Getting its from bits

Getting its frombits which govern the evolution of systems in time, and are expected to be simple, and the initial conditions, which mustbe given from outside. The equations of classical physics can be applied to any number of different types of solar system, having different sizes and shapes. There is nothing in Newton's Frank Wiiczek The inventor of the term ‘black hole’, John Wheeler, has a gift for memorable phrases. ‘Getting its from bits’ Is another of his creations. It refers not to an object, but to a vision of a world derived from pure logic and mathematics. That vision has to a remarkable extent been embodied In modern physics — here is a progress report. he ‘its from bits’ programme’ has a | venerable history, for perhaps the first great quantitative generalization in science was Pythagoras’ discovery of the numerical patterns behind musical sounds. When two strings of a lyre — of the same material, and under equal tension — are played together, they produce a pleasant harmony precisely when their lengths are a ratio of small integers: 2 to 1 for an octave, 3 to 2 for a musical fifth, 4 to 3 for a fourth, and so on. For the followers of Pythagoras, this provided a satisfying example of a principle they held to be completely general, the idea that ‘all is number’. A chain of thought extending over two millennia links this idea to the inspirations of Kepler. Kepler’s three laws of planetary motion are enshrined in textbooks, and provided the foundation for Newton's celestial mechanics. Less publicized is his erroneous ‘zeroth’ law, which was his version of Copernicanism, and the point of departure for his ©WILC.ZEK original research. According to Kepler's zeroth law, which would have pleased Pythagoras, the orbits of the six planets are great circles on spheres alternately inscribed within and circumscribed about the five regular solids. Of course, we now know that there are more than six planets, and Kepler himself was reluctantly forced, by Tycho Brahe’s accurate observations, to abandon circular orbits in favour of ellipses. According to modern views, the number of planets laws of gravity and mechanics, nor for that matter in the other pillar of classical physics, Maxwell’s electrodynamics, that could serve to fix a definite size. Symptomatic of this, there is no way to forma characteristic length from the parameters that govern these theories, namely the gravitational coupling G and the speed of light c. Classical physics is profoundly anti-Pythagorean. Modern Pythagorism and the size of planetary orbits was deter- The most successful theory of modern physics, quantum mechanics, completely changes the situation. Quantum mechanics provides a unique ground-state configuration for each atom and molecule, thus relieving the indeterminacy in the analogous classical theory of solar systems, and making it possible to understand why atoms and molemined more or less accidentally during the complicated process whereby our Solar System condensed out of a gigantic interstellar cules exhibit well-defined, universal chemistry. Ata yet deeper level, quantum field theory, which is the logical extension of quangas cloud. Solar systems around other stars, tum mechanics to include special relativity, which are now beginning to yield their explains why the elementary constituents — secrets to observation, are expected to be very different. electrons and nuclei — exist in myriads of Indeed, classical physics teaches us that the size ofplanctary orbits is not the sort of thing we should aspire to predict. It makes a sharp distinction between the basic laws, single universal field. So, for example, quantum electrodynamics (QED) posits, in addition to the familiar electromagnetic field identical copies, each being an excitation ofa whose excitations represent the formation of photons, an electron field whose excitations Box 1: Afewwords on dimensional analysis _ we sec as the creation of electrons. Dimensional analysisis a time-honoured When Planck introduced his quantum of action, #, he immediately advertised the pos- ‘way to estimate the answer to a. physical question without havingto Solve: or even; perhaps, to fully formulate the governing equations. The 24 \ AAA main Ideais both trivial and profound. tt. is that physical results mustbe . independent of the. choice:of units: Aclassic application of dimensional ‚analysis is to fluid:fiow, Suppose we : ‘are interested in i flow of velocity pi analysis leads to the vague, but extremely useful, principle that reasonably defined quantities should be sibility of a new Pythagorism?. The inability of classical physics to provide a definite scale expressed in appropriate ‘natural units’. of length had been relieved. For Planck observed that from G, cand 4 one can form Inthe system of natural units used in the Planck length: . numbers of the order unity when + ‘particle physics, quantities having dimensions of mass, length and time v2 ii [2 - 10 cm -are:given the dimensions of powers of. - energy (usually in electronvolts), which formisthe Reynolds number, Re = vL/v.. regard these as fundamental. units. So, for example, we can use model Planck proposed that in a complete More generally, by combining appropriate powers of these parameters one can reproduce any unit of measurement needed in the description of the physical world. On the other hand, one cannot combine them to produce a dimensionless pure number. Thus G, ¢ and # provide an ideal, nonredundant system of physical units. From this arises the modern Pythagoras-Planck aircraft in a wind tunnel to study the formulation of physics, not yet attained, programme: flow around real aircraft, by the only additional parameter.to appear compensating with.a larger: v for a smaller L. As long as Re stays the same, would be Newton's gravitational framework in which G, c and # are all profoundly incorporated, and to calculate within that framework all the constants of the flows will differ only by trivial reconstants of nature, expressed in these Planck units, would be calculable pure numbers. FW. -a bodyof size L, in a fluid of viscosity _ “per. unit density. Vi‘Since: v.‘has Pi dimensions of.length? /time, while vof effectively. makes Planck's constant of action #andthe speed:of light, c, both - equal to unity, Essentially all the equations in particle physics contain course has dimensions of lengih/time, the speed of light:c and Planck’s: the only dimensionless-quantity we can constant of action. 4, soit is natural to scalings. In its abstract form, dimensional NATURE | VOL 397| 28 JANUARY 1999 | www.nature.com constant G. In such a theory, all other $A © 1999 Macmillan Magazines Ltd to formulate a theoretical nature, expressed in Planck’s units, as pure numbers. This is a tall order. A particular challenge is that fundamental quantities such as the

size of atoms or the mass of the proton turn Strong insights because it is able to give an accurate and outto have outlandish values (about 10” and The modern theory of the strong interaction, 107", respectively) when expressed in Planck units. If the Pythagoras-Planck which binds atomic nuclei together, is quantum chromodynamics (QCD). This theory has notably advanced us towards getting ‘its from bits, in three distinct ways. First, it accounts, in principle, for those problematical nuclear masses. A full formulation of QCD requires, on the face of it, seven parameters: a pure number a,, analogous to the fine-structure constant, that governs the strength of the strong interaction, plus the masses of six different types of quarks, in addition to # and c. The up, down, strange, charm, bottom and top quarks are the particles that, together with the colour gluons, carry the colour charges of QCD. Although detailed account of high-energy processes, where the calculations become much simprogramme is to succeed, then the standard working assumption of dimensional analysis (see Box 1), that naturally defined entities should be of order unity in natural units, must be profoundly subverted. Asatomic physics developed, some of the spirit of the Pythagoras-Planck programme was realized, but major compromises were required. For many purposes it is a very good approximation to neglect the effects of relativity, and to regard nuclei as infinitely heavy compared with electrons. In this approximation, the fundamental equations of atomic and molecular physics can be formulated in a way that #, together with m, and e, the mass and charge of the electron, appear as the only parameters. From these we can construct a unique unit oflength, the Bohr radius: a= È em This does give the approximate size of atoms — so in this case dimensional analysis is vindicated. In a more accurate treatment of atoms and molecules (such as QED) one must include relativistic effects, and the ability of finite mass protons and nuclei to recoil. The description of these effects brings c, and the finite masses of the proton and other atomic nuclei, into the equations. (Gravity is utterly negligible here, so Gis not required.) Once c is added to the parameters of atomic physics, one can form the fine-structure constant, a, a dimensionless quantity: pler (see Fig. 1). There has also been impressive progress in calculating the masses and properties of the mesons and baryons that take part in strong interactions. These are analogous to atoms formed of quarks, antiquarks and gluons, whereas nuclei — aside from the proton itself — are analogous to complicated molecules. Representative results are shown in Fig. 2 (fora fuller discussion, see Box 2 and ref. 3). These results leave little doubt that correct values of the nuclear masses would emerge from more numerical work, but definitive calculations are probably some years off. even this would be reasonably economical, Second, QCD brings to the fore a proconsidering the amount of data to be correfound property of quantum field theories, what we might call the relativity of charge. lated, that parameter count is grossly unfair to QCD. The up and down