BitStringModel

I attempted to get a 3+1 version of the Dirac equation by treating three dimensions as a "phase space" on the bit string. I believe this paper draft was flawed. I think I was confused between spin and helicity.

I tried to start very simply in accord with my guiding philosophy by allowing only global reversible substitutions. For the particle to sometimes move Left and sometimes Right, it would therefore need two different structures; the simplest approach would be to have one form be a two-pixel particle and the other a one-pixel particle. The reversible substitution would move one pixel structures in one direction and two pixels in the other. This rule would be applied stochastically with another rule that would interchange one and two pixel structures; this would play the role of {$\epsilon$} in the Feynman checkerboard, providing mass. Having a different structure for L and R helicity could presage developing the model to break Parity symmetry as occurs in the Weak interaction.

Although at first I based the dynamics on a substitution rule that would move 1 and 2 pixel structures oppositely, I also had rules X and Y that would translate the particle along, but first interchange 1 and 2 pixel forms. This was vaguely consonant with the idea that the Dirac equation represents a kind of "square root of geometry", in that one must apply not just the X move but {$X^2$} in order to move ALL particles along in the same direction 1 pixel, keeping the same structure. (I will call X a half-step for that reason.) I then found that with my definitions of X and Y moves, that applying them in a loop, that is {$X Y X^{-1} Y^{-1}$} is equivalent to moving the 1-pixel form in one direction and the 2-pixel form in the opposite direction. The translation moves were not accompanied by any passage of time.

First, I though it interesting that this seemed analogous to a particle with angular momentum, in that for the particle to move in the spin direction and go forward in time, it had to go in a loop. Second, because the translation moves X and Y were noncommutative, when the half-steps that interchange 1 and 2 pixel forms are included, motion and time emerge from translations that scramble the bit-string and make it hard to invert simply to the starting configuration

Something else I tried in this paper draft: as some models of the Feynman Checkerboard (Ord and McKeon) developed negative amplitudes from "currents" of forwards in time moves subtracting backwards in time moves, I tried to develop full complex amplitudes. To go back to my guiding philosophy, I imagine the reversible global changes to the checkerboard to be like the derivation of theorems in a formal system. The directions of time within the system are defined through the parity violation. That is, when a 1-pixel structure moves to the right while a 2-pixel structure to the left, that is the Forward direction of time and the inverse is the Backwards direction.

What I tried was introducing an "outside time", imagining someone making changes to the network as one would derive theorems, with their own sense of which theorems were proved first and used to derive others. This would be different from the time internal to the system, which is defined along with the parity sense, in that L helicity should move the particle to the Left over time and R helicity to the Right. In my paper draft I called these Ancestor and Descendent paths, and those together with the time direction gave full complex amplitude.

Later I tried to see a parallel to this in that we write inner products separating bra and ket vectors in an arbitrary way, which parallels the arbitrary choice of complex conjugate. Bra versus ket being like Ancestor and Descendent. But I don't think that matched with the particular way I was trying to introduce complex numbers in this draft.

In the end, I don't think I really got more out of this model than I put in.