theoretical-physics

"... 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. ..."

"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."

"But mathematicians know that the number system we study in school is but one of many possibilities. And indeed, other kinds of numbers are important for understanding geometry and physics. Among the strangest alternatives is the octonions. Largely neglected since their discovery in 1843, in the last few decades they have assumed a curious importance in string theory. And indeed, if string theory is a correct representation of the universe, they may explain why the universe has the number of dimensions it does."

"It is possible to form an analogous theory with seven imaginary roots of (-1)."

"The octonions are much stranger. Not only are they noncommutative, they are also break another familiar law of arithmetic: the associative law (xy)z = x(yz)."

"It is nearly irresistible to ask if the octonions, \mathbb O, the last of the set of four normed division algebras over \mathbb R, have a calling in nature. Certainly several have thought so, but for the most part, the octonions have remained as a well kept secret from mainstream physics. More often than not, the octonions are passed by in haste because they are non-associative, and hence at times temperamental. As we will show, this property is in fact a gift, which will offer a way to streamline some of the standard model’s complex structure."

"Octonions are to physics what the Sirens were to ."

"The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative."

"But perhaps most important, it wasn’t clear in Hamilton’s time just what the octonions would be good for. They are closely related to the geometry of 7 and 8 dimensions, and we can describe rotations in those dimensions using the multiplication of octonions. But for over a century that was a purely intellectual exercise. It would take the development of modern particle physics—and string theory in particular—to see how the octonions might be useful in the real world."

"There is still something in the system which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. ... If with your alchemy you can make three pounds of gold, why should you stop there?"

"Despite its counter-culture status, the octonions have long drawn the curiosity of generations of physicists. The algebra is known to appear without warning in apparently disparate areas of mathematics, within algebra, geometry, and topology. However, despite its ubiquity, its practical uses in physics have remained elusive, due to the algebra's non-associativity, which must be handled with care."

"Besides their possible role in physics, the octonions are important because they tie together some algebraic structures that otherwise appear as isolated and inexplicable exceptions."

"[I]n... 1952, Baade... announced that Hubble’s entire distance scale was in error... According to Baade, the distances formerly assigned to all extragalactic objects must be multiplied by a factor of about two. Later it was generally accepted that this... was probably nearer three. ...[I]t followed that the sizes of all such objects had been underestimated. ...Therefore ...this nebula must be... twice as far away... [T]he average absolute magnitude at maximum brightness of novae... in the Milky Way attain on the average... 7.4, whereas those... in the Andromeda... 5.7... [T]he apparent anomaly could be removed by placing... Andromeda... rather more than twice as far as previously. ...[[w:Distance measure|[E]xtragalactic distances]] had ...been underestimated because of an error in converting... relative distances of s into an absolute scale. ...Baade's revision ...applied only to extragalactic objects... [and] had momentous consequences concerning the size and , for the scale of both was correspondingly increased."

"At last, in 1949, the... ... was ready... Humason... succeeded in photographing the spectra of two remote galaxies in the . These exhibited red-shifts which, on the Doppler interpretation, indicated... one-fifth of the velocity of light. [I]n 1956, with... photoelectric equipment attached... [W. A.] Baum obtained a red-shift... recessional velocity of about two-fifths of the velocity of light."

"An important new survey of the law relating red-shifts and magnitudes published in 1956 by Humason, Mayall and Sandage suggested... that the expansion of the universe may have been faster in the past... so that its age may be somewhat less than that estimated on the hypothesis of uniform expansion. But... caution, for a recent review (1958) by Sandage of Hubble's criteria for constructing the extragalactic distance-scale has revealed that, not only must his Cepheid criterion be corrected but also... the brightest star criterion..."

"As for Hubble’s brightest star criterion, Sandage... has shown that objects in the of galaxies which Hubble believed to be highly luminous stars are... regions of glowing of intrinsic luminosity... two magnitudes brighter... If Sandage’s result is accepted, then the distances of all galaxies beyond those in which Cepheids can be detected... must be augmented by a factor... between 5 and 10... with the result that the rate of increase of velocity with distance will be reduced to between 5O and 100 kilometres per second per megaparsec. Consequently, taking 80 as a rough average... the , if it has expanded uniformly, will have to be increased to about 13-5 thousand million years. If... it was expanding more rapidly in the past... this... might be reduced to about 9 thousand million years."

"I have said that relativity theory predicts a force of cosmical repulsion. ...[R]elativity theory does not talk of anything so crude as force; it describes... curvature of space-time. But for practical purposes... nearly equivalent to the Newtonian force of gravitation... [T]he actual relativity effect is represented with sufficient accuracy by a force of cosmical repulsion... up to the greatest distances... we... observe."