quark masses are very small, and they are the only two quarks that are significant for nuclear physics. By According to modern quantum physics the vacuum, which evolution has selected us to regard asan empty background, isin realitya highly structured, responsive and dynamic medium. Because of the uncertainty principle the ‘vacuum’ contains virtual particles that can, like the molecules in an insulator, arrange themselves to partially screen an inserted charge. If that happens, the charge one measures at smaller distances, inside the screening cloud, or equivalently in higherenergy processes, will effectively increase. The opposite behaviour, antiscreening or putting their masses to zero, and ignoring the other quarks, one obtains an excellent approximate theory containing just one dimensionless quantity, a,. In practice it is very difficult to use QCD to calculate nuclear masses, just as it is very difficult to do self-contained calculations of chemical processes beginning with the Schrédinger equation of quantum mechanics. We have faith in the theory primarily 05 y e? as “lea 0.00735 ic This parameterizes the strength of the electromagnetic attraction between protons and electrons, or equivalently the size of the quantum of electric charge. In the spirit of a,(Q) Planck and Pythagoras, one should not be satisfied to have such a quantity appearing as fundamental in the laws of physics. Rather, one should aspire to calculate it. The pioneers of atomic physics were acutely aware of this challenge. Pauli was fond of saying that the first question he would ask the Almighty would be to explain the value of the fine structure constant. (The joke continues, that after hearing the explanation — from Satan — Pauli thought for a moment, then snapped “Wrong!”.) Thechal- | lenge escalates when we consider the nuclear masses. Indeed, by taking ratios of these masses, ortheratioofany ofthemto theelec- 0. — 1 _ - Y A Energy scale Q/(GeV) 100 tron mass, we can construct many more dimensionless numbers, To satisfy Pythagoras and Planck, we would have to calculate all these numbers, not just take them from experiment. 304 Figure 1 The relativity of charge. Value of the strong coupling constant, a,, established by a variety of experiments (data points) at different energy scales, and compared to the QCD theoretical prediction for a, (solid line). See ref. 4 for detailed references to the experiments. ZA © 1999 Macmillan Magazines Ltd NATURE | 28 JANUARY 1999 | www.nature.com

arithmically on the distance at which they are measured. So, a small coupling will evolve only very slowly. As an illustration, the strong coupling a, is observed to change from a value close to 1 at 107"? cm to about 1/8 at 107 cm, and is predicted to be about 1/25 at 10°” cm. So, whereas thestrong coupling might eventually merge with its weaker brethren, its approach is quite a drawnout affair. When we calculate where the unification takes place, we find a truly remarkable result. The strong, electromagnetic and 14 M(aGseV) 12 weak couplings, which are significantly dif- À ferent when measured at ‘practical’ distances, are calculated to become equal when measured at distances about 17 orders of magnitude smaller — near the Planck unit ofdistance. It is extraordinarily suggestive that the 0.8 Planck scale emerges here. To appreciate why, we must consider extending the notion of the relativity of charge to gravity. The sorts of charges, strong, weak or electromagnetic, 0.6 to which the interactions of the Standard Model of particle physics respond, change only logarithmically with distance, owing to subtle quantum mechanical effects. But 0.4 Figure 2 Comparison of masses of light hadrons (dotted lines) to various lattice simulations (data gravity responds to energy directly, so that it points). These calculations contain just one free parameter, the strange quark mass. Sources of error in runs linearly with energy (or inverse disthe current lattice calculations, which are believed to be responsible for the small residual errors, are tance) scale. From its much inferior strength discussed in ref. 3. at accessible energies, gravity ascends to equality with the other interactions at roughly the Planck scale. Thus we discover asymptotic freedom, though less familiar, with two massless quarks, provides a truly is also possible. In either case, the value of marvellous partial realization of the vision of the charge, or coupling strength, is not an Pythagoras and Planck. Using # and c as units, and with no further inputs — except absolute concept, but depends on how it is measured. Antiscreening is calculated to occur in QCD. The experimental evidence for this behaviour is now quite firm", as you can see in Fig, 1. Because of the relativity of charge, the QCD analogue of Pauli’s question — why is the value of the fine structure constant what the number of colour charges, of which there are three (binary *11'), and the number of quarks, of which there are two (binary ‘10°) — it accurately accounts for all