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SLAGPUB-6509 August 1994 (T/NOY4 Bit-String : a Novel “ of ”’ H. Pierre Noyes Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309

Abstract More conventional TOE’Sare based on the math- We encode the quantum numbers of the standard ematical continuum and the structures of second model of quarks and leptons using constructed bit- quantized relativistic field (QFT). They ignore strings of length 256. These label a growing universe of the flaws of QFT (infinite answers to physically sensible bit-strings of growing length that eventually construct questions, “gauge potentials”, and no a finite and discrete space- with reasonable cosmo- well defined correspondence limit in either classical rel- logical properties. Coupling constants and mass ratios, ativistic field theory, non-relativistic quantum mechan- computed from closure under XOR and a statistical ics or nuclear physics). The most ambitious of these hypothesis, using only ?i,c and m, to f;. our units of theories assume that non-Abelian gauge theories in the mass, length and time in terns of standard (meter- form of “string theory” succeed in explaining “quantum kdogram-second) metrology, agree with the first four gravity”. Comparison with practical metrology is made to seven significant figures of accepted experimental results. Finite and discrete conservation and by identifying h, c and GN~~~in their theoretical commutation relations insure the essential charac- structures. It is then an act of that everything teristics of relativistic quantum mechanics, including else is calculable. Less ambitious TOE’S(eg., GUT’S particle-antiparticle pair creation. The correspondence = grand unified theories) fix the third parameter as limit in (free space) Maxwell electromagnetism and a universal coupling constant at an energy of about a Einstein grawitation is consistent with the Feynman- thousandth of the Planck mass-energy and then “run” Dyson-Tanimwa ”proof.” it down in three different ways to energies a factor of 1015 smaller where these three distinct values are Although currently accepted relativistic quantum identified as the measurable fine structure constant mechanical theories incorporate many discrete phe- (a = e2/hc), weak interaction constant (GpeCmi)and nomena, they are embedded in an underlying space- strong coupling constant a,; because the strong (QCD) time continuum in a way which guarantees the creation coupling “constant” is supposed to diverge at zero of infinities. Despite many phenomenological successes, energy, models must include its energy dependence over they have as yet failed to achieve a a finite energy range. In practice, such theories contain of “quantum gravity”. We believe that these two a fairly large number of phenomenological parameters. difficulties are connected, and that both can be In contrast, we employ a structure in which we need circumvented by basing fundamental physical theory only identify c and m, (the proton mass) in order directly on the computer tools of bit-strings and ti, theory based on bit-strings. This has the to make contact with standard MLT metrology, using further advantage that we can base our model for space the kilogram, meter, and second as arbitrary but fixed and time on finite intervals between events (eg., counter dimensional units. a, GF~~~,GN~~~and a number firings) measured to finite (and fixed in any particular of other well measured parameters can be computed context) accuracy. operational methodology then and the of the fit to evaluated in This a less problematic way. While these comparisons are allows us to avoid such metaphysical questions as whether the “real world” is discrete or continuous [l], very encouraging, with accuracies ranging from four or whether the “act of ” does or does not to seven significant figures, they are not perfect. So require “” [2]. far as we can see the discrepancies could arise from the concatenation of effects we know we have so far By a “” (TOE),we mean a systematic representation of the numerical results not included in the calculations, but we are prepared obtained in high energy particle physics to encounter “failure” as we extend the calculations. and by observational . The representation However, the quality of the results achieved to date we use employs a growing but always finite assemblage lead us to expect that such ”failure” would point to of bit-strings of finite length constructed by a simple where to look for “new physics” in our sense. Since called program universe explained below. we leave no place for “adjustable parameters”, such *Work supported by Department of Energy contract a crisis should be more clear cut for us than in a conventional We do not believe that it is possible DE-AC03-76SF005 15. TOE. Presented at the Workshop on Physics and Computation (PhysComp’94): This Decade and Beyond, Dallas, TX, November 17-20, 1994

