the matter, as the Earth turns towards the East it leaves those Eminencies of Water, which being carried by their own motion slowly towards the East, are as it were unmov'd) in its journey makes the Water swell twice and sink twice in the space of 25 Hours, in which time the Moon being gone from the Meridian of any Place, returns to it again."

"J. Kepler was the first (that I know of) that discover'd the true cause of the Tide, and he explains it largely in his Introduction to the Physics of the Heavens, given in his Commentaries to the Motion of the Planet Mars, where after he has shewn the Gravity or Gravitation of all Bodies towards another, he thus writes: "The Orb of the attracting Power, which is in the Moon is extended as far as the Earth, and draws the Waters under the Torrid Zone, acting upon places where it is vertical, insensibly on included Seas, but sensibly on the Ocean, whose Beds are large, and the Waters have the liberty of reciprocation, that is, of rising and falling"; and in the 70th Page of his Lunar Astronomy,—"But the cause of the Tides of the Sea appear to be the Bodies of the Sun and Moon drawing the Waters of the Sea.""

"Among all the great men who have philosophized about this remarkable effect, I am more astonished at Kepler than at any other. Despite his open and acute mind, and though he has at his fingertips the motions attributed to the earth, he nevertheless lent his ear and his assent to the moon's dominion over the waters, to occult properties, and to such puerilities."

"A law explains a set of observations; a theory explains a set of laws. The quintessential illustration of this jump in level is the way in which Newton’s theory of mechanics explained Kepler’s law of planetary motion. Basically, a law applies to observed phenomena in one domain (e.g., planetary bodies and their movements), while a theory is intended to unify phenomena in many domains. Thus, Newton’s theory of mechanics explained not only Kepler’s laws, but also Galileo’s findings about the motion of balls rolling down an inclined plane, as well as the pattern of oceanic tides. Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism. So, for instance, Freud’s theory of mind relies upon the unobservable ego, superego, and id, and in modern physics we have theories of elementary particles that postulate various types of quarks, all of which have yet to be observed."

"When Gilbert of Colchester, in his “New Philosophy,” founded on his researches in magnetism, was dealing with tides, he did not suggest that the moon attracted the water, but that “subterranean spirits and humors, rising in sympathy with the moon, cause the sea also to rise and flow to the shores and up rivers”. It appears that an idea, presented in some such way as this, was more readily received than a plain statement. This so-called philosophical method was, in fact, very generally applied, and Kepler, who shared Galileo’s admiration for Gilbert’s work, adopted it in his own attempt to extend the idea of magnetic attraction to the planets."

"[I]t was upon... inequality of motions in point of velocity that Galileo built his theory of flux and reflux of the sea; supposing that the earth revolved faster than the water could follow; and that the water was therefore first gathered in a heap and then fell down, as we see in a basin of water moved quickly. But this he devised upon an assumption which cannot be allowed, viz. that the earth moves; and also without being well informed as to the sexhorary motion of the tide."

"The Academy of Sciences at Paris proposed The Tides as the subject for a prize essay in 1740. Four essays were published in consequence at Paris. One essay was by a Jesuit named Cavallieri; this adopted the Cartesian system of vortices. The other essays were by Daniel Bernoulli, Maclaurin, and Euler; these are reprinted in the Jesuits' edition of the Principia, and it is stated that many errors in the original impression have been corrected. ...The second chapter of Daniel Bernoulli's essay contains some lemmas relating to the Attraction of Bodies. ...he determines the attraction at any superficial or internal point of an ellipsoid of revolution which is nearly spherical, neglecting powers of the ellipticity beyond the first. The method used consists in finding accurately the attraction of a sphere, and then approximately the attraction of the difference between the sphere and the ellipsoid on a particle at the pole or at the equator... this method had been previously used by Clairaut. But Daniel Bernoulli seems to claim the method as his own... Although Daniel Bernoulli employed attraction for the purpose of his essay, yet he seems to have had but a weak faith in the principle... Daniel Bernoulli added nothing to our subject; all his results respecting Attraction are included in the formulæ given by Clairaut in 1737. But his theory of the Tides is very important in the history of that subject..."

"This world was once a fluid haze of light, Till toward the centre set the starry tides, And eddied into suns, that wheeling cast The planets: then the monster, then the man."

