Physics

"Consciousness, besides detecting time's arrow, also roughly measures the passage of time. ...but is a bit of a bungler in carrying it out. ...Our consciousness somehow manages to keep ...record of the flight of time ...reading some kind of clock in the material brain ...a better analogy would be an entropy-clock ...primarily for measuring the rate of disorganisation of energy, and only roughly keeping pace with time. ...[I]n forming our ideas of duration and of becoming... [e]ntropy-gradient is... the direct equivalent of the time of consciousness in both... aspects. Duration measured by physical clocks (time-like interval) is only remotely connected."

"Entropy... was discovered and exalted because it was essential to practical applications of physics... But by it science has been saved from a fatal narrowness. ...[T]here would have been nothing to represent "becoming" in the physical world."

"[S]uppose that we had to identify force with entropy-gradient. That would only mean that entropy-gradient is a condition which stimulates a nerve, which thereupon transmits an impulse to the brain, out of which the mind weaves its own peculiar impression of force. ...It is absurd to pretend that we are in ignorance of the nature of organisation in the external world in the same way that we are ignorant of the intrinsic nature of potential. It is absurd to pretend that we have no justifiable conception of "becoming"... That dynamical quality... has to do much more than pull the trigger of a nerve. ...a moving on of time is a condition of consciousness. ...It is the innermost Ego of all which is and becomes."

"The more closely we examine the association of entropy with "becoming" the greater do the obstacles appear. If entropy were one of the elementary indefinables of physics there would be no difficulty. Or if the moving on of time were something of which we were made aware through our sense organs there would be no difficulty. ...Suppose that we had to identify "becoming" with an electrical potential-gradient ...through the readings of a voltmeter."

"Entropy was not in the same category as the other physical quantities ...and the extension ...was in a very dangerous direction. ...But entropy had secured a firm place in physics before it was discovered that it was a measure of the random element in arrangement. It was in great favour with the engineers. ...[A]t that time it was the general assumption that the Creation was the work of an engineer (not of a mathematician, as is the fashion nowadays)."

"The discrimination between cause and effect depends on time's arrow and can only be settled by reference to entropy."

"I am standing on the threshold about to enter a room. ...I must make sure of landing on a plank travelling twenty miles a second around the sun—a fraction of a second too early or too late, the plank would be miles away. I must do this whilst hanging from a round planet head outward into space, and with a wind of aether... I ought really to look at the problem four-dimensionally as concerning the intersection of my world-line with that of the plank. Then again it is necessary to determine in which direction the entropy of the world is increasing in order to make sure that my passage over the threshold is an entrance, not an exit. Verily, it is easier for a camel to pass through the eye of a needle than for a scientific man to pass through a door. And whether... barn... or church door it might be wiser that he should consent to be an ordinary man... rather than wait til all the difficulties in... scientific ingress are resolved."

"I was interested in the concept introduced by Clausius, entropy.., (in addition to energy,) one of the most important variables of nature."

"The first step came from W. Wien, whose displacement law of 1893 is embodied in the shift of the maximum of spectrum energy density, from red to violet, with increasing temperatures. Wien showed that a universal function of the ratio of temperature to frequency must here be in question. The determination of this universal function was the culmination of the insight and consistent labors of Planck (1900), who by postulating the energy quantum, became the creator of modern thermodynamics; for this energy element is a saucy reality, whose purpose is to stay. It not only tells us all we know of the distribution of energy in the black body spectrum in its thermal relations, but it gives us, indirectly, perhaps the most accurate data at hand of the number of molecules per normal cubic centimeter of the gas, of the mean translational energy of its molecules, of the molecular mass, of the Boltzmann entropy constant, even of the charge of the electron or electric atom itself."

"Prigogine was also concerned with the broader philosophical issues raised by his work. In the 19th century the discovery of the second law of thermodynamics, with its prediction of a relentless movement of the universe toward a state of maximum entropy, generated a pessimistic attitude about nature and science. Prigogine felt that his discovery of self-organizing systems constituted a more optimistic interpretation of the consequences of thermodynamics. In addition, his work led to a new view of the role of time in the physical sciences."

"The third model regards mind as an information processing system. This is the model of mind subscribed to by cognitive psychologists and also to some extent by the ego psychologists. Since an acquisition of information entails maximization of negative entropy and complexity, this model of mind assumes mind to be an open system."