~~-gc~DoolMewfII~D to make a “final theory”, and might even welcome a both la(S)I and (a(S)la(S)). Here we introduce the failure serious enough to allow us to abandon this whole Dirac bra (I notation for one dimensional row matrices, approach and turn to more conventional activities. lcet I) for one dimensional column matrices, and bruket We start from a universe of bit-strings of the same (I) for the matrix inner product. Hence, (a(S)lb(S))= length which grow in length by a random bit, randomly Cf=;=,a,b,. Because we interpret the “0” and chosen for each string whenever XOR between two “1” as integers rather than bits, we can define the strings gives the null string; else the resulting non-null operator XOR, symbolized by @, which combines two string is adjoined to the universe. Then recurse. strings to form a third by the elements of the resulting Because of closure under XOR [l],and a mapping string: (a @ b), = (a, - b,)2. This is isomorphic to we present below of the quantum numbers of the 3- the usual meaning of XOR, addition mod 2 or boolean generation standard model of quarks and leptons onto symmetric difference in the sense that the element is 1 the first 16 bits in these strings, we can model discrete if a, and b, differ, and 0 if they are the same. In our quantum number conservation (lepton number, baryon restricted (bit-string) environment, the Dirac notation number,charge, weak isospin, and color) using a bit- then allows us to write la@ bl + 2(aJb)= a + b. string equivalent of 41eg Feynman diagrams. Quarks One consequence of this connection between bit- and color are necessarily confined. known All string states and bit-strings is that if Ja@bl= elementary fermions and bosons are generated, and no a b, then the states are orthogonal in the sense unknown particles are predicted. The scheme implies + that (alb) = 0. Hence given n finite integers ni reasonably accurate coupling constants and mass with the constraint Cr!=,ni S, we can always ratios, calculated assuming equal prior probabilities in < construct an orthogonal state basis from n bit- the absence of further information. The combinatorics and the standard statistical method of assigning qua1 strings ni(S) with a standard representation ni(S) = weights to each possibility provide an alternative O(C~~~lnj)llI(ni)llO(S- C7=n,+lnj). Here we have interpretation of results previously obtained from the introduced the null string O(W) with elements 0, = 0, combinatorial hierarchy, including the closure of these the anti-null string I(W) with elements I, = 1, and bit-string labels at length 256, and the prediction of the bit-string concatenation “II” defined by .:‘Ib ak, IC E Newtonian gravitational constant. Baryon and lepton 1,2,..., sa; bj, j E 1,2,..-,&,IC = sa +j. number conservation then gravitationally stabilizes the Given such a basis, we have n strings which are lightest charged (free) baryon (the proton) and lepton independent under XOR and which can be combined (the electron) as rotating black holes of spin 1/2 and by XOR to produce 2” - 1 distinct strings simply by unit charge. taking them 1,2,.. . up to n at a time. The growing portion of the bit-strings beyond the To generate a growing universe of bit-strings which quantum number conserving labels can be interpreted at each step contains P(S)strings of length S, we use describing an expanding 3-space universe with as an algorithm known program universe which was a universal (cosmological) time parameter. Within as this universe pairwise collisions produce products developed in collaboration with M.J.Manthey . [2,3] Pick any two strings P,(S),Pj(S), E 1,2,. P conserving relativistic 3-momentum (and, when on i,j . . , mass shell, energy) in terms of quantized Mandelstam and compare Pij = Pi @ Pj with O(S). parameters and masses. The baryon and lepton If Pi, # 0, adjoin Pp+1 := Pij to the universe, number, ratio of baryons to photons, fireball time, P := P + 1 and recurse. and ratio of dark to baryonic predicted by Else, for each i f 1,2,.. . ,P pick an arbitrary bit this cosmological model are in rough accord with ai E 0,1, replace P,(S+l):= Pi(S)lla,,set S := S+1 observation. The model contains the free space and recurse. Maxwell equations for electromagnetism and the free Note that, because the arbitrary bits are concatenated only at one growing end of the strings, once the string space Einstein equations for gravitation as appropriate macroscopic approximations for computing the motion length S passes any fixed length L the P(L) strings of a single test particle. present at that time can contain at most L strings which are independent under XOR. Further, since this Bit-strings and program universe portion changes only by XOR once S > L, it can end up containing contain at most - 1 types of Specify a bit-string a(S) by its S ordered elements distinct, non-null strings. Consequently, at any later stage in the of the universe we can always a, E 0,l; and s E 1,2,3,.. . S. If we interpret the symbols “0” and “1” in the strings as integers, we can separate any string into two portions, a label string calculate the norm, or Hamming measure, a(S)by the Ni(L) and a content string Ci(S-L) and write Pi(S)= formula a Cf=,a,. We also call this positive integer Ni(L)llG(s- L).