"The allusion to the "puzzling" problem of [the orbit of] Mars shows that Galileo ought not to have been unaware of the great work of Kepler published in 1609: Astronomia nova... in which the first two of Kepler's laws were formulated. Yet he does not mention here at all Kepler's success in solving the problem, nor his laws, nor his name even, which is brought up... only to criticize his belief in the Moon's attraction [effect upon tides], which is quite reasonably presented in the Astronomia nova and founded on astronomical reasons and not on mystical speculations."

"But to return to Kepler, his great sagacity, and continual meditation on the planetary motions, suggested to him some views of the true principles from which these motions flow. In his preface to the commentaries concerning the planet Mars, he speaks of gravity as of a power that was mutual betwixt bodies, and tells us that the earth and moon tend towards each other, and would meet in a point so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds that the tides arise from the gravity of the waters towards the moon. But not having just enough notions of the laws of motion, he does not seem to have been able to make the best use of these thoughts; nor does he appear to have adhered to them steadily, since in his epitome of astronomy, published eleven years after, he proposes a physical account of the planetary motions, derived from different principles."

"Astronomy teaches the correct use of the sun and the planets. These may be put on a frame of little sticks and turned round. This causes the tides."

"It can rightly be said that symmetry, gauge theories, and spontaneous symmetry breaking have been the three pegs upon which modern particle physics rests."

"Do quarks and galaxies play by the same rules? Physicists believe they should, even though they don't quite know why. For decades, physicists have been searching for a "theory of everything"—a comprehensive description of the laws of nature. In particular, they want to bridge the gap between the large and the small with a quantum theory of gravity—a reconciliation of general relativity with quantum mechanics. String theory appears to be the current best bet..."

"The ideal theory of everything, in the minds of physicists... is a mathematical system of uncommon tidiness and rigor, which may, if all works out correctly, have the ability to accommodate the physical facts... Perhaps physicists will one day find a theory of such compelling beauty that its truth cannot be denied; truth will be beauty... because, in the absence of any means to make practical tests, what is beautiful is declared ipso facto to be the truth. This theory of everything will be... a myth... a story that makes sense within its own terms, offers explanations for everything... but can be neither tested nor disproved... an explanation that everyone agrees on because it is convenient to agree on it, not because its truth can be demonstrated. This... will indeed spell the end of physics... not because physics has at last been able to explain everything... but because physics has reached the end of all things it has the power to explain."

"Any concepts or words which have been formed in the past through the interplay between the world and ourselves are not really sharply defined with respect to their meaning: that is to say, we do not know exactly how far they will help us in finding our way in the world. We often know that they can be applied to a wide range of inner or outer experience, but we practically never know precisely the limits of their applicability. This is true even of the simplest and most general concepts like "existence" and "space and time". Therefore, it will never be possible by pure reason to arrive at some absolute truth. The concepts may, however, be sharply defined with regard to their connections. This is actually the fact when the concepts become part of a system of axioms and definitions which can be expressed consistently by a mathematical scheme. Such a group of connected concepts may be applicable to a wide field of experience and will help us to find our way in this field. But the limits of the applicability will in general not be known, at least not completely."

"Ever since Newton, and especially since Einstein, the goal of physics has been to find simple mathematical principles of the kind Kepler envisioned, and with them to create a unified theory of everything that would account for every detail of the matter and forces we observe in nature. ...The goal was to find not just a single theory that explains all forces but also one that explains the fundamental numbers... such as the strength of the forces and the masses and charges of the elementary particles. ...A unique theory would be unlikely to have the fine-tuning that allows us to exist. But if... we interpret Einstein's dream to be that of a unique theory that explains this and other universes, with their whole spectrum of different laws, then M-theory could be that theory."

"The clearest expression of this modern "religious" mission can be recognized wherever one encounters the ancient Pythagorean search for nature's mathematical symmetry and harmony. This Pythagorean religion was transformed by early mechanists into a search for the mind of the Christian God. That quest has been tempered since the seventeenth century by a concern to find more practical mathematical relationships in nature, but it has not disappeared. ...wherever the religious mission has been retained in its pure form, as, for example, in the quest for a Theory of Everything... fewer women scientists will be found."