"The galaxies exert on one another their ordinary gravitational attraction approximately according to Newton's law. This makes them tend to cling together. So we... have a contest of two forces, Newtonian attraction... and cosmical repulsion... If our theory is right cosmical repulsion must have got the upper hand... Having got the advantage, cosmical repulsion will keep it; because, as the nebulae become further apart, their mutual attraction will become weaker..."

"That the existence of these motions (the "axiom of free mobility") is a desideratum, if not... a necessity, for a geometry applicable to physical space, has been forcefully argued on a priori grounds by von Helmholtz, Whitehead, Russell and others; for only in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained."

"All change is relative. The universe is expanding relatively to our common material standards; our material standards are shrinking relatively to the size of the universe. The theory of the "expanding universe" might also be called the theory of the "shrinking atom". ...[T]ake the... universe as our standard of constancy... he sees us shrinking... only the intergalactic spaces remain the same. The earth spirals round the sun in an ever‑decreasing orbit. ...Our years will ...decrease in geometrical progression in the cosmic scale of time. ... Owing to the property of geometrical progressions an infinite number of our years will add up to a finite cosmic time; so that what we should call the end of eternity is an ordinary finite date in the cosmic calendar. But on that date the universe has expanded to infinity in our reckoning, and we have shrunk to nothing in the reckoning of the cosmic being. ...When the last act opens the curtain rises on midget actors rushing through their parts at frantic speed. Smaller and smaller. Faster and faster. One last microscopic blurr of intense agitation. And then nothing."

"[T]he most satisfying theory would be one which made the beginning not too unæsthetically abrupt. This... can only be satisfied by an Einstein universe with all... major forces balanced. Accordingly, the primordial state of things... is an even distribution of s and electrons, extremely diffuse and filling all (spherical) space, remaining nearly balanced for an exceedingly long time until its inherent instability prevails. ...[T]he density of this distribution can be calculated ...[at] about one proton and electron per litre. ...[S]mall irregular tendencies accumulate, and evolution gets under way. ...[T]he formation of condensations ultimately ...become the galaxies; this ...started off an expansion, which ...automatically increased in speed until ...now manifested ...in the recession of the spiral nebulae. As the matter drew closer... in the condensations... evolutionary processes followed—evolution of stars... of... more complex elements... of planets and life."

"If the astronomers are right, it is a straightforward conclusion from the observational measurements that the system of galaxies is expanding—or, since the system of the galaxies is all we know—that the universe is expanding. There is no subtlety or metaphysics about it ...But are we sure of the observational facts? Scientific men are rather fond of saying pontifically that one ought to be quite sure of one's observational facts before embarking on theory. Fortunately those who give this advice do not practice what they preach. Observation and theory get on best when they are mixed together, both helping one another in the pursuit of truth. It is a good rule not to put overmuch confidence in a theory until it has been confirmed by observation. I hope I shall not shock the experimental physicists too much if I add that it is also a good rule not to put overmuch confidence in the observational results that are put forward until they have been confirmed by theory. So in starting to theorise about the expanding universe I am not taking it for granted that the observational evidence which we have been considering is entirely certain."

"is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements of a configuration are unaffected by the position or orientation of the configuration. ...[M]otions of are the familiar translations and rotations... made in proving the theorems of Euclid."

"The intermediate solutions of Friedmann and Lemaitre are "expanding universes." Both the material system and the closed space, in which it exists, are expanding. At one end we have Einstein’s universe with no motion and therefore in equilibrium. Then... we have model universes showing more and more rapid expansion until we reach de Sitter’s... The rate of expansion increases all the way along the series and the density diminishes; de Sitter’s universe is the limit when the average density of celestial matter approaches zero. The series of expanding universes then stops... but because there is nothing left to expand."

"Within the galaxy the average world-curvature is... thousands of times greater than Lamaître's average for the universe... his formulæ are inapplicable. The result... only the intergalactic distances expand. The galaxies... are unaffected... —s, stars, human observers and their apparatus, atoms—are entirely free from expansion. Although the cosmical repulsion or expansive tendency is present in all of these... it is checked by much larger forces... [T]he demarcation between permanent and dispersing systems is... abrupt. It corresponds to the distinction between periodic and aperiodic phenomena."

"If you think... the shattering of the bubble universe is... tragic... [W]hen the worst has happened our galaxy... will be left intact. ...not so bad a prospect."

"It is scarcely true... that we observe these velocities of recession. We observe a shift of the spectrum to the red; and although such... is usually due to recession... it is not inconceivable that it should arise from another cause."