2 DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned . Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac-. turer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. Bit-string standard model labels Fermi constant is actually evaluated in neutron decay then allows us to predict that GF = m~~/a(256)~. To model the quantum numbers of the standard The correction given in the table is again due to model of quarks and leptons with label strings of length McGoveran; as just noted, he invokes a different theory. 16, we break the label into four segments: LSM = The quark-gluon (QCD) interaction requires a more Lv(2)IILe(4)(ILq(8)((Lg(2).Here g stands for the three careful discussion, which can only be sketched here. generations of neutrinos, charged leptons and quarks Thanks to McGoveran’s Theorem [5] that in any finite with the standard designations (ve,e; u, d), (vP,p; c, s), and discrete theory with a universal ordering operator, (v,,T;t,b). The generation label Lg(2) has only there can be no more than three homogeneous and three non-null possibilities (lo), (01),(11); the most isotropic macroscopic (“spacial” ) dimensions, these are convenient choice depend on an investigation of the exhausted in labeling particles by the three absolutely Kobiyashi-Maskawa mixing angles which has yet to be conserved quantum numbers allowed in current particle carried out. So far as we know, baryon number, lepton physics. There are four candidates: lepton number, number, charge, and the z-component of weak isospin baryon number, charge, and either the uz-component are conserved (and constrained to three independent of weak isospin” or “weak hypercharge”. The extended choices by the generalized GellMann-Nishijima rule). GellMann-Nishijima relation leaves only three of these We use the weak interaction to define the basis strings to specify independently and measure at macroscopic we take to be orthogonal and choose L, : v~ = distances. Any other conserved, discrete quantum (10);~~= (01). Then take Le : PL = (1OOO); PL = number must “compactify“ or be “confined”. The two (0100); e, = (0010); tfR = (0001). For the quarks use flavors of quarks with the L,R and quark-antiquark di- chotomies give us 8 degrees of freedom in our basis, but = q(4)(lq(4); = q(4)110(4)with qE,R L,, Lq, the special role played by the anti-null string reduces the same pattern as the charged leptons. The basic weak interaction vertices are then given these to seven which generate 27 - 1 distinct non-null strings. These can be used to represent the possible by vr, @ fLf = W2L; v~ @ VR = ZO = fz @ fLf where 9 quark and anti-quark states coupled to the color octet stands for any left-handed, charged fermion (i.e f: f: (red, anti-red, yellow, anti-yellow, blue, anti- blue, charged lepton or quark). Similarly, the electromag- black, colorless = 23) of gluons, or the requisite 24 x 23 netic vertices are given by fz @ ft = ~LL;fz @ f; = distinct quark and gluon states needed to describe yc. The 7 is massless, and the 20 is massive (and strong (QCD) interactions. Color, gluons, and quarks unstable), so we need to develop the content labels in are confined, thanks to McGoveran’s theorem. The terms of space-time parameters to distinguish y from label scheme above, appropriately interpreted, provides 20. Note that the 20 and the W couple only to the 2/3 and 1/3 charge and baryon number values for left-handed fermions. The masslessness of the y’s keeps the uptype and down-type quarks, respectively. the transverse photons from coupling directly to the To obtain the pion masses (given the electron coulomb interaction yc,which is represented within the mass-see below), we use Dyson’s contention [6] that electromagnetic sector by (1111). Details, and proof of QED does not allow more than 137 charged particle the conservation laws will be presented elsewhere. pairs of mass m to be specified within a volume whose This scheme gives us 137 distinct charged particles radial dimension is h/2mc. This allows us to model which can couple via the coulomb interaction in one, the neutral pion as fic/e2 = 137 electron-positron and only one, way. Thus, in the absence of further pairs within this distance, and the two charged pions information, the probability of the coulomb scattering as an electron (positron) and anti-neutrino (neutrino) process f* + fF -, yc -, f* + fr is 1/137 and can be included in this ephemeral structure 17). The com- identified as the of e2/fic in the quantum number binatorial corrections due to McGoveran and given in part of the corresponding Feynman diagram. That this the table bring the calculated values reasonably close to leads to a correct description of the energy levels of experiment. We emphasize again that the legitimacy of the (relativistic) Bohr atom, and that the Sommerfeld using his corrections in our context is still controversial. fine structure is the appropriate 1/137 correction to this result has been argued elsewhere. [4] Our present Background needs rethinking before it can be claimed that it leads to McGoveran’s correction to the calculated We have used bit-string physics to provide a theory value of e2/tic given in the table. Similarly, since (in in which the approximation iic/e2 = Z2 - 1 + 23 - the absence of further information) the four-fermion 1 + 27 - 1 = 137 makes sense in terms of a computer (weak) interaction can start from any pair of 16 strings algorithm and information theory. Historically, the and lead to any other pair, the a priori coupling program which originally led to this way probability is 1/(256)2. Examination of how the of calculating a good approximation to the fine