"We are all, in our own way, seekers of the truth... each generation stands firmly on the shoulders of the previous... Whether any of our descendants will ever take in the view from the summit and gaze out on the vast and elegant universe with a perspective of infinite clarity, we cannot predict. But... we are fulfilling our part, contributing our rung to the human ladder reaching for the stars."

"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."

"It is possible that when we finally understand how particles and forces behave at energies up to 1018 GeV, we will just find new mysteries, with a final unification as far away as ever. But I doubt it. There are no hints of any fundamental energy scale beyond 1018 GeV, and string theory even suggests that higher energies have no meaning."

"The theory of superstrings... described the world as a vast symphony of vibrations of infinitesimal strings in a 10-dimensional spade-time. The world would then be governed by a single superforce, melding all four currently known forces, and whose rule would extend over the entire universe. Are we about to reach the goal? ...I am not convinced. It is, for the time being... to be verified experimentally, since it would require phenomenal energies. ...protons have displayed a longevity surpassing initial predictions. ...the unification of everything is predicted at energies that defy human imagination... the theory is shrouded in such a thick mathematical veil that it no longer has any connection with reality. ...as long as physics is not rooted in reality, it is no more than metaphysics."

"We are at the end of our extensive journey into the heart of matter. It has been marked as a relentless march toward Unity. As physics progressed, intimate connections were discovered between phenomena once thought to be completely distinct. A unified description of Nature, sometimes referred to a bit pompously as "the theory of everything," has become the holy grail of modern physics. Symmetry has constantly and reliably guided the physicists' first hesitant steps in this quest for Unity. ...becoming increasingly abstract. ...Now scientists routinely deal with "matter-light symmetry.""

"We need to distinguish between two different ways of viewing the external physical reality: the outside view or bird perspective of a mathematician studying the mathematical structure and the inside view or frog perspective of an observer living in it. ...a mathematical structure is an abstract, immutable entity existing outside of space and time. If history were a movie, the structure would therefore correspond not to a single frame of it but to the entire videotape. ... If a future physics textbook contains the TOE, then its equations are the complete description of the mathematical structure that is the external physical reality. ...is rather than corresponds to...If our external physical reality is isomorphic to a mathematical structure, it therefore fits the definition of being a mathematical structure."

"The true mathematical structure isomorphic to our world, if it exists, has not been found."

"If we assume that reality exists independently of humans, then for a description to be complete, it must also be well-defined according to non-human entities—aliens or supercomputers, say—that lack any understanding of human concepts. Put differently, such a description must be expressible in a form that is devoid of any human baggage like “particle”, “observation” or other English words."

"I will... argue that... our universe is not just described by mathematics—it is mathematics. ...this hypothesis... should... be useful in narrowing down what an ultimate theory of everything can look like. The foundation of my argument is the assumption that there exists an external physical reality independent of us humans. ...Our most successful theories, such as general relativity and quantum mechanics, describe only parts of this reality... In contrast, the holy grail of theoretical physics is a theory of everything—a complete description of reality."

"Our particular laws are not at all unique. ...they could change from place to place and from time to time. The Laws of Physics are much like the weather... controlled by invisible influences in space almost the same way as that temperature, humidity, air pressure, and wind velocity control how rain and snow and hail form. ...The Landscape... is the space of possibilities... all the possible environments permitted by the theory. ...[T]heoretical physicists ...have always believed that the laws of nature are the unique, inevitable consequence of some elegant mathematical principle. ...the empirical evidence points much more convincingly to the opposite conclusion. The universe has more in common with a Rube Goldberg machine than with a unique consequence of mathematical symmetry. ...Two key discoveries are driving the paradigm shift—the success of inflationary cosmology and the existence of a small cosmological constant."

"Back in the 1970s there was a simple dream about how physics would end. A unified theory would be found that incorporated quantum theory, general relativity, and the various particles and forces known to us. This would not only be a theory of everything, it would be unique. We would discover that there was only one mathematically consistent quantum theory that unified elementary particle physics with gravity. ...Because it was unique, this theory would have no free parameters—there would be no adjustable masses or charges. ...There would be only one scale, against which everything would be measured... the Planck scale. The theory would allow us to calculate the results of any experiment to whatever accuracy we desired. ...Looking back, it is clear that the assumption that a unified theory would be unique was no more than that—an assumption. ....we know that there can be no such theory."

