Second Quantization in Bit - String

Using a new fundamental theory based on bit-strings we derive a finite and discrete version of the solutions of the free one particle Dirac equation as segmented trajectories with steps of length h/me along the forward and backward light cones executed at velocity fc. Interpreting the statistical fluctuations which cause the bends in these segmented trajectories as emission and absorption of radiation, these solutions are analagous to a fermion propagator in a second quantized theory. This allows us to interpret the mass parameter in the step length as the physical mass of the free particle. The radiation in interaction with it has the usual harmonic oscillator structure of a second quantized theory. We sketch how these free particle masses can be generated gravitationally using the combinatorial hierarchy sequence (3,10,137, 2127 + 136), and some of the predictive consequences. 1 Bit-String Paths and Trajectories Bit-String Physics, which we ha.ve also called Discrete Physics, [l, 21 grew out of the discovery of the combinatorial hierarchy by A.F. Parker-Rhodes in 1961. [3] A convenient introduction is provided by the Proceedings of the 9th meeting of the Alternative Natural Philosophy Association. [4] Recent work is summarized at the end of this paper. In a technical sense, about all we need from the theory for this paper is the fact that we employ a universe of bit-strings generated by the algorithm called program universe in DP. Define a bitstring a containing W ordered bits by its sequentially ordered elements aur E 0, 1, 20 E 1,2,3, . . . . W, and its Hamming measure a by a = C~=,a, := [a( IV) I. D e ne fi d iscrimination, symbolized by ‘Q”, between two bit-strings by the ordered elements (a $ b)w = (a, b,)2; this is 1 when a, # b, and 0 when a,,, = b,. Starting from a universe of strings of length W, all that program universe does is to pick two strings arbitrarily and discriminate them. If the result is non-null (i.e. the two strings differ), the discriminant is adjoined to the universe and the process begins again. If the two strings discriminate to the null string (i.e 0, = 0 for all w), we concatenate an arbitrary bit to the growing end of each string (i.e. W + W + 1) and the process begins again. We consider two strings a, b and their discriminant a $ b. Given no further information, we now show that the situation can be described by four integers which are invariant under any permutation of the ordering parameter w applied simultaneously to all three strings. Let nlo be the-number of positions where a, = 1, b, = 0, no1 the number of positions where a, = 0, b, = 1, nil the ndmber of positions where a, = 1, b, = 1, and noo the number of positions where a, = 0, b, = 0. Then . a = nlo + nil; b = no1 + ml; la @ bl = nlo + ml (1) *Work supported by the Department of Energy, contract DE-AC03-76SFOO5 15 Contributed to the Workshop on Harmonic Oscillators, Baltimore, Maryland, March 25-28, 1992 / 7210 + 7201 + 7211 + no0 = W (2) Note that the three non-null Hamming measures a, b, ]a $ b] are independent of both no0 and W. Only one of those two parameters can be chosen arbitrarily, subject to the constraint that W 2 7210 + no1 + nil, or no0 2 0. It is the independence of our result from both string length and permutation of the order parameter which allows the statistics of the bit-strings generated by program universe to differ from the binomial distribution usually assigned to Bernoulli sequences, or “random walks”. The “random walk” with which we will be concerned is obtained from our more general model by defining a single, shorter string of length nio + noi by c, = 1 if a, = 1, b, = 0, and c, = 0 if a 2u = 0, b, = 1. Then r = n 10 is the number of l’s and e = noI is the number of O ’s in c(r+e), We now view this situation as describing the “motion” of a “particle” which is taking discrete steps of length h/me in space and h/mc2 in time at velocity fc along the light cones. This is the starting point suggested by Feynman[5] and articulated, for example, by Jacobson and Schulman[6] for a derivation of the Dirac equation in l+l dimensions. If the particle is assumed to start at the origin (0,O) in the Z, ct plane, their boundary condition on the trajectories connecting two events at (0,O) and (~,ct) is x = (r !)(h/mc), ct = (r + e)(h/mc). We tie our model to this same space-time trajectory, but as noted above include an additional degree of freedom. We now classify any string c by the number of bends k(c), which counts the number of times a sequence of l’s changes to a sequence of O ’s or visa versa. As McGoveran discovered,this number is simply computed from the elements of c by k(c) = c,w=;‘(c,+i c,)~. We are interested here in the number of bends in the trajectory string of length r +! = nlo + n ai These strings fall into four classes: RR, LL, RL and LR. For class RR the first and last steps are to the right; it has k + 1 right-moving segments, k left-moving segments and k bends; note that k = 0 corresponds to the forward light cone. Similarly LL has k + 1 left-moving segments, k right-moving segments and k bends. RL and LR cannot have k = 0 and-have k right-moving segments, left-moving segments and bends. This classification is the same as in Jacobson and Schulman, but our statistical treatment is different. In order to distinguish the connectivity we make between the two events from the spacetime trajectories considered by Feynman, we call them paths. It is the interpretation of the additional two parameters nll and no0 that allows us to extend our single particle treatment to an interpretation that has features in common with second quantized relativistic field theory. In the case of a statistically causal trajectory, time ticks ahead at a constant rate. If the particle does not take a step to the right, it must take a step to the left. Although our particle follows the same trajectory in space, if we encounter an example of w corresponding to either nil or no0 it does not move in the single particle configuration space that is all the Feynman approach contains. We interpret this as representing background processes going on in program universe which do not directly affect the particle. In a second quantized relativistic field theory these “disconnected diagrams” are the first to be removed in a renormalization program. Although conceptually crucial to the way we count numbers of paths, they do not enter directly into our calculations. Using light cone coordinates, a bend can be specified by any one of the r positions on the forward light cone and by any one of the f? positions on the backward light cone. However, because of the greater freedom in our string generation, there is no statistical correlation between them. There are rlc ways we can pick a position on the forward light cone and P on the left. All we.need do is insure that the restrictions imposed by the four classes of trajectories given above