3 Table I. Coupling constants and mass ratios predicted by the finite and discrete unification of quantum mechanics and relativity. Empirical Input: c, ti and mp as understood in the “Review of Particle Properties”, Particle Data Group, Physics Letters, B 239, 12 April 1990.

COUPLING CONSTANTS

Coupling Constant Calculated Observed

G-1 hC 1 [2127 + 1361 x [l - &] = 1.693 31.. . x [1.69358(21) x 1p8] mP GFmgltic [2562fi]-1 x [l - &] = 1.02 758.. . x p.02 682(2) x 10-51 Sin2eweak 0.25[1 - &I2 = 0.2267.. . [0.2259(46)] a-Yme) 137 x 11 - &I-’ = 137.0359 674.. . [137.0359 895(61)]

G2,Nrn [(*)2 - 113 = [195]+ = 13.96.. [13,3(3), > 13.9?]

MASS RATIOS

Calculated Observed

137~ = 1836.15 1497.. . [1836.15 2701(37)]

275[1- &] = 273.12 92.. . [273.12 67(4)]

274[1 - &]= 264.2 143.. . [264.1 373(6)]

3 - 7 * 10[1- &] = 207 [206.768 26(13)]

COSMOLOGICAL PARAMETERS

Parameter Calculated

NB IN-y & = 2.328 .... x lo-’’

