Natural units, numbers and numerical clusters

F. Pisano and N. O. Reis Department of Physics, Federal University of Paran´a, Curitiba, PR 81531-990, Brazil

Deﬁned by Lord Kelvin as the science of measurement it is described a fundamental fact of physics. The so called ‘natural’ units represent the unique system of units conveniently used in the realm of High Energy Physics. The system of natural units is deﬁned by the consistent providing of the adimensional unit value to the velocity c of the eletromagnetic radiation in the vacuum, related to the Maxwell classical electromagnetic theory and to the special relativity theory, to the reduced Planck constant ¯h h having the dimension of action, which is the fundamental constant ≡ 2π of the quantum mechanics as well as to the Boltzmann constant kB which has the dimension of heat capacity, being the fundamental constant associated to the statistical mechanics, c =¯h = kB =1. Such prescription is mandatory as we investigate the ultimate constituents of matter and their dynamics, since the atoms, leptons and quarks up to the grand uniﬁcation, strings, superstrings, and the supergravity theories, as well as the total uniﬁcation of the interactions in the M-theory. The one-dimensional character of this phenomenology is totally featured by the use of the natural units, reﬂecting the fact that in order to be possible the investigation of the fundamental constituents of 16 matter it is being necessary the resolution of lengths of the order of 10− cm or to reach energies of the order of at least 1 TeV. The application of this system of natural units results exceptionally adequate in the description of the Universe since the Planck scale, up to the Hubble large scale expanding Universe, resulting in the curious formation of the regular numerical clusters. 06.20.Fn: Units and standards 06.30.-k: Measurements common to several branches of physics and astronomy

1 I. INTRODUCTION h 6.626 068 72 34 h¯ = 10− Js ≡ 2π 2π × 34 The consolidation of speciﬁc systems of units of mea- =1.054 571 596 10− Js (2) × surement was determined by factors related to the envi- ronment, as well as by historical and physiological fac- and the Boltzmann constant, tors. Such fact is possible to be verifyed for instance 23 1 kB =1.380 650 3 10− JK− through the case of the units inch, yard, metrical foot, × 5 1 =8.617 342 10− eV K− , (3) palm and so on. Considering such natural arbitrariety × in the deﬁnition of the patterns of measurement it was deﬁne the system of natural units when are satisﬁed the always necessary to try to establish systems of simple dimensional relations and convenient units such as, for instance, the case of the m.k.s. system, the base of the international system [c]=[¯h]=[kB]=1 (4) of units, or even the case of the c.g.s. system. Naturally, the choice of such standards is arbitrary. Nevertheless, as well as, numerically, also1 the m.k.s. or c.g.s. systems has been becoming com- monly used in the realm of physics and engineering. In c =¯h = kB =1. (5) the High Energy Physics the system of natural units is the exceptionally consistent system not requiring the def- These three universal physical constants are associated initions of the usual patterns of measurement, generically with the electromagnetic Maxwell theory, as well as with used, being possible the deﬁnition of new fundamental the special relativity, c, with the quantum mechanics,h ¯, patterns of measurement. The strongly singular but uni- and with the statistical mechanics, kB, already appearing versal postulate used in this case consists in the fact that in the Boltzmann equation of calorimetry the unique things that can retain their identities in any place of the Universe are the elementary particles, lep- ∆Q =∆Q(Θ) = kB ∆Θ (6) tons and quarks, as well as their interactions described which establishes the relation between the perceptible by the corresponding dynamics given by the general rel- heat quantity ∆Q and the ﬁnite variation ∆Θ of the ab- ativity [1] and by the standard model [2] of the non- solute temperature Θ. The Newton-Cavendish constant gravitational interactions, which consist in the weak and 16 of universal gravitation, by its turn, strong nuclear interactions of short range, 10− cm for 13 11 1 3 2 the weak interaction and 10− cm for the strong inter- G =6.672 599 10− kg− m s− (7) action, and the long range electromagnetic interaction. × In such physics where we are searching the fundamental is the constant appearing both in the Newtonian gravi- laws of the Universe as a whole and the bits of matter, tation and in the general relativity. measurement is essential again, but the quantum and Regarding the c.g.s. or m.k.s. systems the fundamen- relativistic regimes imply, aside from any multicultural tal dimensions are length [L], mass [M], and time [T ] diversity [3] the simplest system of units, the ‘natural’ but taking the mass dimension [M], action [S], and ve- units. locity [V ] as fundamental dimensions, length and time dimensions are [S] II. THE PHYSICAL CONSTANTS AND THE [L]= (8a) NATURAL UNITS [M][V ] [S] [T ]= (8b) As the description of the dynamics of the fudamen- [M][V ]2 tal interactions was being elaborated it became a con- solidated fact that the universal physical constants usu- and being p, q, r real numbers the general dimensional ally play the role of constants of proportionality between relation magnitudes in the equations of physics. The velocity of p q r p q r q+r q 2r the electromagnetic radiation in the vacuum, [M] [L] [T ] =[M] − − [S] [V ]− − (9) in the c.g.s. or m.k.s. units has the natural unit mass c = 299 792 458 m s 1 (1) − dimension [M]n with n = p q r. For action p =1, − − which is an exact value since the meter turn to be deﬁned as the space covered by the light in the vacuum during the time of 1/299 792 458 of 1 second, the reduced Planck constant [4] 1Notice the coincidence with the dimensional value c = 1 1ly yr− . ×