"[I]t was theory that first suggested a systematic recession of the spiral nebulae and so led to a search for this effect. The theoretical possibility was first discovered by W. de Sitter in 1917. Only three radial velocities were known at that time, and they... lamely supported his theory by... 2 to 1. Since then... support is far more unanimous... mainly due to V. M. Slipher... and M. L. Humason... The linear law of proportionality between speed and distance was found by E. H. Hubble. Meanwhile the theory has also developed, and... taken the form... associated with... A. Friedman and G. Lemaître."

"The theory of relativity predicts... a... force... we call the cosmical repulsion... directly proportional to the distance... It is so weak... we can leave it out of... motions of the planets... or any motion within... our... galaxy. ...[S]ince it increases... to the distance we... if we go far enough, find it significant."

"I have said the repulsion is proportional to the distance... Distance from what? From anywhere you like. ...Cosmical repulsion is a dispersing force tending to make a system expand uniformly—not diverging from any centre in particular, but such that all internal distances increase at the same rate. That corresponds precisely to the kind of expansion we observe in the system of the galaxies."

"Euclidean geometry is only one of several congruence geometries... Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces... K may be interpreted as the ' of the surface into the third dimension—whence it derives its name..."

"The cosmological constant['s]... most important consequence: the repulsive force, acting at cosmological distances, causes space to expand exponentially. There is nothing new about the universe expanding, but without a cosmological constant, the rate of expansion would gradually slow down. Indeed, it could even reverse itself and begin to contract, eventually imploding in a giant cosmic crunch. Instead, as a consequence of the cosmological constant, the universe appears to be doubling in size about every fifteen billion years, and all indications are that it will do so indefinitely."

"We should, of course, expect that any universe which expands without limit will approach the empty de Sitter case, and that its ultimate fate is a state in which each physical unit—perhaps each nebula or intimate group of nebulae—is the only thing which exists within its own observable universe."

"De Sitter proposed three types of nonstatic universes: the oscillating universes and the expanding universes of the first or second kiind. The main characteristic of the expanding "family" of the first kiind is that the radius is continually increasing from a definite initial time when it had the value zero. The universe becomes infinitely large after an infinite time. In the second kind... the radius possesses at the initial time a definite minimum value... in the Einstein model... the cosmological constant is supposed to be equal to the reciprocal of R2, whereas de Sitter computed for his interpretation the constant to be equal to 3/R2. Whitrow correctly points out the significant fact that in special relativity the cosmological constant is omitted..."

"One of the few authors to have explicitly connected the physical issue of the expansion of the universe with the philosophical topic of the metaphysical status of space is Gerald James Whitrow."

"All kinds of questions remain. Many have to do with cosmology. How did the universe originate? How did the galaxies become distributed in space like the suds in the kitchen sink..? Why is the cosmological constant apparently very tiny but non-zero and has a peculiar value that leads the universe to expand more rapidly?"

"Red-shifts are produced either in the nebulae, where the light originates, or in the intervening space through which the light travels. If the source is in the nebulae, then red-shifts are probably velocity-shifts and the nebulae are receding. If the source lies in the intervening space, the explanation of red-shifts is unknown, but the nebulae are sensibly stationary."

"In 1917 de Sitter showed that Einstein's field equations could be solved by a model that was completely empty apart from the cosmological constant—i.e. a model with no matter whatsoever, just . This was the first model of an expanding universe. although this was unclear at the time. The whole principle of general relativity was to write equations for physics that were valid for all observers, independently of the coordinates used. But this means that the same solution can be written in various different ways... Thus de Sitter viewed his solution as static, but with a tendency for the rate of ticking clocks to depend on position. This phenomenon was already familiar in the form of gravitational ... so it is understandable that the de Sitter effect was viewed in the same way. It took a while before it was proved (by Weyl, in 1923) that the prediction was of a redshifting of spectral lines that increased linearly with distance (i.e. ). ..."

"[W]e stress... the wide range of validity exhibited by s in theoretical physics. ...[I]t has ...been demonstrated how they can be employed to derive equations of optics, dynamics of particles and rigid bodies, and electromagnetism. In addition, physicists have succeeded in formulating the laws of elasticity and hydrodynamics as variational principles, and even Einstein's law of gravitation was included in this category by Hilbert, who found a scaler function... for which \partial\int\mathfrak{h}\,dx_0\,dx_1\,dx_2\,dx_3=0 is equivalent to Einstein's law. This function has been called the "curvature," an identification which induced Whittaker to describe Hilbert's principle in the laconic words, "gravitation simply represents a continual effort of the universe to straighten itself out.""