Mdark/Mvis M 12.7

NB - NB (2127 + 136)2 = 2.m.. x 1078

M 4~10~~m, P/ Pcrit Mcrit

4 structure constant was started by Ted Bastin and Clive Then the Parker-Rhodes calculation of mp/mecan Kilmister in the 1950s using a somewhat different set be carried through using either the electromagnetic of computational . They were joined by John coupling constant (a = e2/hc) or the Fermi (weak) Amson, Gordon Pask, and Fredrick Parker-Rhodes. coupling constant. Self-consistency then requires that In 1961 Parker-Rhodes invented the combinatorial the two mass ratios so calculated be equd, to a hierarchg [SI and in subsequent work crystallized his good approximation. Independent of the details, of indistinguishables [9]. He starts from bit the “structural” factor, in my interpretation of the strings of length 2, and maps closure under XOR formulae is the same in both calculations. using square matrices and binary arithmetic. His To be more explicit, we refer to our original abstract scheme gives the three levels discussed above discussion of the Parker-Rhodes mp/mecalculation that we have been able to relate to the neutrinos, (Ref. [3], main text). There we noted that dimensional charged leptons, and quarks of the standard model. analysis shows that mp/me = 137/(z(1- z))(l/y) Mapping the 127 sets of level 3 which close under where “137” =tic/e2 and the geometrical factor comes XOR using 16 x 16 matrices yields strings of length from a statistical calculation of the creation and 256 and 2127- 1 sets. Parker-Rhodes’ sequence cannot reabsorption of proton-antiproton pairs in 3space, due be extended because the 2562 independent mapping to the coupling of the electron to proton-antiproton matrices available cannot begin to cover this number pairs via virtual prays with probability 1/137. If we of sets. use the Fermi interaction due to virtual weak vector By extending the Dyson to gravitation bosons rather than yrays to create and reabsorb the (Ref. [9]), the termination of the mapping can be same proton-antiproton pairs, thus replacing e2/fic attributed to the formation of a black hole containing, by GFm$/hc, appropriately normalized, the rest of initially, 2127 + 136 w 1.7 x w hc/Gmi the calculation remains the same. In particular, the gravitating baryons of protonic mass within ti/mpc. geometrical (or “structure”) factor (z(1 - z))(l/y) is This assemblage is unstable due to Hawking radiation identical. Since both the electromagnetic and the weak (which our model produces in quantized form), but calculations are perturbative in the respective coupling if the system started from charge e, spin ih and constants, the two calculations must specify the sanae baryon number 1, our baryon-number conserving mass for the electron in ratio to the proton mass, up to the neglect of terms of order [e2/hc] [G~m$/hc]. model insures that it will end up as a gravitationally x stabilized proton. From the Zurek-Thorne point of Equating the two results, the geometrical factor cancels view [lo],the 2127+136 baryon-antibaryon pairs which out and we arrive at the same connection between are lost in this process are just the number of bits of e2 and GF which weal-electromagnetic unification at information lost in forming this charged, rotating black the “tree level” also predicts. Calculated parameters hole. Similarly, lepton number conservation allows us are given in the table. It is to be emphasized that to view the electron as gravitationally stabilized. This our understanding of weak-electromagnetic unification in turn allows us to use the electron mass in the way does not requireindeed give no indication of the we did in calculating the pion and muon masses given of -the unobserved “Higgs boson.” in the table. We have now identified five universal constants Content strings, discrete space-time, (ti,c, mp,G, e2). Dimensional analysis allows US to and cosmology take any three from experiment in order to connect our scheme to the arbitrary units of mass, length, One distinction (an advantage from my point of and time in which physicists express the results of view) of using program universe rather than the measurement. Taking these to be h,c and mp and Parker-Rhodes mapping to generate our strings is that applying the McGoveran corrections we get the fine it continues to generate growing content strings after structure constant to seven significant figures and G in the labels of the known particles are gravitationally agreement with experiment, as indicated in the table. stabilized and fixed. We now sketch how these can We have now shown why, in our model, the be used to represent particle scattering processes to electron and proton are stable. But their mass ratio requisite accuracy. will still depend on their electromagnetic and weak We have shown above that la@ b/ = a + b - interactions. Parker-Rhodes originally computed the 2(alb). It follows that la 61bl f a + b. Further, the mp/meratio using an interesting mixture of classic smallest value that la 6I bJcan take on occurs when and combinatorial ideas (Ref. [ll]). McGoveran [ll] the maximum number of cancellations between the has supplied an alternative combinatorial argument, non-null elements in the two strings take place, i.e., which appears to me to attribute most of the mass when the maximum number of instances of a, = 1, of the proton to the strong (QCD) interactions. b,,, = 1 and hence a,,,- b, = 0 occur. Clearly, in

