SLAGPUB-6509 August 1994 (T/NOY4 Bit-String Physics: a Novel “Theory of Everything”’ 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 theories (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, unobservable “gauge potentials”, and no a finite and discrete space-time 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 laws and by identifying h, c and GN~~~in their theoretical commutation relations insure the essential charac- structures. It is then an act of faith 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 consensus theory 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, information 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 quality of the fit to experiment 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 observation” does or does not to seven significant figures, they are not perfect. So require “consciousness” [2]. far as we can see the discrepancies could arise from the concatenation of effects we know we have so far By a “theory of everything” (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 experiments to encounter “failure” as we extend the calculations. and by observational cosmology. 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 algorithm 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 symbols “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, set number, ratio of baryons to photons, fireball time, P := P + 1 and recurse. and ratio of dark to baryonic matter 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.