arXiv:physics/0110060v3 [physics.class-ph] 13 Sep 2002 r aou ntenme ffnaetlconstants fundamental of number the on Trialogue Abstract: Veneziano Gabriele Okun B. Lev Duff J. Michael Keywords: zero. in advocates MJD argues curr while GV our theory), constants, explain three superstring returne to with we decided approach 2001 and traditional of diverged the summer still the views In our in cafeteria. that debate CERN our famous started the We of physics. in constants dimensionful c IS/SS2008 SISSA/ISAS E-mail: France Orsay, 91405, Sud Th`eorique, Universit`e Paris Physique de and Laboratoire Switzerland, 23, Geneva CH-1211 CERN Division, Theory E-mail: Russia 117218, Moscow, ITEP E-mail: USA Laborato MI, Arbor, Randall Ann Physics, Theoretical for Center Michigan [email protected] [email protected] [email protected] hspprcnit ftresprt rilso h number the on articles separate three of consists paper This oeso unu rvt,Sadr Model. Standard Gravity, Quantum of Models ulse yIsiueo hsc ulsigfrSISSA/ISA for Publishing Physics of Institute by Published tp/je.is.tacieppr/hp3022 /jhttp://jhep.sissa.it/archive/papers/jhep032002023 ry n oiin.LOdevelops LBO positions. ent ao fa otto(within two most at of favor umr19 nteterrace the on 1992 summer otesbett find to subject the to d Accepted: Received: ffundamental of ac ,2002 9, March 2002 4, March hep032002023.pdf

S JHEP03(2002)023 JHEP03(2002)023 5 4 5 5 6 6 7 7 8 9 9 3 4 3 9 3 15 16 18 10 G. Veneziano y (QST) 12 L.B. Okun – 1 – = 1 G , = 1 ~ , = 1 c c ~ G 3.1 The meaning3.2 of The meaning3.3 of The status of 4. The cube of theories 5. The art of putting 6. International system of units 7. Remarks on Gabriele’s part II8. Remarks on Mike’s part III9. Conclusions II Fundamental units in physics:1. how many, Introductory if remarks any? — 2. Three questions and one answer 3. Fundamental units in QFT+GR 4. Fundamental units in old-fashioned quantum5. string theor Fundamental units in modern QST/M-theory6. The disagreement with Lev 7. The disagreement with Mike 8. Time variation of fundamental units? 13 I Fundamental constants: parameters and units — Contents 1. Introduction: parameters and units 2. Stoney’s and Planck’s units of3. L, T, The M physical meaning of units JHEP03(2002)023 27 23 24 19 ts ns 25 – 2 – M.J. Duff Party — III A party political broadcast on behalf of the Zero Constan 1. The false propaganda of2. the Three Constants The Party false propaganda of3. the Two Constants Three Party fundamental theories? 4. An operational definition 5. The operationally indistinguishable world6. of Mr. Tompki What about theories 19 with7. time-varying constants? Conclusions 20 27 JHEP03(2002)023 , ~ ion I sug- f standard nsionless (like bsence of the latter [1]. eoretical model at hand ed for such dimensionful ss M with corresponding is article contains no new latter as fundamental (or l constants of Nature. It ysical quantities. Theoret- ourse with the evolution of no mixing angles. It blows of accurately defined terms colleagues of mine, I believe d, involve measurements, i.e. c units in order to reproduce (or basic) units. nsionless quantities and their ess ratios, such as the famous andard dimensionful units and nstants which cannot be . But experiments, from which name of C.F. Gauss. In spite of to confusion and proliferation of — quantum of action and angular ~ . This article is concerned with these G – 3 – Lev B. Okun , the quantum of action (and of angular momentum) c — velocity of light, c — Newton’s gravitational constant). To clarify the discuss G 137 and similar gauge and Yukawa couplings in the framework o / There are two kinds of fundamental constants of Nature: dime 1 ≃ c ~ 137) and dimensionful ( / / 2 1 e Physics consists of measurements, formulas and “words”. Th It is clear that the number of such constants depends on the th On the other hand the term “fundamental constant” is often us expressed in terms of more fundamental ones, because of the a ≃ = momentum, and gest to refer tobasic) the units. former It as is fundamentalin necessary parameters an and and experimentally sufficient the meaningful toical way have equations the three describing basi dimensions the ofsolutions physical depend all world on ph deal dimensionless fundamental withthese parameters theories dime are extracted andcomparisons by with which standard they dimensionful could scales.hence be without Without teste certain st conventions physics is unthinkable. 1. Introduction: parameters and units There is noseems well reasonable established to consider terminologyα as for fundamental the the dimensionl fundamenta dimensionful constants which I propose to call fundamental and the Newton gravitational coupling constant formulas, it deals mainly withthat “words” an because, adequate unlike language many or is the crucial uses in (i.e. physics. actuallywrong misuses) statements. The of absence ill-defined terms lead 2. Stoney’s and Planck’s units ofThe L, three T, basic M physicalmetric dimensions: units: cm, length sec, L, gram, are time usually T, associated with and the ma model of elementary particles or its extensions. At present this numberup is with a the few inclusion dozens, of if hypothetical one new particles. includesconstants neutri as the velocity of light and hence depends onphysics. personal At preferences each and stage it of changes this of evolution c it includes those co α Part I Fundamental constants: parameters and units Abstract. JHEP03(2002)023 n α as c ] or action /T 2 has been derived ML S ). At first sight t has been derived by c , when the standards S . ] = [ ncy of a certain atomic rm “electron” and mea- and m s which cannot be calcu- /c ically close to each other, t depend on the number S cm and sec are connected S l um and mass. l orld without gravity it still MvL and to bject in Nature, the photon would not be diminished in a given to us by Nature herself e: it is the basis of relativity hree corresponds to the three = c till necessary and sufficient to e Nature herself suggests er ignorance). This “negative” α ~ S t depend on the dimensionality t ] = [ ian. , J leads to new phenomena, unknown S because its fundamental character and . v /l c 2 α e c/G ~ = 2 p c S = . The expression for m G P , he introduced [3] as universal units of Nature – 4 – does not depend on the choice of units, whereas ~ The gram is the mass of one cubic cm of water. √ m α , 1 , introduced as universal units of Nature for L, T, is the maximal e/ 2 e c c = P S /m ]. (Here [ ] denotes dimension.) m ~ , 3 /T ] and angular momentum [ = 2 P t G/c L/T is more fundamental than depends explicitly on the units of length and time and hence o , ML √ is not only the speed of light in vacuum. What is much more c c c c e P c . ] = [ = ] = [ v α /m because the value of T S on dimensional grounds: ~ √ t 2 c , e = 2 Mv P l and G/c c ] = [ √ , e S ET = m In the Introduction we defined as fundamental those constant Enormously more universal and fundamental are An important step forward was made in the middle of XX century In the 1870’s G.J. Stoney [2], the physicist who coined the te When M. Planck discovered in 1899 One can easily see that Stoney’s and Planck’s units are numer S l metre was defined in 1791 as a 1/40,000,000 part of Paris merid 1 ] = [ S conventions. However has not only atheory “negative” definition, which unifies but space also and a time, “positive” as on well as energy, moment the numerical value of fundamental unit of velocity. lated at our present leveldefinition of fundamental applies knowledge (or equally rath to parameters and to units (to looks superior to in newtonian physics and described by relativity. Therefor of cm and secline. were defined in terms of of wave-length and freque 3. The physical meaning of units The Gauss units werewith “earth-bound” the and size “hand-crafted”. and The rotation of the earth. as units of velocity [ express the dimension of any physicalbasic quantity. entities The (notions): number t space, timeof and space, matter. being It the doesand same no nature in of spaces fundamental ofwould interactions. be any three. dimension. For instance, It in does a no w sured the value of elementary charge tremendous changes in physics, three basic dimensions are s for L, T, M: significant is the fact that it is the maximal velocity of any o [ being only one of such objects.world without The photons. fundamental character The of fact that their ratios being from M: 3.1 The meaning of It is important that equating the Coulomb and Newton forces. The expressions for JHEP03(2002)023 . ) c J S G G , 0 or c ~ J ), en wave- /c ). (As is well . ~ as the unit of But the bridge GUT 2 ~ . When 2 S and M c = y or SUSY, is badly 2 q o energy. This comes to use hen quite seriously by ( (actually by 1 uction. The majority of α .) 