5 this case la 63 b( = la - bl and we have proved that theory of the Faddeev-Yakubovsky type, using sums la - bl 5 la@bl 5 a + b. It follows immediately rather than integrals, is tedious but straightforward. that any three strings which discriminate to the null The Mandelstam invariants emerge in a natural way, string-a 63 b @ c = O-specify a triangle with integer as does crossing symmetry. Details will be presented sides given by the three Hamming measures a, b, c. This elsewhere. follows from the that these three integers satisfy We have seen that program universe can generate the triangle inequalities la - b( 5 c 5 a + b, cyclic labels which can in turn be used it to model on a, b, c, as we have just proved above. Similarly, the quantum numbers eventually associated with the constraint a @ b @ c 63 d = 0 and the restriction “electric charge” or “visible matter”. This tentative a + b + c < d specifies (up to a choice of chirality) identification closes when we have reached the 127 a tetrahedron with arbitrary integer edges a, b,c. We labels of level 3 and the 256 labels which fix a particular have already shown that a,b, b,c and c,a specify representation of the standard (3 generation) model triangles with the third side of length la 63 bl, Ib 63 cI of quarks and leptons. During this process we will and IC 63 a1 respectively. The fourth side is obviously encounter the 10 labels of levels 1 and 2, statistically, also a triangle with sides of length la 63 b], ]b 63 c] and 12.7 more frequently than the labels associated IC @ a/. If we interpret the corners of the tetrahedron with electric charge. Once label length exceeds 256 and we start constructing our model of space-time, as counter firings, and assume that (for example) the time it takes a light signal to go along side a to the abc with the gravitational constant also fixed, this class vertex of the tetrahedron and back along side b is the of 10 uncharged labels can form gravitationally bound same the time it takes a particle to go along the third “dark matter” composed of neutrinos, photons and as gravitons.The 10 labels can now specify two spin states side of length la63bl; this speed is V,b = yc< for the neutrinos, two for the spin photons, five for c. Clearly this construction has a Lorentz invuriunt 1 the spin two gravitons, and a tenth for our quantized kinematic significance for finite and discrete rotations version of Newtonian gravitation. Thanks to lepton and boosts which preserve the integer character of number conservation, this uncharged (dark) “matter” the description [12]. The non-commutativity of can coalesce around finite numbers of neutrinos or finite rotations then implies the non-commutativity (but not and!) anti-neutrinos. We have yet to of position-velocity measurements, which in turn compute the fluctuations in program universe which implies relativistic commutation relations. These will (statistically) predict the distribution of masses in commutation relations in turn allow us to model a finite this dark matter. and discrete version of relativistic quantum mechanics The random choice of the concatenated bits when the minimum length interval between events is in program universe guarantees that, to a first hlmc. approximation, all strings will contain an equal number To give this construction a dynamic significance, we of zeros and ones, and hence that the universe will have recall that the energy-momentum structure for four about the same number of fermions and antifermions. particle processes with four-momenta pi = (&,&), We expect these to annihilate to (eventually) photons, i E 1,2,3,4 (c = 1) can be specified by the three but fermionic labels cannot persist unless there is scalar Mandelstam invariants s,t,u [13]. These are departure from equality of zeros and ones by at least defined for the process 1 + 2 -, 3 + 4 by s = (pl + one unit for the total collection of labels. Since we have ~2)~= (213 + ~4)~= rn? + 2E1& - 2P1 62 + mi; (256)4 possible types of baryon-number conserving t = (pl-p3)2 = = m;-2E1E3+2pi-p’3+mf; collisions which lead to photons, we expect the baryon u = (pl - ~4)~= (p2 - ~3)~= m: - 2E1E3 + 2pi - to photon ratio to be 256-4, which is about right. & + mi. Note that s, t,u are not independent because After fireball time (transition from an optically thick (on shell) energy-momentum conservation guarantees to an optically thin universe) we expect a conventional, that s + t + u = m: + 4+ mf + m:. Assume matter dominated expansion. Using the number of that energies and momenta are integers when measured distinct labels (2’27 + 136), the number of baryons in units of some smallest mass Am, and that the expected from the pairwise collisions is the square of individual particle invariants on mass shell are given by that number. If this is all in what is observationally s, = Ei (Ei + 1) - p’i .$i = mz. Here finite and discrete believed to be baryonic matter, we expect this to Lorentz boost invariance is preserved for any integer be about 2% of the critical mass, which is about value of p in the range -E 5 p 5 +E in the same what is observed. Whether we can meet the power way that the finite and discrete angular momentum spectrum fluctuations in the background radiation components in units of h preserve the rotational observed by COBE and the light element isotope invariance of j(j + I) - j,” = (1/2)(j+j- + j-j+) = mass ratios (“deuteronomy”) remains to be seen. j2 + ji. Representation in terms of bit-strings and Preliminary values of the parameters are collected in construction of a four-particle relativistic scattering the cosmological section of the table.