the ‘its’ of nuclear physics, and much else besides. ‘Its from bits, to be sure! that all the coupling strengths become equal simultaneously. Even in the absence of a detailed theory, we find here a concrete, semi-quantitative indication that all of the basic forces arise from a common source. Time Space it is? — receives a startling answer: “It’s anything you like, at some distance or other”. We can simply declare it to be, say, 1/10, thereby defining the distance where it is 1/10. This is the phenomenon of dimensional transmu- Getting it all — or hitting a wall? tation’. A dimensionless measure of the quantum of charge, the coupling ‘constant’ has been transmuted into a unit of distance. ras-Planck programme? This brings me to my third and final point. The relativity of charge, which plays such a central role in The approximate QCD theory with two QCD, applies as well to the other interactions massless quarks appears, naively, to bea famof the Standard Model of modern physics — ily of theories, each with a different value of the coupling, and none defining a scale of distance. But because of dimensional transthe weak and electromagnetic interactions Although QCD accounts admirably for the strongest forces in nature, it is certainly not a Theory of Everything. What, if anything, does it portend for the full Pythago- Electron Photon DIDI Da FA Quark ba units they use to measure length. This differ- (although for them it is a much smaller effect). This brings up the possibility that all the couplings — that is the quanta of each of the strong, weak and electromagnetic charges — might have a common value when meaence in units matters for comparison of sured at exceedingly small distance scales (or purely QCD quantities to non-QCD quantiequivalently at high energies), despite their disparate values at currently accessible scales, There are several other pieces of evidence contributing to electron-electron scattering in pointing QED, by exchange ofa mutation, it turns out to be a family of perfectly identical theories that differ only in the ties, such as the ratio of the diameter of the proton to the Bohr radius, but it does not affect dimensionless quantities within QCD itself, such as ratios of nuclear sizes or nuclear masses. So QCD, in its slightly idealized version NATURE | VOL 397 | 28 JANUARY 1999 | www.nature.com toward this possibility, as I Giuon Figure 3 Feynman graphs. a, The simplest graph virtual photon. Ina more described in these pages last year“. accurate calculation, b, one must allow for For our present discussion, what is crucial is that the inverse couplings depend logmultiple exchanges. c, A typical contribution to FA © 1999 Macmillan Magazines Ltd the interaction of quarks in QCD.

This example of how vastly different scales emerge, provides critical insight for antiscreening) is small. An outstanding the vexing problem, fundamental for the its quantum size (Compton wavelength) But this is related, by the relativity of charge, to the exponentially smaller distance where unification takes place. Putting this idea into Pythagoras-Planck programme, of how to generate extremely large (or extremely small) dimensionless numbers. Any dynamical effect due to a large coupling automatically generates an exponentially large ratio of defined by 41 m,),....,¢ According to QCD, the an equation, we find: scales, if the fundamental coupling (before That occurs when a, is measured to be unity. example is the proton mass, or equivalently proton’s Compton wavelength is essentially determined by the dimensionally transmuted length where the strong interaction becomes strong and holds in the quarks. Box 2:Bit proliferation- crunching thenumbers Mproton a exp( kl unifica) Mptanck for the proton mass in Planck units. Here, Mpiande = (Ae1 G)'?= 10" porn is the Planck mass unit, Q, ,i¢eq = 1/25 is the common value of the strong, electromagnetic and weak cou- È are always violent, but dome plings when they unify, and k= 11/2 risacalculable numerical factor that characterizes extremely complex. outputs, there: and short-lived, quantum fluctuations: “must be a lot of logical processing inin the colour version of electric and the antiscreening. This formula works remarkably well. Suddenly one sees ‘out- If simple input parameters are to.give. between. Here! describe the 14 computationalmachinery that processes “and ‘01’ into: tables.of nuclear 2 magnetic fields, even in:what evolution has designed us to regard as ‘empty’ space (for otherwise we'd always be landish’ numbers like 10% from the perspective of exp( — 1/a) — which is actually considerably bigger — and they no longer distracted), ..i appear quite so daunting. and calculatein quantum:field theories waytoa ld these all up. The only really Although all of these developments justify optimism, it remains conceivable that the properties. | The traditional way to visualize is by means:of Feynman:graphs, successful approach has been to ‘its from bits’ programme will hit a wall. A which follow the tracks of