2 p p+r q =2,r = 1, and for velocity we must have p =0, [A]NU =[A]SI (¯h− c ) (17b) q =1,r = −1withn = 0 for both of such magnitudes. The mass or− energy, length, and time have the p =1, when the fundamental unit is a length, and q = r =0,andq =1,p = r =0,andalsother =1,p = p+q+r q = 0 attributions with the respective mass natural unit [A]NU =[T ]− , (18a) dimensions n =+1, 1, 1. For any physical magnitude − − A with the c.g.s. or m.k.s. units p 2p q [A]NU =[A]SI (¯h− c − ) (18b) p q r [A]=[M] [L] [T ] (10) for the time as natural unit, which comply the linear al- gebra of fundamental units [5]. The conversion equations and if we choose as fundamental unit the mass [M]the allows us to convert any physical quantity to NU, or back natural units (NU) and the International System of units, to the SI. The conversion is realized by setting explicitly SI, from the French terminology Syst`eme International the [A]SI as every SI unit which it contains. The dimen- d’Unit´ees,oftheA quantity are related according to sional pattern of some fundamental physical magnitudes α x y in natural units is [M] [A]NU =[A]SI [¯h] [c] (11) ≡ 1 1 2 1 0 1 [E]=[M]=[Θ]=[L]− =[T ]− (19) where [¯h]=[M][L] [T ]− and [c]=[M] [L][T ]− .Tak- ing now [A] writen as in Eq. (10) with the [¯h]and[c] which can be veriﬁed in the following sequence, dimensional relations the Eq. (11) becomes (i) [E]=[M]: 2 α p+x q+2x+y r x y it follows from the mass-energy relation E = Mc ; [M] =[M] [L] [T ] − − . (12) (ii) [E]=[Θ]: The values of α, x,andy are obtained by realizing that take the energy-temperature Boltzmann equation, two quantities are equal just when the potencies of the E(Θ) = Q(Θ) = kBΘ where Θ is the absolute tempera- ture; three fundamental units are the same. Therefore, 1 (iii) [E]=[L]− : [M]α =[M]p+x, from the energy-momentum relativistic equation for a 0 q+2x+y massless particle, E = pc,wherep = p is the scalar [L] =[L] , (13) momentum, and the de Broglie relation |λ |= h/p, associ- 0 r x y [T ] =[T ] − − , ating the wave character (λ) of the matter (p), it follows that λ = hc/E or E = hc/λ; and to know , and in terms of the , ,and 1 α x y p q r (iv) [E]=[T ]− : dimensional exponents it is just necessary to solve the 1 being [c]=[L][T ]− = 1 in natural units, then from system of algebraic equations, 1 1 [E]=[L]− we see that [E]=[T ]− . All the diversity of the world [6] is found at dif- p + x = α, ferent sizes, energy, temperature or any other magni- q +2x + y =0, (14) tude. We ﬁnd the nucleons and atoms from the Fermi 15 r x y =0, length, 1 fermi 10− m, to the Angstr¨om scale, 1 A˚ 10 ≡ 9 ≡ − − 10− m, virus between (20 and 300) 10− m, bacteria resulting and cellule at 1 µm. Atoms form molecules× and there are molecules which contain hundreds of atoms such as α = p q r, proteins, the DNA and RNA extended up to several − − x = q r, (15) twelves of angstr¨oms. Today are synthetized molecules − − containing until 104 atoms. The human being is the most y = q +2r, evolued structure, being perhaps the conscience of Uni- verse [7]. It is composed by 1028 105 mol of atoms, be- and we can write the dimension of the quantity A in a ' mass natural unit as ing hydrogen, carbon, nitrogen, and oxygen the majority of them. Even if more than one hundred of chemical el- p q r [A]NU =[M] − − (16a) ements have been catalogated in the Mendeleev periodic classiﬁcation just four of them are the essential ones to with the conversion factor build up the basic molecular structures that allows life and intelligent life. q r q+2r [A]NU =[A]SI (¯h− − c ) (16b) given in terms of theh ¯ and c universal constants. III. THE NUMERICAL CLUSTERS The same general procedure gives the following results,

p+q+r With the fundamental constants c, G,¯h, kB,andthe [A] =[L]− , (17a) NU electron mass