6 Electromagnetism and gravitation References The Feynman-Dyson [14] -Tanimura [15] proof [l] H.P. Noyes, “OPERATIONALISM REVISITED: of the Maxwell and Einstein equations for free space Measurement Accuracy, Scale Invariance and the electromagnetic and gravitational fields makes sense Combinatorial Hierarchy,” Physics Philosophy In- when we note that field strength per unit mass terface (in press). or charge is defined by the acceleration of a single [2] H.P. Noyes, “STAPP’s QUANTUM DUALISM: The James/Heisenberg Model of Consciousness,” test particle. [16] This allows us to use a scde invariant space-time theory based on measuring finite in Mind-Body Problem and the Quantum; Proc. ANPA WEST 10, F. ed. 112 Blackburn and discrete intervals between events (such as counter Young, firings) and measurement accuracy Ax, At down Ave., Menlo Park, CA 94025. &ed [31 J.Amson, Appendix in T.Bastin, H.P.Noyes, C.W. to intervals which involve (directly or indirectly) the Kilmister, and J.Amson, 1nt.J. Theor.Phys., 18, 455 production of particlegparticle pairs. This lower (1979). bound on classical, relativistic scale invariance arises [4] M.J. Manthey, in H.P. NOY~S,“ON THE CON- because of the complete breakdown of the concept STRUCTION OF RELATIVISTIC QUANTUM of “test particle” and hence of “classical, relativistic MECHANICS: A Report”, SLAC-PUB- field” once the phenomenon of pair creation prevents 4008, June 1986, pp. 101-110. us from giving meaning to what we mean by a single [5] H.P. Noyes and D.O. McGoveran, Physics Essays, particle. The Dyson paper is admittedly paradoxical, 2, 76 (1989). and Tanimura’s paper is at least as ambiguous. [6] D.O. McGoveran and H.P. Noyes, Physics Essays 4, However, by using “scale invariance bounded from 115 (1991). below” and a significant extension of the calculus [7] D.O. McGoveran and H.P. Noyes, in Discrete and of finite differences to non-commuting coordinate Combinatorial Physics (Proc. ANPA 9), H.P. Noyes, shifts, L.H.Kauf€man and HPN (in preparation), ed., ANPA WEST, 409 Leland Ave, Palo Alto, CA believe we have succeeded in providing a rigorous 94306; see also Ref. 5, p. 91. underpinning for these derivations, showing that our [S] F.J. Dyson, Phys. Rev. 85, 631 (1952). theory contains electromagnetism and gravitation as [9] H.P. Noyes, “Non-Locality in Particle Physics,” macroscopic approximations. SLAC-PUB-1405 (1975). [lo] T. Bastin, “On the Scale Constants of Physics,” Stvdia Philosophim Gandensia, 4, 77 (1966). Caveat and conclusion [ll] A.F. Parker-Rhodes, “The Theory of Indistinguish- ables,” Synthese, 150, Reidel, Dordrecht, Holland We warn the reader that detailed and rigorous (1981); - “The Inevitable Universe” (unpublished). mathematical proof of some of the statements made [12] W.H. Zurek and K.S. Thorne, Phys. Rev. Lett. 54, above is still missing. We wish to thank David 2171 (1985). McGoveran for pointing out to us that this caveat [13] D.O. McGoveran, in Alternatives in Physics and is particularly relevant for the use we make of , M.J. Manthey, ed., ANPA 12, F. Abdullah, the corrections he derived in the context of the City Univ., London EClV OHB, pp. 15-17. combinatorial hierarchy construction. For him, [14] H.P. Noyes, “The RQM Triangle”, in Reconstructing constructing our bit-strings using program universe the Object; Proc. ANPA WEST 8, F. Young, ed., and bringing in the identification of the labels form ANPA WEST, 112 Blackburn Ave., Menlo Park, “o~tside’~--- i.e., from known about quantum CA 94025. number conservation in particle physics - amounts to [15] “Review of Particle Properties,” Phys. Rev. D 45, creating a diflerent theory. While we have confidence 1 June 1992, Part 11, p. 111.50. that mixing up the two approaches in the way we have [16] F.J. Dyson, Am. J. Phys, 58, 209 (1990). can, eventually, be justified in a compelling way, it may [17] S. Tanimura. Annals of Physics, 220, 229 (1992). well turn out that our confidence in this outcome is [18] “On the Measurability of Electromagnetic Fields”, in Marshal Memorial Volume, E.G. Sudarshan, ed., overly optimistic. World Scientific (in pres). To summarize, by using a simple algorithm and detailed physical interpretation, we believe we have constructed a self-organizing universe which bears a close resemblance to the one in which physicists think we live. It is not “self-generating”-unless one grants that the two postulates with which Parker-Rhodes begins his book on the “inevitable universe”, namely: “Something m-sts! ” and “This statement conveys no information” suffice to explain why our universe started up. 7