2 c rticles with half-integer q = ) one simplifies relativistic 00) — to special relativity, c 9, 10]); and it is known now v/c ’s potential at sub-millimetre d in the literature. Note that sic for the structure of atoms, restored near the Planck mass GUT esent only as wishful thinking. α -time. s, ( = nd superstrings. ght-seconds, the length does not β 0) — to quantum field theory, ( , where ~ ues depend on c , is at present different from that of P GUT as a conversion factor between time and t α is not as firm as that of , c √ / P G l P l – 5 – , = P s and a natural unit of the action λ m J units in classifying theories was first demonstrated in is is also fundamental in the “positive” sense: it is the G (usually it is defined as ~ , c ~ , (bosons) tend to be in the same state (i.e. photons in a laser, c J ~ G in units of v as a conversion factor between frequency and energy or betwe and its derivatives, , the quantity ~ c G , the whole realm of quantum mechanical phenomena appears. ~ . S , because the quantum theory of gravity is still under constr ~ become identical to time, just as momentum is not identical t It is natural when dealing with quantum mechanical problems By expressing The cube is located along three orthogonal axes marked by Symmetry between fermions and bosons, dubbed supersymmetr The role of Particles with integer The characteristic length of a superstring 0) — to non-relativistic quantum mechanics, ( 2 and ~ kinematics. On the other hand the role of distance or between massin and spite rest-energy of is the often possibility overstate of measuring, say, distance in li from the pseudoeuclidian nature of four-dimensional space 3.2 The meaning of Analogously to length and momentum is often overstated. J known, the fundamental parameters are “running”: their val 3.3 The status of The status of and between superstrings and experimentalRecent physics surge exists of at interest pr distances to demonstrates possible that modifications the of position Newton of (0 quantum of the angular momentum experts connect their hopes with extra spatial dimensions a The vertex (000) corresponds to non-relativistic mechanic 4. The cube of theories The epistemological role of a jocular article byM. Bronshtein G. [5, Gamov, 6], D. A.as Ivanenko Zelmanov the [8, and cube 7] of L. and theories. Landau others [4], (see e.g. t [ (fermions) obey the Pauliatomic exclusion nuclei principle and which neutron is stars. so ba broken at low energies, but(in many particular theorists in believe superstrings that it and is M-theories). or Rubidium atoms in a drop of Bose-Einstein condensate). Pa are close to JHEP03(2002)023 , .) P s ties λ ,m ~ in equa- c, , G ~ c, . The Boltzman P is a critical value. k ues of dimensionless reductionism”, which ,m tical predictions with ~ f fundamental particles number of fundamental c, he point of view of pure conversion factor Moreover it stresses their e units is realized by putting e transition from a few (or adically differs from ones. The SI might be useful ative ones. Electric current is onal System of Units (Syst´eme molecules in one gram-molecule, des 7 basic units (metre, second, , for which of the flux of photons. ” (which mean “substituted by”) is played by the string length makes natural their use in dealing k ful, while the right-hand sides are → P P l ,t with that of ”) the three basic units P k , l ← P m /mole. As for unit of optical brightness or =1 23 – 6 – G 10 joules/kelvin) is the average energy of an ensemble , × 23 − =1 02 . 10 ~ and hence of × , = 6 38 . , G A ) — to futuristic quantum gravity and the Theory of Every- =1 ~ (or any of them) in the so obtained dimensionless equations N G c c, = 1 ~ c k , G ~ ) = 1 in all formulas. However one should not take these equali s c, λ (or G = 1, ~ It is interesting to compare the character of (Boltzman constant The above arguments imply what is often dubbed as a “moderate can be explained in terms of a few fundamental interactions o It is necessary to keep in mind that when comparing the theore variables become dimensionless. In practice the use of thes The absence of = 1, constant is anone) important particle conversion systems factor to many which particle signals systems. th However it r illumination (candela), it is obviously expressed in terms fundamental parameters will be ultimately calculated. 5. The art of putting thing, TOE. There is a hope that in the framework of TOE the val and thus expressed indimensionless parameters. terms of three basic units and a6. certain International system of units An approach different from the aboveInternationale — underlies d’Unit´ees SI) the [11, Internati 12]. Thiskilogram, System ampere, inclu kelvin, mole,from candela) the and point 17 of derivative physics view four of out technology ofnumber and its of metrology, seven moving but basic electrons from units per t are second. evidently Temperature deriv is up to a tions because all measurements involve standard scales. in this case means that all physical phenomena — to general relativity, ( universality and importance. experimental results one has anyway to restore (“ does not diminish the fundamental character of these units. as there is no physical quantity with the dimension of of particles. Mole is triviallycalled connected with Avogadro’s the number number of The universal character of with futuristic TOE. (In the case of strings the role of too literally, because their left-handdimensionless. sides It are would be dimension more proper to use arrows “ In such natural units all physical quantities and c instead of equality signs “=”. JHEP03(2002)023 s = 12 , λ − 0 ~ µ 10 0 c, , while ε × it is the S k 85 . and ”). = 8 J 0 , whereas ε 2 omical choice is to hep-ph h to his section 2, in tivity , whereas for ”, “ concludes section 2 with h in SI is secondary with ike, but that is a purely sides of physics, the most sionful constants [14, 15]; P owing statements: 1. that red directly [16]. Gabriele nsionless numbers. Accord- as a unit of as fundamental units. ntal units. According to my onfusing” and the set ,m ~ ndamental constants with the ~ and statements do not appear l.” Mike explained to me that ical” and in other parts of the of which can be communicated on factors is fundamental.” , G t not in text-books (see critical c, ~ is not a fundamental unit (LO)]; = 1. According to my section 5, physics c, c ) explicitly in the equations, while ~ newton/(ampere) s = 1. According to the SI standard 0 are as good for the role of fundamental 17 c, λ µ − F = 10 G 0 × ε 57 . – 7 – . ~ = 12 ), as Gabriele is doing in part II and refs. [14, 15, 16]. 0 ~ µ = 1. As these equalities cannot be considered literally, I instead of ”), and not physical one (“ G . 2 s = G λ ~ = hep-th c as units of velocity and length. s λ and c In fact, at the end of section 2 Mike writes “that the most econ (In response to the above paragraph Mike added a new paragrap I also cannot agree that the electron mass, or It seems to me inconsistent to keep two units ( In the framework of SI vacuum is endowed with electric permit Of course, if you forget about the pedagogical and practical . This is caused by electrodynamic definition of charge, whic 2 /c use natural units where therehis are natural no units conversion are factors at al unit as the Planck mass or which he ascribed to me the view that one cannot put believe that Mike uses thea same statement: three “Consequently, none units of as these I units do. or However conversi he 8. Remarks on Mike’s part III In section 4 of Mike’shelp part of III an he alien introduces with aing whom to definition it Mike, of is only fu possible those constants toto are exchange fundamental the only the alien. values dime Thus Mikesection concludes 5 that above, there this exist actually no corresponds fundame to the use of not using According to my section 5 above, this corresponds to using substituting by unity the third one ( only one. 3. that massesadmits unlike in length section and 6 time that intervals his are two units not can measu be “pedagogically c economical way is nottheoretical to approach (“ have fundamental units at all, like M this definition is allowedexposition to of use SI in in scientific ref. literature, [13]). bu 7. Remarks on Gabriele’s part II I note with satisfaction thatin some his of part the of originalin arguments this string Trialogue theory II. there Among2. is them room that there units only of are for action the two are and foll arbitrary not [which three means dimen that respect to the current. In electrostatic units farad/m and magnetic permeability 1 The role of conversion factor is only a secondary one for is “most practical”, but hepart considers II the he latter insists “not on econom using JHEP03(2002)023 d Michigan Center of to make the following , are the same as in our ion 5 of his paper Mike iscussing their points of should be explicitly pre- ~ s ng. We, theorists, commu- s) three fundamental units ycle, while the everyday life ft of this article and helped is changed, while dimensions unctions and constants have e critically analyzed. In these rms. onful units and compare their c e without standardized dimen- ch was used in ref. [10], where s, students, and non-physicists. ms (and hence the everyday life) n of fundamental physics. Other ivistic kinematics, the velocity of s appreciated, as well as Humboldt , which is the ratio of electron velocity in α – 8 – = 1 in relativistic eqations, but must understand that this c is chosen as the unit of velocity.) c The “alien definition” of fundamental constants is misleadi Concerning Mike’s criticism of my article [10], I would like hydrogen atom to that of light. It is not clear to me why in sect disagrees with these considerations. 