"All of this picture of the expansion is exciting, pleasant, coherent, well in order. But what if the s are not to be interpreted by the Doppler-Fizeau law in the classical mechanical view, or general relativistically, by the fact that the ratio of the of a photon (as measured by a co-moving observer) to the space radius of curvature is independent of ? Not speaking of quasars, the first indications for non-Doppler redshifts for a galaxy have been provided... What if not all galaxies were formed at the dawn of the Big Bang; what if some are being formed now? Then, at least, the can be anything larger than the age of our own Galaxy..."

"A book, too, can be a star, explosive material, capable of stirring up fresh life endlessly, a living fire to lighten the darkness, leading out into the expanding universe."

"The definition of inflation is extraordinarily simple: it is any period of the Universe's evolution during which the scale factor, describing the size of the Universe, is accelerating. This leads to a very rapid expansion of the Universe, though perhaps a better way of thinking of this is that the characteristic scale of the Universe, given by the Hubble length, is shrinking relative to any fixed scale caught up in the rapid expansion. In that sense, inflation is actually akin to zooming in on a small part of the initial Universe."

"This model of the expanding universe I shall call the substratum. It achieves in the private Euclidean space of each fundamental observer the objects for which Einstein developed his closed spherical space. Although it is finite in volume, in the measures of any chosen observer, it has all the properties of an infinite space in that its boundary is forever inaccessible and its contents comprise an infinity of members. It is also homogeneous in the sense that each member stands in the same relation to the rest. This description of the substratum holds good in the scale of time in which the galaxies or fundamental particles are receding from one another with uniform velocities. This choice of the scale of time, together with the theory of equivalent time-keepers... makes possible the application of the Lorentz formulae to the private Euclidean spaces of the various observers. It thus brings the theory of the expanding universe into line with other branches of physics, which use the Lorentz formulæ and adopt Euclidean private spaces. ...[T]here is no more need to require a curvature for space itself in the field of cosmology than in any other department of physics. The observer at the origin is fully entitled to select a private Euclidean space in which to describe phenomena, and when he concedes a similar right to every other equivalent observer and imposes the condition of the same world-view of each observer, he is inevitably led to the model of the substratum which we have discussed."

"The ideas that prove to be of lasting interest are likely to build on the framework of the now standard world picture, the hot big bang model of the expanding universe. The full extent and richness of this picture is not as well understood as I think it ought to be, even among those making some of the most stimulating contributions to the flow of ideas."

"If the general picture, however, of a Big Bang followed by an expanding Universe is correct, what happened before that? Was the Universe devoid of all matter and then the matter suddenly somehow created? How did that happen? In many cultures, the customary answer is that a God or Gods created the Universe out of nothing. But if we wish to pursue this question courageously, we must of course ask the next question: where did God come from? If we decide that this is an unanswerable question, why not save a step and conclude that the origin of the Universe is an unanswerable question? Or, if we say that God always existed, why not save a step, and conclude that the Universe always existed? That there's no need for a creation, it was always here. These are not easy questions. Cosmology brings us face to face with the deepest mysteries, questions that were once treated only in religion and myth."

"The most far-reaching implication of general relativity... is that the universe is not static, as in the orthodox view, but is dynamic, either contracting or expanding. Einstein, as visionary as he was, balked at the idea... One reason... was that, if the universe is currently expanding, then... it must have started from a single point. All space and time would have to be bound up in that "point," an infinitely dense, infinitely small "singularity." ...this struck Einstein as absurd. He therefore tried to sidestep the logic of his equations, and modified them by adding... a "cosmological constant." The term represented a force, of unknown nature, that would counteract the gravitational attraction of the mass of the universe. That is, the two forces would cancel... it is the kind of rabbit-out-of-the-hat idea that most scientists would label ad-hoc. ...Ironically, Einstein's approach contained a foolishly simple mistake: His universe would not be stable... like a pencil balanced on its point."

"In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems may be ignored; and if we further assume that the total matter in the world has but little effect on its macroscopic properties, we may consider them as being determined by the solution of an empty world."

"An "empty world," i.e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy."

"The solution of (1), which represents a homogeneous manifold, may be written in the form:ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)where \kappa = \sqrt \frac{\lambda}{3}. If we consider \rho as determining distance from the origin... and \tau as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world..."

"Why should not the space be there already, and the material system expand into it..? ...[I]f the speed of recession continues to increase outwards, it will ere long approach the speed of light, so that something must break down. The result is that the system becomes a ... such a system cannot expand without the space also expanding. ...[E]xpansion of space has often been given too much prominence ...and readers have been led to think that it is more directly concerned in the explanation of the motions of the nebule than is... the case. ...If we adopt open space we encounter certain difficulties (not necessarily insuperable) which closed space entirely avoids; and we do not want... speculation as to the solution of difficulties which need never arise. If we wish to be noncommittal, we shall naturally work in terms of a closed universe of finite radius R, since we can at any time revert to an infinite universe by making R infinite."