"What is entropy? ...a measurable physical quantity just like the length ...temperature ...the heat of fusion ...or the specific heat of any given substance. At ... ...the entropy of any substance is zero. When you bring the substance into any other state by slow, reversible little steps ...the entropy increases by an amount computed by dividing every little portion of heat you had to supply ...by the absolute temperature at which it was supplied ...and by summing up all these small contributions."

"[H]igher animals... feed upon... the extremely well-ordered state of... foodstuffs. After utilizing it they return it in a... degraded form—not entirely degraded... for plants can... use... it. (...[Plants] have their most powerful supply of 'negative entropy' in the sunlight)."

"Clausius proposed... The entropy of an isolated system increases in any spontaneous change."

"Energy is needed to replace not only the mechanical energy of our bodily exertions, but also the heat we continually give off... And that we give off heat is not accidental, but essential. For this is precisely the manner in which we dispose of the surplus entropy we continually produce in our... life process."

"Kelvin... formed... the view that... the essential component of the steam engine is the cold sink—the surroundings into which waste heat is discarded. The crucial part of the engine... didn't have to be designed or constructed... inverting common sense. ...Kelvin's conceptual somersault led him to promote ...the central ...cold sink to a universal principle ...all viable engines have a cold sink ...not ...[in] those words but ...[in] essence ...Take away the cold sink and the engine stops ..."

"It is my thesis that the physical functioning of the living individual and the operation of some of the newer communication machines are precisely parallel in their analogous attempts to control entropy through . Both of them have sensory receptors as one stage in their cycle of operation: that is, in both of them there exists a special apparatus for collecting information from the outer world at low energy levels, and for making it available in the operation of the individual or of the machine. In both cases these external messages are not taken neat, but through the internal transforming powers of the apparatus, whether it be alive or dead. The information is then turned into a new form available for the further stages of performance. In both the animal and the machine this performance is made to be effective on the outer world. In both of them, their performed action on the outer world, and not merely their intended action, is reported back to the central regulatory apparatus. This complex of behavior is ignored by the average man, and in particular does not play the role that it should in our habitual analysis of society; for just as individual physical responses may be seen from this point of view, so may the organic responses of society itself. I do not mean that the sociologist is unaware of the existence and complex nature of communications in society, but until recently he has tended to overlook the extent to which they are the cement which binds its fabric together."

"Rudolph Clausius... noticed a common feature of nature and had the stature... to publish [in 1850, Über die bewegende Kraft der Wärme (On the motive force of heat)] what others might think a simpleton's observation: heat does not flow from a cooler to a hotter body... [I]n this and subsequent papers he developed this... into a quantitative principle..."

"Clausius... summarized... the First and Second Laws: ...[T]he energy of the world is constant; the entropy strives towards a maximum."

"Carnot was... wrong about his perception of the steam engine, but... the essence... shone through... his fundamental misconception... that heat is a fluid—caloric—that flows from a hot reservoir [source] to a cold sink... [to] turn an engine... as [does] a waterwheel... by water. ...He ...considered heat ...neither created not destroyed as it flowed ...[H]e was able to prove ...efficiency of an idealized steam engine ...that ignores friction, leaks ...[etc.] is determined only by the temperatures of the ...source and ...sink ...independent of ...pressure and ...working substance [e.g., water, steam or air]. ...[T]he hot reservoir should be as hot as possible and the cold... as cold as possible. All other variables were fundamentally irrelevant."

"When is a piece of matter said to be alive? When it goes on... moving, exchanging material with its environment... When a system... is not alive... all motion usually comes to a standstill... as a result of friction... [T]he whole system fades away into a dead, inert lump of matter. A permanent state is reached, in which no observable events occur. The physicist calls this the state of thermodynamic equilibrium, or of 'maxiumum entropy'."

"[T]he statistical concept of order and disorder... was revealed by... Boltzmann and Gibbs... This too is an exact quantitative connection...entropy = k\log Dwhere k is the... and D is... the atomistic disorder of the body... The disorder... is partly... heat of motion, partly... atoms and molecules being mixed at random... e.g., sugar and water molecules... The gradual 'spreading out' of the sugar over all the water... increases the disorder D, and hence (since the logarithm of D increases with D) the entropy. ...[A]ny supply of heat increases the turmoil of heat motion, that is ...increases D... [W]hen you melt a crystal... you... destroy the neat and permanent arrangement of... atoms or molecules and turn the crystal lattice into a continually changing random distribution."