9. Conclusions It is obvious thatare using the proper only language possibleconclusions (terms basis are and for viable a semantic only selfconsistent through descriptio the improper usage ofAcknowledgments te I am very gratefulview. to I Gabriele am indebted andto to Mike Val improve for Telegdi it. who their criticallyTheoretical patience Physics read The during the in the hospitality dra completion d of ofAward this article and the i the CERN grant of Theory RFBR Division 00-15-96562. an world), then electromagnetic and opticalwould properties change of drastically ato because of change of of atoms are not changed (mass and charge of electron as well a means that one can (and should!) put remark. The statementphysical that meaning only in dimensionless asented theory variables, in does f dimensionless not form. mean Sometimesratios one that with can every use ratios problem dimensi ofentertaining other stories dimensionful by O. units. Volberg [17] Thisstories, and approa in G. Gamov order [18] to wer light demonstrate was the assumed to peculiarities be of ofremained relat the the order same of as that ours. of a In car, ref. or [10] even I bic have shown that if nicate not with aliens, butSuch with communication is our impossible experimental and colleague sionful physics is units, unthinkabl without conventions.. JHEP03(2002)023 (2.1) mental ssary and sufficient. e emergence of funda- of (dimensionful) fun- quantum string theory antities, typically repre- ], whose way of looking ke some trivial introduc- ing from. be to their content, even myself arises from. I have g. y, and modern string/M- ile, in section 7, the same what surprising conclusion sagreement on these issues se days. Here, rather than I tried to incorporate in my he words than in the physics, nful constants following Lev’s arious theoretical frameworks and summarize them stressing e of time-varying fundamental vely. In sections 6 I will try to , ... e p m m , – 9 – c 2 e ~ = Gabriele Veneziano α I summarize my previous work on the question of how many funda constants, with dimensions of space and time, were both nece Physics is always dealing, in the end, with dimensionless qu senting ratios of quantities having the same dimensions, e. two uncontroversial. In sections 3, 4 and 5 I describe how I see th After reading those papers of mine once more, I still subscri The rest of this note is organized as follows: In section 2 I ma • tory statements that are hopefully dimensionful constants (fundamental units)such are as needed in renormalizable v QFTtheory. + I GR, will old-fashionedbetween also Lev string try Okun, theor to Mike underline Duff, where and myself past appears and to present1. be di originat Introductory remarks Some fifteen years agodamental I constants in wrote string athat theory, short where I letter came [14] toSomewhat on the later, the some I number becameat aware the of question of S.subsequent defining Weinberg’s work fundamental 1983 on constants paper the in subject [1 physics [15, 16]. if I might haverepeating expressed the details some of specific mywhere, arguments, points in I differently will my try opinion, the to thethe organize disagreement impression between that, Lev, in Mike thebut and end, this the is disagreement is what more we in should t try to find out. mental units (the name I willsuggestion) adopt in for fundamental QFT+GR, dimensio in(QST), the and old in the Nambu-Gotopoint so-called formulation at Polyakov of formulation, the respecti originwill of be disagreement done between w.r.t. myself Mike.units. and Section Lev 8 wh briefly discusses the issu 2. Three questions and one answer Let me start with two statements on which we all seem to agree: Part II Fundamental units in physics: how many, if any? Abstract. JHEP03(2002)023 , 2 3 in Q A ~ have (2.2) , also c ~ matter and j and 2 c /q and A i . Obviously, q c he framework we rs to be an obvious lementary particles number the ratios of various ce only ding to S. Weinberg’s speed of light, e, answers al) relativity the ratio we compute ) is a q tio of any physical quantity tions therefore analyze fore that both d Model (SM)) plus General f various terms in the action in QM, the ratio of the total he string revolution of 1984. out FC’s at the dawn of the q/u , q ) and ( does matter (large ratios, for instance, u ) appear to be more fundamental units of ~ q ~ so that q/u second and or = ( – 10 – c q centimetre of the same kind ). q q u (e.g. /u 2 q ( / ) name q /u 1 is a q q u = ( definition [1]? 2 : How many units (fundamental or not) are necessary? : are there units that are more fundamental than others accor : are units arbitrary? : a qualified yes. within three distinct frameworks, and provide, for each cas 3 2 1 /q to a fixed quantity 2 1 3 Q Q Q and try to answerand them interactions. in the context of different theories of e q speed and action thanvelocities any other. in In a newtonian givenof mechanics problem each only matter. velocity appearing By in contrast, the in problem (speci to the (universal) It is customary to introduce “units”,q i.e. to consider the ra Let us now ask the following three questions where At the QFT level of understanding A a special status as the mostIndeed, basic let’s units apply of S. speed Weinberg’s and criterion action. [1] and ask: can correspond to a semiclassical situation). It appears there matters. Likewise, in classical mechanicsmatter, only the the ratios overall o normalization beingaction irrelevant while, to the (universal) quantum of action terms of more fundamental units? Within QFT the answer appea Q I hope we agree that the answer to the first question is yes, sin I think that the answer to the other two questions depends on t • • • and are considering (Cf. Weinberg, ref. [1]). The next three sec and these ratios do not depend on the choice of units. respectively. 3. Fundamental units in QFT+GR Quantum Field Theory (QFT) (or more specificallyRelativity the (GR) Standar represent the stateWeinberg’s of 1983 the paper art in [1]string HEP reflects revolution. before therefore t I the would summarize attitude it ab briefly as follows: JHEP03(2002)023 n as c (3.1) (3.2) cified in the P , . S M dx dV 1 silly” according nits, but we can m erything that is te unit for mass. e action principle and action, then, nciple at least, in would be replaced e third question is e? Why not less? . This forces us to , vide the action by, ear as concepts that ~ ≡− instead of ut how to go from the dx ce a separate unit for ˜ dV three [13], and for how asses are pure numbers in terms of F its ical theories [10]. fundamental in the sense p = ervals. Let us recall how ed by an arbitrary factor. m ≡− a separate units for distances F or ⇒ e = dt m  ma ) x ). However, in the presence of many ( . If we have already decided for c ⇒ t ˜ V dt × , it is much less − ~ 2  2 v ) ˙ x x ( 1 2 and – 11 – V  c ). − Z α 2 ˙ x ≡ , does not look like a particularly good choice since it m P dt 1 2 M , and of dimensionless constants, without also introducing   ~ ) , x c Z ( m V = would define therefore a fundamental unit of time (the Compto − S by varying the action 2 ~ , i.e. in terms of something more fundamental (and of some spe ˙ x ~ 1 2 ma  = and Z c and would have dimensions of F = ˜ /m S ~ : most probably three 3 SM + GR framework. Thisonly is identify a two bit fundamental strange: ones. we SoWhy seem why to not three? need more? Why three not u to mor present This understanding is oftemperature, because physical for it electric phenomena) looks current toin and unnecessary introdu the resistance, (and x, etc., y even“ or andseven z units directions. of the I International referthree System to fundamental of Lev units Units for define (SI) a the down discussion to so-calledAnd abo “‘cube” why not of less, phys say justare two? qualitatively Well because different mass from, ormass say, energy emerges app distances in or classicaland mechanics time (CM). get int We can base CM on th is very hard, even conceptually, to compute, say, by some mass or lengthindeed scale. three. Unlike Hence in it the looks case that of the answer to th of SW. The Planck mass, obvious, however, which mass or length scale, if any, is more but, as it’s well known,If classically we the had action only one can species be of rescal particles in Nature we could use, It is quite clear, I think, that in QFT+GR we cannot compute ev observable in terms of No physical prediction would change byprovided using units we in redefine which forces m accordingly! In this system of un no. Had wewithin chosen a instead given some theory,terms we other of would arbitrary be units able of to speed compute them, in pri A dimensionless constants such as unit of velocity, particles of different mass, we cannot decide which mass to di which choice is most fundamental. I think there is evenQFT a is deeper affected reason by why UV QFT+GR divergences needs that a need separa to be renormalized wavelength of the chosen particle divided by • JHEP03(2002)023 , p e m m by a (4.1) s ~ viously it , and has to ales (be it a tring theory. P M y (QST) or ) originates from a ~ rinciple, to compute p it as a pure number sary to introduce by , , isfy SW’s criterion of in in the renormalized eeds two fundamental ) as unit of mass and . s us to replace ll understand how m c rary. Perhaps one day, units of energy (energy h here were a single particle, , the units of action (hence (Area) d disappeared? My answer was Z ~ 2 − s λ = ~ S , appears in the Nambu-Goto (NG) action: T , (see below for an example). Of course, I could – 12 – s λ , i.e. one fundamental unit of speed and one of length. (Area) s d and λ has already been implicitly used in order to talk about c in terms of two other physical quantities defining more in terms of something more fundamental! Z c s s T and λ λ c = , which turns out to be fundamental both in an intuitive sense s S λ and and c c ! , but in QFT+GR it is not. We do not know therefore which of the s , is more fundamental and the same is true for the electron mas λ P p M m and or c etc. etc. P F M ) have also changed. And what about my reasoning in QFT+GR? Ob G : yes, : the above two constants are also sufficient! ~ 2 3 With string theory the situation changeshence because a it is single as mass. if t Indeed,classical a parameter, single the string tension A The apparent puzzle isthen clear: (and still where is): hasbeing it our replaced changed by loved its energy dress! divided byof Having tension, adopted i.e. new by length) mechanism known as dimensional transmutation,through and string is arbit theory or some other unified theory of all interactions, we wi two, is related to This was the conclusion of my 1986 paper: string theory only n dimensionful constants A having computed down-to-earth units of space and time, but this would not sat introduce a cut-off which, in principle, has nothing to do wit The best we canparticle do, mass in or QFT+GR, aconsider is mass the to extracted ratio from take the any ofthat, strength one any in of of other general, a these physical we force have mass mass no to sc way the to compute, chosen even un in principle for be “removed” in the end. However,theory. remnants of the In cut-off rema QCD, for instance, the hadronic mass scale (say instead compute does not hold waterFurthermore, any the more: absence For ofhand one, UV a QFT divergences cut and makes off. GR it get unneces unified in s and in the sense of S. Weinberg. Indeed, we should be able, in p any observable in terms of where the speed of light well defined length, the area of a surface embedded in space-time. This fact allow • • 4. Fundamental units in old-fashioned quantum string theor JHEP03(2002)023 , α for and ory, (4.4) (4.3) (4.2) ~ not together. elow). 