3 =0510 999 06 MeV 2 (20) which is much larger than the Avogadro constant, = me− . c− , NA 23 1 6.022 141 99(47) 10 mol− . In the atomic scale, elec- 19 where 1 eV 1.602 177 33 10− joule, which is the tric forces are dominant× but in large scale such forces ≡ × lightest massive elementary fermion among all the ele- anulate each other and the gravitation becomes domi- mentary particles of the electroweak standard model, the nant. If a nucleon with the mass Mn of almost 1 GeV is mass of the proton a black hole in rotation, of Schwarzschild [8], its radius 1 2 2 is obtained by equalling the kinetic energy 2 mv ,tothe Mp+ 938.271 99 MeV c− (21) ' gravitational potential energy GmMn/r. Considering and of the neutron, the limit v = c, the Schwarzschild radius is

2 Mn Mn0 939.565 33 MeV c− (22) r =2G , (31) ' S c2 and the electron electric charge corresponding to the gravitational lenght of a nucleon 19 e = 1.602 177 335 10− coulomb, (23) − × Mn 52 ag = G 10− cm (32) we can obtain adimensional numbers such as c2 '

Mn0 Mp+ in which we do not take into account the factor 2. If a 1839, 1836 (24) nucleon of classical radius a e2/(m c2) reaches the m m e− e− ' e− ' ≡ gravitational lenght ag, the ratio between these lengths or even the Sommerfeld ﬁne structure constant will be

2 2 e 1 a e 40 αem (25) = 0.2 10 (33) ≡ hc¯ ' 137 ag GMn me− ' × as well as the dimensional quantities such as the radius that shows that a nucleon is much larger when com- of the Bohr level of the Hydrogen atom pared to what it would be if it were collapsed in a Schwarzschild black hole. We classify two adimensional h¯ 8 a0 0.5 10− cm 0.5 A˚. (26) numerical groups. The ﬁrst one near to the unit, such ≡ m c ' × ≡ e− as 1 1836 or 2 ¯ 1 137 and their in- me− /Mn / e /hc / Realize the absence of the universal constant of gravita- verses. Another' cluster of adimensional' numbers involve tion G in these three last relations which cannot be com- the constant of gravitation G, centered around 1040 as bined withh ¯ and c to form a gravitational ﬁne structure e2 1 constant analogous to αem. The characteristic ondula- 40 2 = (0.2 10 ), (34a) tory length of the electron at velocities near to that of GMn 1836 × × the light c, by its turn, is given by the reduced Compton hc¯ 137 = (0 2 1040) (34b) wavelength 2 . , GMn 1836 × × ¯ 1 e2 λe− h 40 λe = = αem a0 a0 (27) =1 (0.2 10 ), (34c) − 2 137 GMnme × × ≡ π me− c ' − hc¯ 40 and when all energy of an electron, 2, is electrostatic = 137 (0.2 10 ), (34d) me− c 2 GMn me− × × potential energy kind, e /a,then 2 e 40 2 2 = 1836 (0.2 10 ) (34e) 2 e Gm × × m c = , (28) e− e− a 1 always proportional to G− . Let us denotate this cluster resulting the electron classical radius, of numbers by 1. N 2 e 13 a = 3 10− cm 3 fermi (29) m c2 e− ≡ × ' IV. THE CHANGE OF UNIVERSAL PHYSICAL When we build up numbers involving the constant G CONSTANTS it is possible to obtain numbers that are centered in a cluster of the order of 1040 as the ratio between the forces With the scrutiny of the numerical clusters we have, of the electrostatic and gravitational interactions among this way, a cluster centered in 0 1 being this last 40 N ∼ a proton and an electron, one centered in 1 10 . Considering the time varia- tion [9] of the universalN ∼ physical constants, the time func- 2 Felectr e 40 tional behavior of the gravitation universal constant G = = 10 (30) 1 Fgrav GM + m ' G(t) t− , suggested for the ﬁrst time by Dirac, [10,11] p e− ∼