particles crunch the numbers (see ref. 3 for particularly serious possibility is that we will in.space and time. a review): To do that, one first replaces continuous:Space-time by a lattice, converge on a unique set of basic equations for physics — many physicists believe that and restrictsattention to:a finite box. Thedetailsare very intricate-and such equations will emerge from investigations into superstring theory — but that the scattering of particles, by Clever, but one must check that the these equations will contain consistent solupinto kl:all'the ‘approximations involved in discretizing tions describing many basically different ‘and ‘boxingare: not too severe.In possible worlds. There might, for example, be valid solutions describing worlds with different electron/proton mass ratios, or different numbers of quarks. Twenty years ago, Particles that are not observed - those that are neither in the:initial nor final state- are virtual ‘particles. Feynman graphs describe ssible ways: practice; about 10° points are used, to ensure accuracy at.thé few per quantum:electrodyn: cent level. Sums over so many variables s (QED). itis. almost always agoo approximation: in-QED to use only the simplest possible. graph to describe:the: interaction, and are employed. Heroes working on numerical QCD have pushedthe’ an excellentapproximation to.use only a few. In quantum:chromodynamics sl © (QCD), ón the.other hand,theprobability that more:‘complicatedgraphs such. frontierofhigh-speed parallel processing, often designing and. constructing their own computi ng When floating point multiplications per become impractic = An entirely different.approach is calculates the masses of observed: hadrons;mesons, baryons or; in ences of the sorts mentioned above. Many principle; Nuclei and ‘gluebalis’ particulars of what we commonly regard as the most basic features of the world would then hinge on an accident of history (that is, necessary: The particle picture, epitomized. by Fey man: graphs;is an easy-to-calculate approximation for limited purposes, but the: gluons). by. dropping appropriate (bound states made purely of colour fundamental equationis:of QCD are. mixtures ofquarks, antiquarks and formulatedinterms offields fillinggluons into the roiling medium of d the simplest,: which amplified patch we emerged from). Attempts to calculate the electron mass from first principles might be as futile as attempts fields at one space-time point, and and perhaps the most profound, way: together, and how fast theymove. to calculate the shape of the Solar System, or the anatomy of frogs. Still, wemusttry. O The particles we see are the resonant Frank Wilczek is at the Institute for Advanced Study, modes, which can persist as coherent School of Natural Sciences, Olden Lane, Princeton, ‚measuring how long they hang to state the theoryis to give the rule which:governs the probability amplitudes for different configurations of the fields. This:tuletis easily stated: mathematically, is verysymmetrical, entities fora reasonable amount of and relates only:th fields at nearby:: - of hadrons are found, quite titerally,. space-time points (thatis,it is local). asthe: frequencies one can sound on The difficultyis that when.one applies. an exoticgong, constructed to purely New Jersey 08450, USA. time. In these calculations, the masses the rule, one finds that many different “ configurations «occur with substantial result that would surely have pleased probably They:reflect thatthere “Pythagoras. Hes which posits that the entire observable Universe expanded from a small patch early on, has made it plausible that the known Universe is homogeneous not for any fundamental reason, but just because we are only sampling a small patch of reality. With this in mind, we need only travel sufficiently far, or wait sufficiently long, to encounter differ- Second), Within this framework, one Substantial:cont space.and time, Universe is a very big place (volume ~10' in Planck units), But inflationary cosmology’, machines. At the moment, two different teraflop machines are devoted full time to QCD calculations (1 teraflop=10” as Fig. 3ccontribute to the interaction is notparticular! sm many complic one might have objected, against this possibility, that if there were other solutions, we should have seen regions of the known Universe where they are realized. After all, the -cahnot:be done analytically, so Monte Carlo sampling techniques e-mail: wilezek@sns.ias.edu 1. Misner,C., Thorne, K. & Wheeler, |, in Gravitation Ch, 44 (Freeman, New York, 1973). 2. Planck, M, S.-B, Pruss, Akad, Wiss, 440-480 (1899), 3. Burkhalter, R. http://xxx.lanl.gov/abs/hcp-12/9810043 mathematical specifications. It is a 4. Schmelling, M. httphoolunl.govf«bs/hep-<x/9701002 5. Coleman, $. & Weinberg, E. Phys. Rev. D7, 1888-1910 (1973). FW. 6. Wilczek, F. Nature 394, 13-15 (1998). 7. Linde, A. Inflation and Quantum Cosmology (Academic, San Diego, 1990). $A © 1999 Macmillan Magazines Ltd NATURE | VOL 397 | 28 JANUARY 1999 | www.nature.com