3 sitting at a A 2 , and at allows the L /c s the new Planck ), appear in the , but not s . As noticed by and λ 3 but the transition − xt section. Clearly, 2 essary. c antities and a pure , /M gy levels of atoms in ~ A e are e and ke second quantization here are definite hopes ǫ N ory would only contain ): 2 work, the gravitational m s n 2 c ve s G α 2 λ E al forces in a geometrical ) λ = α ,L rst-quantized string theory 2 p 1 c e e . However, if we argue that  L ǫ = ~ 2 m 1 ( ≫ m P 2 l r 1 n , − 2 on a string of length  2 1 − 1 1 n 2 r L L . These are given, to lowest order in =  ~  2 2 1 2  c (in the above string units, i.e. as lengths) in 2 2 s ~ e p g )= , and some in principle calculable dimension- s n  m = λ – 13 – e E 2 m − a 2 and and m 1 n P 2 c E ( − M 1 ~ (which are like the “modern” units of length, and time), , and the Planck length = c = n 12 E mn /ω ω the UV cutoff. We can express this by saying that, in string the induced by a string of length and 1 and of a dimensionless parameter, the string coupling (see b 2 is c 12 a , , 2 s from it. A simple calculation gives (for s λ c/ω is the (dimensionless!) string coupling discussed in the ne λ r s g fundamental units, separately. We may view string theory as an extension of GR th N play the role of fundamental units of length and time. instance answer [14]. Obviously, if we follow Weinberg’s definition, distance less ratios (such as the electron mass in string units, we see that, once more, only The situation hereNovikov reminds and Zeldovich me [19, part of V,two that ch. 23, of par. 19], pure such quantum a the gravity first quantization provides the UV cutoffwell needed defined. in order Furthermore, to in ma quantumto string be theory able (QST), to t compute both the answer doesnumber. not contain anything elseAnother but more geometrical familiar qu exampleterms is of the the computation electron of mass, the its ener charge, and less fundamental than the electron charge, mass and Weinberg argues, convincingly I think, that the quantities where And indeed theconstant, most amazing outcome of this reasoning is that terms of introduction of all elementaryway. particles No and wonder all then fundament to findLet that us only consider geometrical units foracceleration are instance, nec within the string theory frame G by what we arefrequencies really measuring are not energies by themselves, 5. Fundamental units in modern QST/M-theory We now turn to the samein set of the questions within presence the of context of background fi fields. Here I will attempt to gi JHEP03(2002)023 i ) es ty α µν (5.1) (5.2) G h converts µν G actually provides z 2 d wo-dimensional Rie- µν  G (say for the Hydrogen remains the same. In ) ply stems from the fact mean something (in the µ 2 mically. As a prototype, X ) s ( y convention-independent X λ φ the string length parameter T) can be compensated by X wer is a bit subtle. String . Similarly, it should allow )
or to compare them to those tation value of ground fields whose vacuum φ γ metric The dimensionless action (i.e. ( s le parameters of string theory, is the so-called dilaton. Finally, and λ R ,... φ c are pure numbers whose possible 2 is also dimensionless. − s 2 )+ ν ) and , λ X X gives the dimensionless parameter — X 2 c ( µ − s φ λ µν X 2 of the previous section via: G c µν , the flat minkowskian metric, then it will ν s − G λ and (∆ is clearly dimensionless, while the dimensions µν X η does)? What is, in other words, the difference β i φ values of ∂ – 14 – φ i 1, are the string coordinates as functions of the . h µ and φ τ h − X c ) define genuinely different theories, values of is clearly of fundamental importance. However, in = diag α α for some physical length ∆ ∂ s i in terms of the VEV of and ) ν λ αβ absolute (or . Both σ α — that controls the strength of all interactions (e.g. X X ), with respect to which the the partial derivatives are de- γ 2 ( i ,...,D ) s  ∆ are such that φ 1 g and µν µ h γ X , σ, τ G µν c X − h as in the example discussed above, the actual proportionali G = ( ∆ √ = 0 is the so-called string metric and is proportional to µν z in more conventional notation) reads: µν µ Z η ~ G µν 2 1 µν ), G appearing in (5.2) can be reabsorbed by a GCT. This is why it do G ≡ = i ∼ s 2 σ, τ ) λ ( S µν µ , i.e. the X and thus define the same theory as long as (∆ G h X ) are, respectively, the metric and scalar curvature of the t and X γ = c ( µ R X numbers. Thus, through its non trivial dimension, the The exponential of the expectation value of fundamental units of space andIf time the that VEV we are of after. The mere finiteness of The difference between the two kinds of constants, if any, sim and metre/clock αβ world-sheet coordinates dimensionful and dimensionlessexpectation alike, values are (VEV) replaced weconsider hope by the to back be bosonic able stringthe to in action determine a divided dyna gravi-dilaton by background. The beautiful feature of this formulation is that all possib fined. Furthermore, γ where of the metric components mann surface having coordinates lengths and time intervals into pure the and thus also the string-loop expansion. Instead, the expec same way in which the actual value of known as the string coupling automatically introduce the constants our context, the real question is: do the actual values of theory should allow to compute between dimensionful and dimensionless constants? The ans to compute (∆ atom). Calling that pure number soin many centimetres cm would fix but, of(physical) course, quantity is this just would (∆ be just a convention: the trul that, while different values of values distinguish one theory (or one vacuum) from another. that are related bya a General GCT Coordinate on Transformation (GC particular, if constants not make sense to talk about the JHEP03(2002)023 ν X . In ∆ via a α µ µν X the most η ∆ µν g , a meaningful X eason to introduce ut having just two istent string theories amental constants of ifferent value of , a truly fundamental sically distinct) choices and energy are logically µν he standard definition of inkowskian, in particular dinary Quantum Gravity angular momentum. But, cal GR only G he question of fundamental ent non-perturbative vacua pared (see section 7). nt string vacuum, we would ure numbers whose absolute cts, i.e. to complicated bound le, yet unknown, “M-theory”. ing also, through QM, an UV classical strings), only ratios of ting on the use of the same (or since we can identify the energy using. Therefore I do agree that cal dimensions for momenta and is, for each single ∆ would not be reducible to 2 , i.e. the values of some physical length i ) 2 ) µν X X G area and the Planck constant is the unit of h the – 15 – has not really disappeared: it has actually been ~ proportional to the de-Sitter metric, stressing the fact i has disappeared from the list. He claims that, without ~ µν disappears in favour of a dimensionless parameter similar s G h λ define at present, within QST, the most practical (though not s . The action is naturally λ ~ or , and T ~ physically meaningful. Note, incidentally, that in Classi , c . Thus, as stressed in [15, 16], my early statement in [14] abo is i I would concede, however, that, given the fact that momentum To rephrase myself: within the NG action there seems to be no r In conclusion, I still stand by my remark in [15] that the fund To summarize, QM provides, through the string metric The situation would be very different if φ h , there is no unit of momentum, of energy, and, especially, of distinct from lengths andof times reciprocal) for units ordinary for objects, boththe sets insis set can be pedagogically conf promoted, in string theory, to a grander role, that of provid cutoff that hopefully removesand both the the ubiquitous infinities singularities of of Classical QFT GR. and or metre/clock allowing us tovalue reduce space-time distances to p economical) set of fundamental units. a tension Nature are, in QST, theof constants its of VEV’s the does vacuum.correspond QST How to many allow? perturbations (phy around WeWe different now still vacua believe of do a not that sing know,M-theory all has. however, known how Until cons many then, physically Iunits inequival do in not QST, think but we Ifind can would really that be answer we t very need surprised more if, than in one