4 c it is possible to verify, sistematically, in this case that the L = (37) Hubble H unit group 0 1 is not aﬀected. However, we verify 0 that all theN elements∼ of the cluster are shifted be- 1 where the universal constant = 100 km s 1 Mpc 1 neath 1040. It is also consideredN the variation of H0 h0 − − 2 is the Hubble present expansion rate of the Universe with the elementaryN ∼ electric charge2 e, in such a manner that the adimensional ignorance parameter [4] =(071 it diminishes with the expansion of the Universe. It is in- h0 . 0 07) 1.15 and 1 pc (parsec) 3 262 ly 3 086 1016 m.± teresting to observe that the elements of the unit group . 0.95 . . In a energy× natural unit ' ' × 0 as well as the ones of the cluster 1 will spread-out. N N 33 H0 =2.13 h0 10− eV. (38) × V. THE HIGH ENERGY PHYSICS AND THE 44 From the Planck time scale tPlanck 10− sec up to the COSMOLOGY Hubble time [16] '

1 10 17 On dimensional grounds it can be established the tHubble = H0− 10 yr 10 sec (39) Planck scale, inaccessible to any conceivable experiment, ≈ ' Energy: there are 61 orders of magnitude, the same as the spread of time in seconds to be scrutinyized since the Big Bang 1 5 2 19 to the age of the Universe. Taking the value L EPlanck = hc¯ /G 1.2 10 GeV, (35a) Hubble ' × 15 109 ly, in terms of the electron classical radius, a', × Mass: there is a third cluster,

1 M =(¯hc/G) 2 2.1 10 8 kg, (35b) LHubble 40 Planck − 2 5 10 (40) ' × N ≡ a ' × Time: coincident with the 1 cluster. Such coincidence, 1 = 1 N N 5 2 44 2, seems to indicate an unknown deep connection be- tPlanck = hG/c¯ 5.4 10− sec, (35c) N ' × tween the universal physical constants and the structure Length: of the Universe. Perhaps such large numbers realize the 1 = 2 coincidence not so occasionally. The establish- 1 N N 3 2 35 ment of the correlation lPlanck = hG/c¯ 1.6 10− m, (35d) ' × numerical cluster 1 elementary particles Density: N ←→

5 and c 96 3 ρPlanck = 2 5.1 10 kg m− (35e) hG¯ ' × numerical cluster 2 large scale Universe, N ←→ involving the c, G,and¯h universal constants. In natural accordingly to the Dirac large numbers hypothesis, the units, the Newtonian gravitational constant G is associ- and clusters ought to be connected by a simple ated with the Planck energy scale according to 1 2 NmathematicalN relation. Particularly,

1 19 = EPlanck 10 GeV. (36) 1 = 2 . (41) √G ≈ N N ≡N Gravitation is closely connected to the Planck scale since Thus, it is possible to verify the existence of a numerical series, EPlanck is the characteristic scale of gravity quantiza- tion [12]. However, the Planck scale is invisible to our 0 1 1 3 2 , ± 2 , ± , ± 2 , ± , (42) highly selective detectors which are looking up to the N N N N N kTeV or PeV scale, [13] with the preﬁx ‘peta,’ P 1015, i.e., n/2 being n =0, 1, 2, 3, 4, only, all along for the and very beyond in the Cosmic Ray Physics [14,15]≡ on the ± Universe,N with contained in the interval 1038 way of a systematic phenomenology to ﬁnd new physics. 1042. Some numbersN of such a series can be easily≤N inter- ≤ The horizon of the observable Universe is determined by preted such, for instance, the Hubble length, density of a nucleon 2 = N2 = (43) density of the Universe 1 N N 2As a matter of fact, the quantum of electric charge is the to the cluster , 1 N charge of the down-like quark ﬂavors with 3 e for a particle state. − | | Compton length of a nucleon 1 = 2 , (44) Planck length N1

5 or here. N.O.R. is grateful to the CNPq (Brazil) for ﬁnan- cial support. Planck mass 1 = 2 , (45) nucleon mass N1