"Let me come finally to the question that is of more direct interest to the audience here: ‘what are the scales of string unification?’ or put more provocatively, ‘will we see strings and extra dimensions at the LHC?’ (this is also discussed in Peskin’s talk ...) The conventional (and conservative) hypothesis is that the string, compactification and Planck scales lie all to within two or three orders of magnitude from each other, and are hence far beyond direct experimental reach. The non-gravitational physics at lower energies is thus described by a 4d supersymmetric quantum field theory (SQFT), which must at least include in it the MSSM. This conventional hypothesis is supported by the following three solid facts : (i) Softly broken SQFTs can indeed be extrapolated consistently to near-Planckian energies without destabilizing the electroweak scale ; (ii) the hypothesis is (almost) automatic in the weakly-coupled heterotic string theory, and (iii) the minimal (or ‘desert’) string-unification assumption is in remarkable agreement with some of the measured low-energy parameters of our world."

"The most natural expectation away from asymptotic limits in moduli space of supergravity theories is the desert scenario, where there are few states between massless fields and the quantum gravity cutoff. In this paper we initiate a systematic study of these regions deep in the moduli space, and use it to place a bound on the number of massless modes by relating it to the black hole species problem. There exists a consistent sub-Planckian UV cutoff (the species scale) which resolves the black hole species problem without bounding the number of light modes. We reevaluate this in the context of supersymmetric string vacua in the desert region and show that even though heuristically the species scale is compatible with expectations, the BPS states of the actual string vacua lead to a stronger dependence of the cutoff scale on the number of massless modes. We propose that this discrepancy, which can be captured by the “BPS desert conjecture”, resurrects the idea of a uniform bound on the number of light modes as a way to avoid the black hole species problem. This conjecture also implies a stronger form of the Tadpole Conjecture, which leads to an obstruction in stabilizing all moduli semi-classically for large number of moduli in flux compactifications."

"Since 1974, the guiding principle for building extensions of the standard model has been the so-called desert hypothesis. ... Its premise is that there is no new physics between the energy scales of electroweak unification (109 GeV or 1 TeV) and the vicinity of the Planck mass MPl (1019 GeV). That implies an enormous "desert," 16 orders of magnitude wide, where one would expect to encounter nothing new. MPl is the mass at which a particle's Compton wavelength becomes equal to its Schwarzschild radius—a realm where one can't do without a quantum theory of gravity. ... The leading contender for the elaboration of the desert picture has been the supersymmetric extension of the standard model ..."

"The surprising discovery of Newton’s age is just the clear separation of laws of nature on the one hand and initial conditions on the other. The former are precise beyond anything reasonable; we know virtually nothing about the latter. ,,, … how can we ascertain that we know all the laws of nature relevant to a set of phenomena? If we do not, we would determine unnecessarily many initial conditions in order to specify the behavior of the object."

"Inflation has attracted cosmologists because of its potential to free the standard big bang model from its worst flaw, the need for special initial conditions and, in particular, the requirement of initial acausal homogeneity. Naturally one must check whether inflation itself depends critically on initial conditions. Several “no hair” theorems and perturbation calculations have indicated that inflation is stable, and that it will take place when the initial conditions are perturbed. This has led to the belief that inflation will start in any generic universe."

"In the Green-function treatment of particle motion, a unit impulse is represented by a force F(t) = δ(t–t′) and is the analogue of the unit point source in spatial problems. The initial conditions play the role of boundary conditions."

"The next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions."

"I do think... that you would find it would lose nothing by omitting the word "vector" throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell."

"Here as he walked by, on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i^2 = j^2 = k^2 = ijk = -1 & cut it on a stone of this bridge."

"Quantum theory may be formulated using s over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. ...[P]roblems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". ... This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. ...There are precisely four 'normed division algebras': the real numbers \mathbb{R}, the complex numbers \mathbb{C}, the quaternions \mathbb{H} and the octonions \mathbb{O}. Roughly speaking, these are the number systems extending the reals that have an ‘absolute value’ obeying the equation |xy| = |x| |y|. Since the octonions are nonassociative [their use] proves difficult... except in a few special cases. ...[I]nstead of being distinct alternatives, real, complex and quaternionic quantum mechanics are three aspects of a single unified structure."

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

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

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

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

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

"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."

"'(2). But the question arises, what special connexion has the number Four with mathematics generally, or with that branch of mathematical science in particular, to which the "Lectures on Quaternions" relate?"