any unit consiste of length6. and The one disagreement of with time. Lev Lev cannot accept (part I) that pure number. is an invariant. However, inquantities the of classical case this (and type even matter for while in QST, (∆ as I said in the previous two sections, ~ constants should be consideredin valid the if NG the formulation vacuum of of QST. QST is m to that, in such a vacuum, even area needed to convertcanonically the conjugate action variables, into this alengths would number. (or lead for However, to times by identi and t of energies). a string For strings with its that’s length, fine, but when it comes to ordinary obje ref. [20] I gave the example of of an alien: only the dimensionless numbers (∆ GCT. That would mean a really different world, like one with a d or speed in those units are physically relevant and can be com JHEP03(2002)023 r , in and B c k is different c a fundamental ly tell an alien not . ime. Mike seems to n will be able to tell d the speed of light. T is s the idea that we do y communicate to an ion factors, like , everything into pure definitions of cm. and e ancestors (or descen- c g. worlds with different and the velocity of the y the H atom) and ask: artificially, a conversion in cm/s for any of those le from ours. I conclude, two physically equivalent she does, then she should c c our al. sing to give momentum a unit nitions of cm/s) or else, again, her world is ke’s alien I have also changed the alien’s 10 10 × = 3 a relevant number to be compared with the c is the same but atomic properties differ. No is c – 16 – c in these units? If the alien cannot even identify the s λ and check whether they agree with ours? I claim the s λ and c and , so that energies are now measured in ergs, in GeV, or whateve by saying that c T agrees with ours, she will also be able to tell us whether her a fundamental unit because the or α express this by saying that, in the two different worlds, is 3 c either agree with ours. s s. in terms ofwhich a are physical your system values of she can possibly identify (sa electron in the H atominstead, are that twice as large, is indistinguishab answer to be: yes,us we whether can, her and, to theλ same extent that the alie In order to do that, we “simply” have to give the alien velocity of our electron in units of not like ours. It thus looks to me that the alien story support have, in our own world,agree some with fundamental me units on of the length alien’s and reply, t but then concludes that to distinguish left fromalien right. our values Then for he asks: can we similarl The alien story Mike quotes an example, due to Feynman, on how we could possib order to convert any quantity into any other and, eventually unit because a completely rescaled world, in which both experimental result will be able to distinguish about these statements since a rescaling of all velocities is inessenti system then she lives in a different world/string-vacuum; if come up with the same numbers (e.g. Reducing fundamental units to conversion factors Mike’s second point is that these units can be used as convers in atomic units, alien’s. Incidentally, the same argument candants) be of applied ours, either or to to som inequivalent string vacua. A value of which differs from ours wouldratios really of mean different the worlds, velocityWe e. of may the electron in the Hydrogen atom an To stress that my alien’s reaction is different from that of Mi • • 3 gender. other than length.factor, In the order string tension to do that we introduce, somewhat states of fundamental strings or branes, it looks less confu we wish, different choices being related by irrelevant redefi 7. The disagreement with Mike Two issues appear to separate Mike’s position from my own: JHEP03(2002)023 . 2 ~ ) 2 has X 2 ) of pure (∆ X / ) without 2 ) 1 ck constant µν X G uantum unit of dering space as As I stressed in in terms pure number. candidate for the GR it looks most f length and time umber (∆ quantities (and of ntal length, proper- QM, a fundamental rod (and clock) that able to tell the alien k, at present, more like (∆ intervals. scaling ifference, for instance, damental in the sense rescaling of their size, to provide the missing quantum mechanically, The important point is . There is, in relativity, r by choosing arbitrarily ces (as compared to the and quantum mechanics and also depends on our e three units form a very ps not the convenience, of heory, a fundamental unit a relevant tion are completely arbitrary to convert degrees Kelvin into to pure numbers, while, keeping not space and time intervals equivalent – 17 – express (1) in the O . With QST, the third unit naturally emerges as being the physical length or time interval, , allows to ~ s λ and single to time or as both being c . However there appears an interesting option to do away with and s c λ matters. In string theory this allows us to identify the Plan ~ equivalent S/ a meaning in itself.I would In be the able absence to rescale of this any number fundamental arbitrarily units (e.g. o by re changing physics. Onlywould ratios matter. of Because two of lengths QM,gives, in however, out there of the is any problem, a fundamental MeV, centimetres into seconds, or everything into numbers. that there are unitsthat, that when are a arbitrary quantity and becomes units that are fun latter units, dramatic new phenomena occur.having or It not makes a having huge aties d fundamental of length. physical Without systemsatoms a would would fundame be not have invariant a underwhich characteristic an atom size, overall to and we uselength, would as we be can a un meaningfully metre. talkfundamental about length, By short of contrast, or course). with large distan aGoing fundamental back q to the discussion at the end of section 5, the pure n with the string length eliminatinga the third necessity, unit but besides perha those needed toI measure also lengths agree and time section with 2, Mike it is that easy tosome all convert unit. any that quantity I into matters a only pure addprovide, are numbe to in pure this the numbers. string observationfundamental that theory, relativity than units any of other.corresponding length units) and is The a time number matterunderstanding of which of of choice loo distinct the and convenience, underlying physical useful physical to laws. reducethird this Within unit number QFT after to + three, but there is no obvious On this particular point,a therefore, I fundamental tend unit to ofunit agree of speed with action Lev (its (a maximal minimalof uncertainty); value); length there there is, (the in is, characteristicthird string fundamental in size t unit of of strings). theconvenient system three-constants except system. QST that, Thes classically, appears the(and units of the ac same isonly true therefore of mass, energy etc.), while, string length numbers. numbers. However, I do insist that the point is Going further down,being say from two to one or to zero, means consi the two units JHEP03(2002)023 2 ) X is always does), and as a trivial , so far), of for an atom /T c φ units, is kept ) 2 ) T not ( X E is believed to grow time-dependence of 2 ss than from those of energy, ) or (∆ has a physical meaning on to distinguish space maller than it is today X α 2 also wish to acknowledge . However, this is not at ) s from conversion factors. nd the variation in time of he ratio α of an atom so that (∆ oughts to this issue and for X relative e (∆ µ nts the time variation of some inistered by the “Fondation de X s work. ing theory. For instance, in the uch as “time variation of a fundamental can depend on time, in QST, if represents the wavelength of light coming to us α – 18 – µ (1), this may have left an imprint of short-distance X O as a fundamental unit of speed and c , has no meaning, unless we specify what else, having the same c )). If it ever approached values for an atom and for the light coming from the galaxy. 30 This is what distinguishes, inWhile my opinion, I fundamental see unit noand reason thus to to distinguish introduce the Boltzmann’sfrom units constant, time of I and temperature see to every reas introduce conversion factor. Another clear difference is that, while t 2 the same, we do observe, in Nature, a variety of speeds (all le lengths, and of frequencies. ) is accompanied by a corresponding variation of the ∆ However, I claim that, in principle, the time variation of (∆ We do believe, for instance, that in a cosmological backgrou X (10 µν O physics on the CMBR spectrum. Acknowledgments I am grateful to Lev and Mike for having given their serious th pushing me to clarify mythe point support of of view a to “ChaireL’Ecole them, Internationale Normale and Blaise during to Sup´erieure”, Pascal”, the myself. adm final I stages of thi should be (and indeed is being) tested. For instance, the sam with the expansion of the Universe if ∆ all an absolute theoretical necessity (e.g. from a distant galaxy.(∆ The observed red shift only checks the for each one of thephysical two length systems w.r.t. separately because the it fundamentalearly unit represe Universe, provided by this str quantity( for the CMBR photons was much s remains constant. The same is usually assumed to be true for 8. Time variation of fundamental units? I think that the above discussion clearly indicates that the unit”, like have an intrinsic physical meaning. G fixed. Only the time variation of dimensionless constants, s JHEP03(2002)023 (1.1) bjects there are three , the velocity of light, nics, special relativity there were three basic ~ nsionful constants, such imum quantum of action to combine uantities, such as electric l prejudices and it seemed , namely the Planck length ould all be re-expressed in c units: centimetres, grams (or perhaps metres, kilograms m s (1.2) nstant, ensionful constants determined kg , were somehow accorded a less 35 44 S” system). 