"[F]inding the T.O.E. would in no way mean that psychology, biology, geology, chemistry, or even physics had been solved or in some sense subsumed. The universe is such a wonderfully rich and complex place that the discovery of the final theory... would not spell the end of science. Quite the contrary: The discovery of the T.O.E.—the ultimate explanation of the universe at its most microscopic level, a theory that does not rely on any deeper explanation—would provide the firmest foundation on which to build our understanding of the world. Its discovery would mark the beginning, not the end. The ultimate theory would provide an unshakable pillar of coherence forever assuring us that the universe is a comprehensible place."

"[A]greement with observed facts" never singles out one individual theory. There is never only one theory that is in complete agreement with all observed facts, but several theories that are in partial agreement. We have to select the final theory by a compromise. The final theory has to be in fair agreement with observed facts and must also be fairly simple. If we consider this point, it is obvious that such a "final" theory cannot be "The Truth."

"All science has one aim, namely, to find a theory of nature. We have theories of races and of functions, but scarcely yet a remote approach to an idea of creation. We are now so far from the road to truth, that religious teachers dispute and hate each other, and speculative men are esteemed unsound and frivolous. But to a sound judgment, the most abstract truth is the most practical. Whenever a true theory appears, it will be its own evidence. Its test is, that it will explain all phenomena. Now many are thought not only unexplained but inexplicable; as language, sleep, madness, dreams, beasts, sex."

"Concerning such dimensionless constants] I would like to state a theorem which at present cannot be based upon anything more than upon faith in the simplicity, i.e., intelligibility, of nature: there are no arbitrary constants of this kind; that is to say, nature is so constituted that it is possible logically to lay down such strongly determined laws that within these laws only rationally completely determined constants occur (not constants, therefore, whose numerical value could be changed without destroying the theory)."

"[T]he single equation of nature, aimed at by Lagrange and Hamilton, by Weber and Maxwell in their several ways, has... reached a more profound significance and now even holds dynamics, awkwardly it is true but none the less inexorably, in its grasp. That it is not complete, that it never can be complete, is admitted (for the absolute truth poured into the vessel of the human mind would probably dissolve it); but that it is immeasurably more complete to-day than it was yesterday is as incontrovertably true as it is inspiring."

"If all you have is a hammer, everything looks like a nail."

"Symmetry is one of the great unifying themes in physics. From cosmology to and from to , symmetries determine which shapes, interactions, and evolutions occur in nature. Perhaps the most important aspect of symmetry in theories of physics, is the idea that the states of a system do not need to have the same symmetries as the theory that describes them. Such spontaneous breakdown of symmetries governs the dynamics of , the emergence of new particles and s, the rigidity of collective states of matter, and is one of the main ways emerges in a quantum world. The basic idea of spontaneous symmetry breaking is well known, and repeated in different ways throughout all fields of physics."

"Mathematical gauge theory studies connections on principal bundles, or, more precisely, the solution spaces of certain partial differential equations for such connections. Historically, these equations have come from mathematical physics. Gauge theory as a tool for studying topological properties of four-manifolds was pioneered by the fundamental work of Simon Donaldson in the early 1980's. Since the birth of the subject, it has retained its close connection with symplectic topology, a subject whose intricate structure was illuminated by Mikhail Gromov's introduction of pseudo-holomorphic curve techniques, also introduced in the early 1980's."

"There is nothing mysterious about spontaneous symmetry breaking. There are many examples in physics. Hold a drinking straw between the palms of your hands and you have a physical system that can be described by equations possessing rotational symmetry. Press your palms together and the straw bends. The symmetry is broken. You cannot necessarily predict how the straw will bend; it could bend "up," "down," "sideways," or in any other direction. This unsymmetrical situation, however, is the stable solution of perfectly symmetrical equations."

"The for scalar electrodynamics in Fermi gauges are shown to imply a homogeneous first-order partial differential equation for the effective potential involving only the gauge parameter and the external scalar field. Spontaneous symmetry breaking is consequently a gauge-invariant phenomenon. Also observable quantities, including masses, physical coupling constants, and elements, of a theory with spontaneous symmetry breaking are found to be invariant, if a change in the gauge parameter is accompanied by a suitable change in the ground-state expectation value of the scalar field. The generalization to a non-Abelian gauge theory is briefly indicated."