1 for 2 ,and N

mass of the Universe 1 3 = 2 = 2 , (46) Planck mass 1 2 N ×N N [1] A. Einstein, Sitzungsberichten der Preussischen Aka- or still, demie der Wissenschaften 11, 778 (1915); Annalen der Physik 49, 769 (1916). The original references can be 1 3 extension of the Universe 2 found in The Collected papers of A. Einstein (Princeton = 2 = 2 (47) Planck lenght N1 ×N N University Press, 1989). [2] S. L. Glashow, Nucl. Phys. 22, 279 (1961); A. Salam and 3 for the cluster 2 , and ﬁnally, J. C. Ward, Phys. Lett. 19, 168 (1964); S. Weinberg, N Phys. Rev. Lett. 19, 1264 (1967); A. Salam, in Elemen- total number of nucleons in the Universe = tary Particle Theory: Relativistic groups and Analytic- 2 80 ity, Nobel Symposium, No. 8, edited by N. Svartholm 1 2 = 10 (48) N ×N N ≈ (Almqvist and Wiksell, Stockholm, 1968); S. Weinberg, which is the Eddington number, the largest number of the Rev. Mod. Phys. 52, 515 (1980); A. Salam, Rev. Mod. whole physics together with the total number of photons, Phys. 52, 525 (1980); S. L. Glashow, Rev. Mod. Phys. 52, 1089. 539 (1980). The original references for the color sector are O. W. Greenberg, Phys. Rev. Lett. 13, 598 (1964); M. Gell-Mann, Acta Phys. Austriaca, Suppl. IX, 733 (1972); D. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 VI. CONCLUSIONS (1973); S. Weinberg, Phys. Rev. Lett. 31, 494 (1973); H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. 47 The simple but meaningful algebraic combinations of 365 (1973). the universal physical constants are grouped in regular [3] Robert H. Romer, Units–SI-Only, or Multicultural Diver- numerical clusters building up a ﬁnite geometrical series sity? Am. J. Phys. 67(1), 13-16 (January, 1999); Letters 20 80 with a step of 10 and oposite extremes 10± . Introduc- to the editor, Am. J. Phys. 67(6), 465-470 (June, 1999). ing the system of natural units in which [c]=[¯h]=[kB]= [4] Particle Data Group, D. E. Groom et al., Review of 1, the High Energy Physics allows a one-dimensional Particle Physics, The European Phys. Journal 15, 1-878 system of units. Such inevitable one-dimensionality is (2000). On line data are available from the Particle Data the natural direct result of the Universe ultimate con- Group website http://pdg.lbl.gov. stituents search, according to Georgi “our pride and our [5] A. Maksymowicz, Natural units via Linear Algebra,Am. curse” [17]. Admitting the inverse time variation of J. Phys. 44, 295-297 (1976). the gravitation universal constant, the whole cluster of [6] Victor F. Weisskopf, Of atoms, mountains, and stars: numbers centered around the unit is not afected, even A study in Qualitative Physics, Science 187, 605-612 though the cluster centered in 1040 is deviated below (February, 1975). this scale. All clusters spread-out with the elementary [7] Joe Rosen, The Anthropic Principle, Am. J. Phys. 53(4), electric charge reduction. 335-339 (April, 1985); The Anthropic Principle II,Am.J. Phys. (5), 415-19 (May, 1988); V. Agrawal, S. M. Barr, All surprises and the new physics islands must be in 56 J. F. Donoghue, and D. Seckel, Anthropic considerations some speciﬁc places on a natural unit onedimensional in multiple-domain theories and the scale of electroweak scale. Such islands contain an apparently inexhaustible symmetry breaking, Phys. Rev. Lett. 80, 1822-25 (1998); source of new physics [18] such as the continuous con- Viable range of the mass scale of the standard model, statations of the deterministic chaos [19] due to a hy- Phys. Rev. D 57, 5480-92 (1998). persensible initial conditions dependence of the classical [8] S. Weinberg, Gravitation and Cosmology (Wiley, 1972). dynamical system, as well as the cellular automata pos- [9] Edward Teller, On the change of physical constants, sibility [20] and the uncertainty principle [21] even for Phys. Rev. 73, 801-802 (1948). classical systems [19] due to an irreducible limitation on [10] P. A. M. Dirac, Proc. Roy. Soc. A133, 210 (1937); Nature the knowledge of the properties of the physical system. 139, 323 (1937); New Basis for Cosmology,Proc.Roy. Soc. A165, 198 (1938). [11] R. H. Dicke, Dirac’s Cosmology and Mach Principle,Na- ACKNOWLEDGMENTS ture 192, 440-441 (1961); P. A. M. Dirac, Nature 192, 441 (1961). F.P. should like to express his gratitude to Professor [12] See a clear description and the perspectives in the re- J. Polchinski for your short but very elucidative reply to view article by Joseph Polchinski, Quantum Gravity at a question which has motivated the scrutiny contained the Planck Length, Int. J. Mod. Phys. 14(17), 2633-2658

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