8 k − − − 10 10 . In terms of length, mass, and time . 10 2 G × × × − 1 : T − P 1 616 390 . . 177 T − 1 . − M MT = 1 = 5 3 2 = 2 3 5 L L LT – 19 – /c /c ~ ~ c/G ] = ] = ] = Michael J. Duff G ~ G c ~ . Here we present the platform of the Zero Constants [ G [ [ c p p p = = = or Boltzmann’s constant and P P P e . According to Veneziano’s Two Constants Party, there are 2 ; in special relativity there was a maximum velocity given by T L λ M ~ G and the Planck time P ; in classical gravity the strength of the force between two o M c According to the manifesto of Okun’s Three Constants Party, Later, as an undergraduate student, I learned quantum mecha Yet later, researching into quantum gravity which attempts , the Planck mass P , and Newton’s constant, only two: the string length c the velocity of light Part III A party political broadcastParty on behalf of the Zero Constants Abstract. fundamental dimensionful constants in Nature: Planck’s co and newtonian gravity. Ingiven quantum by mechanics, Planck’s there constant was a min Party. 1. The false propaganda of theAs Three a Constants young Party student of physics in high school, I was taught that quantities in Nature: Length,charge Mass or and temperature, Timeterms occupied [21]. of a these All lesser basic otherand status three. q seconds, reflected since in As the they a three-letterand name c result, “CGS” seconds system there in were the alternative, three but basi still three-letter, “MK Once again, the numberas three the seemed charge important of and the other electron dime was determined by Newton’s constant of gravitation fundamental role. Thisentirely fitted natural, in therefore, perfectly to withthree be my basic told high units, that schoo first theseL identified three a dim century ago by Max Planck their dimensions are JHEP03(2002)023 , s λ G and (2.1) (1.4) (1.3) (1.5) D and M 1 − , the action c (denoted , ct 2 ~ , mass λ = D 0 h L x 4 le [14, 15, 16], self-styled has the four-dimensional ent unified framework, I tants of Nature have been ], with axes , Gabriele begins with the V be regarded respectively as t, apart from the velocity of 6]. Lev Okun is their leader. 0 limits of the full quantum tivistic quantum mechanics, dulus fields coming from the hoice of vacuum but does not 4 spacetime dimensions? Un- mensional cube rather than a asic units is due to Gauss [27], so an. member. → D ) arty, although this was long before the D > − 2 G 5 . − c 3 T D then appears as − 1 − , 4 D 1 3 − to a length coordinate V , ~ 2 − − D G t 1 c c ~ : 0, and ( D L 1 − ~ ~ cT 1 ≡ G D D D D -dimensional Planck length − → = G G G G = D ) – 20 – M 2 1 2 4 = = = − 1 λ G 2 2 2 ]= , c − − − ~ D D D D G [ depend on D D D 0, ( T L M G → ) subscript), the , G P 1 − c was appreciated. 0, ( ~ → and requires only one dimensionful parameter, the string lengt ) c , the dimensions of ~ c , G 1 is the volume of the compactifying manifold. Since are given by − V D and , c T ~ Adherents of this conventional view of the fundamental cons What about Kaluza-Klein theories which allow for quantum mechanics, relativity and gravitation into a coher ~ It seems that the choice of length, mass and time as the three b 4 interpretation as the vacuuminternal expectation components value of of the scalar metricintroduce tensor, mo any it more depends fundamental on constants the into c the lagrangi dubbed the “Three Constants Party” byUntil Gabriele recently Veneziano I [1 was myself, I must confess, a card-carrying 2. The false propaganda of theMy Two faith Constants in Party the dogmaleader was of shaken, however, the by rebel paperstwo-dimensional Two Nambu-Goto by action Gabrie Constants of Party. alight As string. still needed a He to notes string convert tha the theorist time coordinate like gravity. Note, once again thatsquare we or are some dealing figure with of a a three-di different dimension. which neatly summarizesnewtonian how gravity classical and mechanics, relativisticthe non-rela quantum ( field theory can After compactification to four dimensions, and hence (dropping the learned about the Bronshtein-Zelmanov-Okun (BZO) cube [10 where by Gabriele). time divided by we could declare himsignificance to of be the founder of the Three Constants P JHEP03(2002)023 new (2.7) (2.5) (2.4) (2.3) the six-dimen- appearing in the ~ n fact, simpler than G . So the fundamental s open the number of to just two was made and the vacuum expec- sh to update Gabriele’s 11 2 L just two: length and time. λ en the TOE would require . tum gravity. The Einstein- ∼ 3 . Of course quantum gravity 6 λ 6 h, c, G gth ven-dimensional Planck length c onal action for the M2-brane, λ λ there is no dilaton. Consequently, . In superstring theory, the ten- ∼ , 2 and is the Regge slope. This is because 3 λ ′ λ P α i L φ 3 and e 6 h ~ Area (2.2) 2 c /cT cT 2 2 ~ only in the combination 2 λ 1 , Gabriele would say that we are still left with λ = (6d-volume) (2.6) ~ (3d-volume) 2 = = 6 6 – 21 – 3 λ 6 and will be shorter for string coupling less than 1 3 = 6 2 3 1 3 λ λ λ 2 2 10 λ and ~ for λ S = L = 3 G 6 λ 3 ~ S ~ S φ involves ~ rather than will be different in different vacua but does not introduce any 3 φ λ is the tension of the string and ′ πcα is the membrane tension. Alternatively, we could start with 2 is the fivebrane tension. Eleven-dimensional M-theory is, i / 3 6 T T = 1 2 T However, even if we substitute In the light of the 1995 M-theory [22] revolution, we might wi A similar argument for reducing the three constants previously by Zeldovich andHilbert Novikov [19] action divided with by regard to quan length in M-theory is sional action of the dual M5-brane, and where and where where dimensional Planck length is given in terms of the string len the Nambu-Goto action takes the form square of the Planck length, and so we need only ten-dimensional superstring theory in thisthe respect, three lengths: since membrane length, fivebraneare length all and ele equal [23] up to calculable numerical factors unity [26]. tation value of the dilaton field where the corresponding parameter is the membrane length where the corresponding parameter is the fivebrane length does not pretend todimensionful constants be required the for a TOE TOE. and so this argument still leave argument by starting with the corresponding three-dimensi So if this wereonly to two describe fundamental the dimensionful theory constants of everything (TOE), th Once again, the vev of constants into the lagrangian. the number two. This also reduces the number of basic units to JHEP03(2002)023 . = 1 cannot c ast adopt the Indeed, no less an ) wing, which prefers not afeteria between Lev, 5 ts Party). If you want 2 = 1 (just as you cannot c, λ c nts as you like; the more ( eed. t is real physics and what alled the Seven Constants ot distinguish length from and others. There at the ee Constants Party, length tions such as at Texas A&M, I continued re are no dimensions at all units at the same time, and f these units or conversion , the international body that h), kilogram (mass), second e is to use natural units where , put nergy to temperature nd joined what Gabriele would temperature), mole (amount of ental constants to be synonymous with re “derived” [27, 28]. The attitude S R , 2 2 ω are nothing but conversion factors e.g. mass c Gm ~ kT . mc 2 G = = = = – 22 – E E E S and R c , ~ has dimensions. In my view, this apparent dimensionless and you may quite literally set it equal to one c is c ofeec ´neaedes Poids et G´en´erale Mesures Conf´erence ) wing of the Two Constants Party is different from Gabriele’s , G = 1, you must trade in your membership card for that of (or at le ~ In these units, c = 1, if you want to follow the conventions of the Seven Constan 6 Our attitude was basically that Gabriele’s claim led to many heated discussions in the CERN c contradiction arises from trying to use two different sets of Incidentally, Lev (part I) objects in his section 5 that equa k This ( In this respect, I take the the number of dimensionful fundam 6 5 habits of) the Twotime. Constants Party, whose favorite units do n energy to mass to introduce a separate unit for mass. the number of fundamental (or basic) units. or energy to frequency different units you employ, the more different constants you n As such, you may have as many so-called “fundamental” consta no different from Boltzmann’s constant, say, which relates e Gabriele and myself. We went roundthese and round arguments in at circles. lunchtime Back College Station conversations Hilton, with we Chris eventuallycall reached Pope the a consensus Zero a Constants Party [16]. be taken literally because really goes to the heartis of mere my convention. disagreement In with theand Lev units time about have favored different by wha dimensions members and of you cannot, the therefore Thr to length, in the formula for the Schwarzschild radius put authority than the administers the SI systemParty, of decreeing units, adheres that to(time), seven what ampere units might (electric be are current), c substance), “basic”: kelvin candela (thermodynamic (luminous intensity), metre(lengt while theof rest the a Zero Constants Party isthere that the are most economical no choic conversionfactors is factors fundamental. at all. Consequently, none o In the natural units favored by the Zero Constants Party, the to put JHEP03(2002)023 7 this (3.1) , say, x c (3) as a ntractions O sics. achts and zebras constants such as iteness of 1) as a symmetry , (3 O . 