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

"'(3). One general form of answer... is... that in the mathematical quaternion is involved a peculiar synthesis, or combination, of the conceptions of space and time; and that while TIME is usually pictured or represented by metaphysicians under the figure of a line—a single stream with its ONE current—an unique axis of progression, SPACE is, on the contrary, imagined or conceived in connexion with THREE distinct axes, three lines at right angles to each other... height, length, and breadth. In time, we have only the forward and the backward, looking before and after. In space, there is not merely the contrast between the directions of upward and downward, but also between those of southward and northward, and again between westward and eastward. Time is said to have only one dimension, and space to have three dimesions. The former is an unidimensional, the latter a tridimensional progression. The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space," or "space plus time": and in this sense it has, or at least it involves a reference to, four dimensions. In an unpublished sonnet to Sir John Herschel, entitled "The "(...Greek ...equivalent to the Latin Quaternio), the author of the Lectures introduced the two following lines... an expression of the view... in the foregoing remarks..:"And how the One of Time, of Space the Three, Might in the Chain of Symbol girdled be.""

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the... development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes."

"'(4). Those who are entirely unacquainted with mathematical science may yet derive, from what has been above remarked, a sufficient preliminary insight into the nature of the speculations and inquiries to which the "Lectures on Quaternions" relate. A philosophical, if not a technically scientific, knowledge of the author's general aim, and of the idea which has guided him, may in this way be easily attained. But a very moderate acquaintance with the conceptions of geometry will suffice to render intelligible, from another point of view, the importance which the author attaches to the number Four in mathematics."

"[T]he common solution, using three Euler's angles interpolated independently, is not ideal. The more recent (1843) notation of quaternions is proposed instead, along with interpolation on the quaternion unit sphere. Although quaternions are less familiar, conversion to quaternions and generation of in-between frames can be completely automatic, no matter how key frames were originally specified, so users don't need to know—or care—about inner details. The same cannot be said for Euler's angles, which are more difficult to use."

"While translations are well animated by using vectors, rotation animation can be improved by using the progenitor of vectors, quaternions. ...By an odd quirk of mathematics, only systems of two, four, or eight components will multiply as Hamilton desired; triples had been his stumbling block."

"Yet, though few, if any—Clerk-Maxwell perhaps only excepted—ever possessed the same almost magical quality of physical insight, none could be more strict than Lord Kelvin in requiring demonstration freed from untenable assumptions or undemonstrable hypotheses. Daring as he was, at least in his earlier days, in the application of analytical methods to the phenomena of nature, he was in several ways very conservative. For example, he never would countenance the use in physics of the method of quaternions. At the British Association Meeting at Cambridge in 1845, he had met Hamilton, who there read his first paper on Quaternions. One might have thought that the young enthusiast would have readily welcomed a new and ingenious method of symbolic analysis: but it was not so. He would not use quaternion notation or quaternion methods himself, nor did he admit the into his work."

"(5). As early as the first book of Euclid's Elements, an attentive student is (or may be) led to consider the relative length, and also the relative direction, of one straight line as compared with another. Thus when Euclid shows, in his very first proposition, how to construct on a given base AB an equilateral triangle ABC, he virtually teaches how, when one line AB is proposed or given, to draw a new line BC (or AC), which shall in length be equal to the given one, and in direction shall make with it an angle of sixty degrees, namely, the angle ABC (or BAC), which is the third part of 180 degrees, or of two right angles."

"I first became personally acquainted with Tait a short time before he was elected Professor in Edinburgh… It must have been either before his election or very soon after it that we entered on the project of a joint treatise on Natural Philosophy. He was then strongly impressed with the fundamental importance of Joule’s work… We incessantly talked over the mode of dealing with energy which we adopted in the book, and we went most cordially together in the whole affair. … We have had a thirty-eight years’ war over quaternions. He had been captivated by the originality and extraordinary beauty of Hamilton’s genius in this respect; and had accepted, I believe, definitely from Hamilton to take charge of quaternions after his death, which he has most loyally executed. Times without number I offered to let quaternions into Thomson and Tait if he could only show that in any case our work would be helped by their use. You will see that from beginning to end they were never introduced."

"[Q]uaternions form the appropriate algebraic basis for a description of nature whenever we have to deal either with pseudoreal group representations or with co-representations of Wigner's Type II. The context in which quaternions arose historically, in a study of the three-dimensional rotation group, can now be seen to be an extremely special case of this general principle. Every group which admits pseudoreal representations equally admits a natural description in terms of real quaternions."

"'(1). The word "Quaternion" requires no explanation, since... it occurs in the Scriptures and in Milton. Peter was delivered to "four quaternions of soldiers" to keep him; Adam, in his morning hymn, invokes air and the elements, "which in quaternion run." The word (like, the Latin "quaternio," from which it is derived) means simply a set of four, whether those "four" be persons or things."