What Is Happening at the Bottom

We imagine that the cells that underlie physics are very small.  Lets assume that the distances between such cells is somewhere between a Fermi, 10^‑13cm,  and  Planck's length;  =1.6x10^‑33cm.  If it's down near Planck's length, particles such as electrons, would be enormous entities consisting of structures spread out over volumes containing more than 10⁶⁰ bits.  The momentum information would be spread out over volumes greater than 10⁷⁵ bits.  Because of superposition only a tiny fraction of the information in that volume would actually represent the momentum of the particle.  While it is difficult to give strong arguments as to what the scale should be, many interesting consequences are independent of the scale.    We call such informational models of physics "Digital Mechanics".  At first glance it seems that Digital Mechanics [5] must be at odds with things we know about physics. However, efforts to reconcile the two make steady progress.

The utility of Digital Mechanics rests on the question of Finite Nature.  If Finite Nature is true, then we may be able to derive QM from Digital Mechanics.  If Finite Nature is false, then QM cannot be derived from Digital Mechanics.  If Digital Mechanics ever becomes a successful model, it might allow for the development of a purely mechanistic, deterministic substrate for Quantum Mechanics.

The advantage of Digital Mechanics is that simple universal digital systems can do anything that can be done by any digital system.  This is also the disadvantage.  It is hard to think about the properties of the members of a class when each member can do everything.  The field of Computer Science has very few examples of useful or meaningful analytic solutions as to what some digital system will or won't do.  On the contrary, there is a celebrated proof that, in general, there are no analytical shortcuts that can tell the future state of some general computation any quicker than doing the computation step by step (this is the so called "halting problem" for Turing Machines [6] ).  There are normally no solutions in closed form.  There is not yet any good hierarchy of concepts that express complex behavior in terms of simpler behavior, as is done in physics.

On the other hand, good programmers synthesize programs that are hierarchical.  Programmers start with simple subroutines and build on them to create more complex programs and data structures.  Its just that most any old computational structure can produce complex behavior that defies any analysis other than running the program to see what happens.  A typical Macintosh, with an 80 megabyte hard drive is a universal computer with about 10^200,000,000 states; it is clearly capable of some pretty complex misbehavior.

There are exceptions; for example the use of cellular automata for lattice gas models [7] .  In this case very simple digital systems behave, in the limit, like systems easily modeled by differential equations.