1) symmetry rather than ionless worldvolume coor- ed 2 , tal” units because there are he three space coordinates, limit. Although of less historical . If I could paraphrase Lev’s hould not be included as a used to accepting (3 about. dz c pecial relativity and gravity. 2 is more than just a conversion ntum mechanics that there is O Lev (part I) makes the, at first z ~ c mensions in some units, such as ine element becomes: → ∞ city not be included as a fundamental embodies a fundamental physical + x 1). In this respect, the conversion c ofeec ´neaedes Poids G´en´erale Conf´erence et 2 c e-no¨cnrcinto the Galileoner-In¨on¨ucontraction group, with dimensions length/xylophone, , e dy z (2 to a special status is misleading. For 2 c e same footing as the Galileo group. So, in my y O . c G ′ ξ + d and appearing in brane actions (2.2), (2.4) and 2 λ and y d axes. This requires the introduction of three Both are, in Gabriele’s terminology (part II), c dx = λ c 2 – 23 – z , 8 ξ x , x is for the benefit of people for whom treating time c ~ c ensures that we have + ct and , by c 2 ξ = y dt 0 , 2 x c x − in = c 1) rather than merely limit. However, this is by no means unique. There are other co 2 , for special treatment has more to do with psychology than phy ds (3 c O = 1 may be imposed literally and without contradiction. With → ∞ c · · · are on an equal footing. = x c G , as “dimensionful constants” so as to distinguish them from = , related to the old ones, ′ and (3). This is certainly true. But it is equally true that the fin c ξ , let us introduce three new superfluous units: xylophones, y O c = ~ To drive this point home, and inspired by the However, I think this elevation of Similarly, the “fundamental” lengths I am grateful to ChrisTo put Pope this more for rigorously, this the admits Poincar´egroup example. a Wig 7 8 , which are dimensionless in any units. , c, G, k . . . obtained by taking the to other subgroups. For example, one is obtained by taking th and α 3. Three fundamental theories? Lev and Gabriele remain unshaken insight their reasonable, beliefs, point however. (echoed by Gabriele in part II) that understanding, I will still refer~ to constants which have di importance, these other subgroups areopinion, mathematically the on singling th out of the conversion factor becomes irrelevant. We have become so Mesures but to be perverse we could do so. symmetry that we would not dream of using different units for t factor. It embodies aa fundamental minimum physical non-zero principle angular of momentum. qua Similarly, principle of special relativity that there is a maximum velo example, the appearance of point of view it mightthree be to fundamental say physical that there theories:According are three to quantum “fundamen this mechanics, point s basic unit of (or, view, equivalently, Boltzmann’s temperature, constantconstant.) should for example, s as a fourth dimension is unfamiliar. But once you have accept to measure intervals along the superfluous “fundamental” constants, length/yacht and length/zebra, respectively, so that the l Lev’s point is that the finiteness of ensures that we have merely factors equally “silly”. Both can be set equal to unity and forgotten (2.6) can be removed fromdinates, the equations by defining new dimens JHEP03(2002)023 [18] means kT = E . Indeed, particle rsion factors, in the factors is important inue to go round and hem as you like. h/mc w, no significance should asic units. Similarly, we fundamental constants”. ordinary fate of universal = not a basic unit and that e go further? Another way c unit by trading distances orate investigations, would l laws as those of our own pressing particle masses as C role of mere unit conversion able redefinition of physical e. In practice, the net result o that of a black hole whose ev that we can whittle things his sense, the black hole acts Mr. Tompkins in paperback ple, the minimum number of R measured by the time it takes nverse mass, mass and inverse onstants determining the limits cepts of space, time and motion, do away with clocks in favor of ogressively incorporated into the way to ask what is the mimimum kinds of measuring apparatus: rods, George Gamow, has been accepted, we do not need both clocks – 24 – c should not be treated as fundamental, for example, = 1 but we need not use that language. c . We are, in effect, doing just that when we measure ct · · · = = x G Probably Lev, Gabriele and I would agree that = 9 c = ~ It seems to me that this issue of what is fundamental will cont J-M. Levy-LeBlond [29] puts it like this: “This, then, is the So I would agree with Lev that the finiteness of the conversion The reason why we have so many different units, and hence conve I am grateful to Chris Isham for this suggestion. 9 light to travel that distance, and rulers. Clocks alone are sufficient since distances can be is to point out that once the finiteness of around until we can all agree on an operational definition of “ that we can dispenseBoltzmann’s with constant is thermometers, not fundamental. thatdown Let temperature to us length, is agree mass with and L to time or argue rods, that scales the and clocks. conversion factor Can w interstellar distances in light-years.rulers. Conversely, It we is may can thus do superfluous away with to rulers havewith as both masses basic length apparatus using and and the length formula time as for as a the b basi Compton wavelength constants: to see their natureimplicit as concept common synthesizers background be of pr physicalfactors ideas, and then to often play tounits.” a be finally forgotten altogether by a suit 4. An operational definition “If, however, we imagineworld, but other with worlds, different with numericalof values the applicability for of same the the physica old physicalat c concepts, the which new modern and science correctbecome con a arrives matter only of after common very knowledge.” long and elab (minimum angular momentum, maximum velocity) but,be in attached my to vie their value and you can have as many or as few of t first place is that, historically, physicistsscales, used clocks, different thermometres, electroscopes etc.number of Another basic units,basic therefore, pieces is of to apparatus. ask what is, in princi theorists typically express length,mass, mass respectively. and time Finally, units wedimensionless as can numbers, i do namely the awayCompton with ratio wavelength scales of equals a by its particle ex Schwarzschildas mass radius. our t So rod, in scale, t is clock, as thermometer though etc. we all set at the same tim JHEP03(2002)023 1) , (3 O . ~ ns calls right and left 11 lly meet the alien, of nition is fine, but does ts are given by vacuum s that can be performed him to perform a cobalt calculate their values in or different constants of ately, after the discovery mpkins [18], we will ask ft or right” and “matter an imagines that we can violation of parity in the tants to be fundamental, ceived difference could be at he oth vacua respect lculation is too hard, but e cobalt behaves the same scriminating between what e purpose of this section to t his left hand (or tentacle). re the same lagrangian but ot new, but, in my opinion, e that we are both described en Feynman’s criterion is not ntal, whereas all dimensionful mine a left handed thread. In s definition, the dimensionless at would allow us to conclude tant or his Newton’s constant , say by ten orders of magnitude, ngly obvious. A similar point of view rg [17] and Gamow [18] imagine a out resolving the “he or she” ambiguity. lculate the numerical value of hysics World from a respectable physicist – 25 – 10 , are not. G and c , ~ In a similar vein, let us ask whether there are any experiment In physics, we frequently encounter ambiguities such as “le My apologies to those readers to whom this was already blindi I will follow Feynman and assume that the alien is a “he”, with 10 11 are the same as what60 we experiment call and right tell and himthis left. that way the However, we spinning we can electrons cancourse, deter agree ask we should on beware what shaking is handsHe with left would him be and if he made right. holdsof of ou CP When antimatter violation and we we annihilate eventua could with also us! eliminate this Fortun ambiguity. which would tell usnature whether as the ours. alien’sotherwise If universe not. the has answer the In iswhether particular, same there yes, is and we in inspired shallunambiguously principle define by that any these Gamow’s experimental his cons Mr. difference velocityare th To of different light, from ours. his Byexplained Planck’s “unambiguously” away cons by I a mean difference thatdevoid in no of conventions. per theoretical (Of course, assumptions.way ev for We the have alien as to forby assume us a that etc. TOE th To be (perhaps concrete,expectation M-theory) we values might in imagin of which scalar thelive fields. fundamental in constan The possibly alien different and vacua. we thus Let sha us further assume that b may be found in [32].who On believed the that other a hand, legitimate I ambition once of read a a TOE letter would in be P to ca terms of more fundamentalbecause constants, we not do not justnot know because resolve of the the anything dispute ca more betweenpropose fundamental. Gabriele, one Lev that This and does. defi me.parameters, I such It will as conclude is the that, th fine accordingconstants, structure to including constant, thi are fundame Weinberg [1] defines constants to be fundamental if we cannot symmetry.) 5. The operationally indistinguishable world ofThe Mr. idea Tompki of imaginingthe a early universe literature with is very different confusing.universe constants in For is which example, the n Vol’be velocity ofand light describe is different all from sorts ours of weird effects that would result: or antimatter”. Let usare begin genuine by differences recalling Feynman’s andcommunicate way what with of are some di mere alienweak being conventions. interactions [30]. we Feynm would have If no it way were of deciding not whether for wh the JHEP03(2002)023 d, 12 [18] 2. √ , which have h sacred to the G exceeds exceeds twice the c 2 ~ ~ 13 / / and 2 magines a universe in ng I am objecting to! 4 constants can be oper- e t us identify him with c ntal physical constants: y determine a difference : no experimental infor- culable or whether some , however, is entirely a matter , = me ell us how many such di- e ~ r effect easily noticeable by hysics. α = = 1, just as we are. To use nd the alien could exchange tandard model, but the aim or he speed of light, but simply Mr. Tompkins in paperback significant and meaningful. ~ t, differences in dimensionless c E is different from ours. Choosing to ant distances of planets to the sun) ecay of the proton to a hydrogen α e on this point, but I think the first two it in the top ten [24]. , the photon emitted by an atom would ould be a matter of convention in a world with reement. If, for example, the alien tells us that rather than E . Any perceived difference are all merely ; and the quantum constant c G c E George Gamow, = – 26 – are the electron mass and charge respectively. In k e of the Two Constants Party or the seven constants c and m . This universe is demonstrably different from ours. But, and + e , then we can be sure that his 2 + λ e + + H , where 2 H → that is ten orders of magnitude smaller than ours is, to my min → mc p ; the gravitational constant k p c fails the Feynman test. c ofeec ´neaedes Poids G´en´erale etConf´erence Mesures By contrast, in his critique of Vol’berg and Gamow, Lev [10] i fundamental constants. Second, he assumes a change incomparisons. these After all, an inhabitant of such a universe (le In this one sentence, Gamow manages to encapsulate everythi I believe that these two examples illustrate a general truth “The initials of Mr. Tompkins originated from three fundame In his section 7, Gabriele (part II) claims to disagree with m Indeed, participants of the Strings 2000 conference placed 12 13 meaningless. There is nothat experimental would information allow us that to we draw a any conclusion. which the binding energy of an electron in a hydrogen atom have a momentum electron rest energy 2 of the such a universe it would be energetically favorable for the d atom and a positron in my opinion, the correctthat conclusion in has this nothing universe to the do with dimensionless t fine structure const parameters like the fine structure constants are physically differences in convention rather than substance. By contras sentences of his section 8he indicate observes that the we are decay actually in ag First, he takes it as axiomatic that there areationally three defined. I for one am mystified by such Feynman’s alien) is perfectly free to choose units in which Three Constants Party, the attribute this effect (or any other effect) to a difference in mation that we andin the dimensionful alien quantities. could exchange No can matter unambiguousl whether they are the the velocity of light to argue that in his universe, for the same energy the equation to be changed bythe immensely man large factors on in the order street.” to make thei of convention, just as theno difference CP between violation. left and So right w Of course, our currentmensionless knowledge constants of Nature the requires. TOEof is There M-theory insufficient are is to 19 to t inare reduce the the this S result number. ofremains cosmological Whether one accidents of they (like the the are top ratios all unanswered of cal questions in fundamental p JHEP03(2002)023 se are G Indeed, and 17 . For exam- c , G ~ e dimensionless and . For example, the c G , ~ s at all, these standard e recent paper presenting rmining the fundamental ]. rac [25], have nevertheless of scalar fields. he speed of light might be t from a different epoch in ation in the physical laws time-dependence would be nging ts appearing in a Theory of ns in ds. In this way of thinking, erviewer on whether she believed hen sensible to ask whether nclusion that y I know to implement time ot necessarily wrong, are fre- tios, rather than dimensionful l force over cosmological times ontrast, the number and values e constant [34]. As a result, a ons required before a star reaches its are currently in vogue as as an c mensional spacetime symmetry is the vac- e, the cosmological constant can receive a anging from Fahrenheit to Centigrade that ul to Fred Adams for this example. 15 – 27 – are changing in time. (For some reason, time-varying c are fundamental devolves upon the issue of whether the or G 16 times lighter than the black hole discussed in section 3, who G 19 and c , ~ So, in my opinion, one should talk about time variations in th 14 In the context of M-theory which starts out with no parameter Unfortunately, this point was made insufficiently clear in th On the other hand, many notable physicists, starting with Di The only other possibility compatible with maximal four-di This point of view isOne also could taken then in sensibly [33]. discuss a change in the number of prot I am reminded of the old lady who, when questioned by the TV int is not as popular.) Indeed, papers on time-varying 17 14 15 16 replacing parameters by scalarvarying fields constants is of the Nature.constants only in sensible a The wa TOE was role also of emphasized scalar by Gabriele fields7. [14, in 15, Conclusions 16 dete The number and values ofEverything is fundamental a dimensionless legitimate constan subjectof of dimensionful physical constants, enquiry. such By as c astrophysical data suggesting afront time-varying page fine article in structur thechanging New over cosmic York Times history. [35] announced that t model parameters would appear as vacuum expectation values proton is approximately 10 Compton wavelength equals itsthis Schwarzschild dimensionless radius. ratio could It change is over time. t ple, any observed change inshould the be strength of attributed the to gravitationa changing mass ratios rather than cha alternative to inflation.quently I presented believe in that ais these misleading best ideas, way described while and inconstants. n that terms of the time-varying time-vari dimensionless ra parameters of the standard model but not about time variatio uum expectation value ofcontribution a from the 4-index vev field of strength. the M-theory For 4-form exampl [31]. ~ Chandrasekar limit for gravitational collapse. I am gratef causing it!”. in global warming, responded: “If you ask me, it’s all this ch our own universe andthe we issue stumbled of across whether his historical recor 6. What about theories with time-varyingSuppose constants? that our “alien” came not from a different universe bu changing in time. Accordingmerely to convention, without our physical criterion significance. above, any such entertained the notion that results of any experiments could require the unambiguous co JHEP03(2002)023 x mber of , Planck G 18 interesting and h , c to have the value to c R40899. choice of units to the k mperature. ince his original paper ate communication). ure intervals along the matical Institute, Oxford, ut that the Greek root of specially Gordy Kane, and v Okun and Gabriele Vene- axis in fathoms. The same l Mitra, Jose Sande Lemos, declared seven. The most economical correspondence with Thomas z stants with colleagues at Texas nstants is fundamental. is misnamed. The hospitality of ledged. hat by adding , thus emphasizing its role as a noth- Trialogue so – 28 – by definition two 19 has changed over cosmic history is like asking whether the nu and not c through ofeec ´neaedes Poids G´en´erale etConf´erence Mesures Warren Siegel (private communication) makes the following means axes in nautical miles and intervals along the ,. . . is a quite arbitrary human construct, differing from one y G , c dialogue 299,792,458 metres/second exactly, introduced not only a basic length, mass and time but also a te observation was made independently by Steve Weinberg (priv and ing but a conversion factor. , This research was supported in part by DOE Grant DE-FG02-95E h in So asking whether the value of By analyzing Planck’s papers [3] Lev came to the conclusion t 2. In 1983, the 1. Planck was actually a member of the Four Constants Party, s 3. Sailors use the perverse units of section 3, when they meas 19 18 litres to the gallon has changed. contradicts his definition of natural units [36]. the High Energy Theory Group,during Imperial the College preparation and of the this Mathe article is gratefully acknow Note added. points: next. There is nothing magicchoice about is the zero. choice of Consequently, two, none three of or these dimensionful co Acknowledgments My interest in thisziano. subject was I am fired also by gratefulA&M, conversations for with especially discussions Le on Chris fundamental Pope, con at at Imperial the College, University especially ofDent, Chris Michigan, Harald Isham. e Fritsch, I JamesWarren have Glanz, Siegel enjoyed Robert and C. Steve Helling,dia Weinberg. Saiba Alejandro Rivero points o JHEP03(2002)023 , Mr. (1928) ussian). smic 60 and String e theory of the , Ya.I. Perel’man , in (1986) 199. (1899) 440-480; Phil. Trans. R. 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