I'm familiar with the Ising model. I used simulated annealing to search for minimum error in complex irregular pattern matching.
Stephen Wolfram in his book, A New Kind of Science, strongly promoted binary Cellular Automata as a path for all physics. His chapter on physics was replete with the words, maybe, perhaps, etc. I bought the book, read the final chapters, returned it, and trashed it in an Amazon review. Less knowledgeable reviewers were in awe, but the more knowledgeable were not.
Interesting idea. Jumping also sounds like hidden variables that are too fast to be useful in any prediction except as a probability distribution.
If it is really at the plank level, experiments would be more than difficult.
I think that it may also be an idea for resolving entanglement. Alice's particle is still in the same jumping phase as Bobs. Since photons have a zero space and time existence, they will be in the stochastic synchrony in the time frame of a photon's (zero) existence wherein the photon will be captured the instant it is created.
I once read that the wave function for an electron also traveled at the speed of light, while the phase velocity plods along. I tried looking it up to verify it but had no luck. But that would also explain entanglement for heavier particles.
See if this logic makes sense to you:
A Hilbert space is mappable 1-1 with a Cantor space. If we talk about "paths", they are evolutions on the Hilbert space. "Between" these points, on the dust, there are intervals that are hidden "to us". They're not hidden "variables" exactly, it's weirder than that.
Let's say the Hilbert space is what we can "observe", whereas the missing intervals are unobservable. So, we're living in Flatland. What do we "see"? Well, we have a weird notion of "virtual particles" and things like that. We "see" what looks to us like uncertainty, but what we really have is "holes" in our spacetime.
The idea of "holes" is not new. Positrons started out as holes in the Dirac sea, yes? However what I'm proposing can be formally described with Morse functions. The holes are an infinite sequence of Betti numbers. The idea is, if you "tile" a space and then remove a tile, you've broken symmetry. Penrose covered tiling quite nicely.
What we can "see" in the Hilbert space amounts to a projection. To "fill" the projection we only need to map the visible portion, and such a map is provided by the spectral basis with coordinates e^2πi. A spectral Cantor measure is a probability measure that maps directly to Hilbert space.
A probability measure in R^d is called a spectral measure if it has an orthonormal basis consisting of exponentials. In this paper we study spectral Cantor measures. We establish a large class of such measures and give a necessary and sufficient condition on the spectrum of a spectral Cantor...
ui.adsabs.harvard.edu
(In the discrete case we could - possibly ha ha - use Walsh functions or something like that).
So we just need to understand the Hausdorff dimension and the Euler characteristic. Everything else stays the same, the unitary operators still work and you can derive instantons and etc.
In quantum computing there are the theories no hiding, no cloning, and no deleting. These are constraints on the symmetries. They tell us that information is a lot like energy, it can't be created or destroyed. When information appears it has to "come from" somewhere, and when it disappears it has to go somewhere. This is why the set theoretic view is so important. Stone's representation theorem tell us the relationship between Hilbert space and Cantor space is isomorphic to that of (countable, atomless) Boolean algebras. These relationships are not to be taken lightly, ultimately they PREDICT that information transfer must accompany every movement (evolution) of energy in a Hilbert space.
Information transfer can be quantified with TREES made of subsets of the space. This means the following:
1. Let us say we have a very fast stochastic process bounded on both sides by 0 and some maximal number of steps. Define S as a non empty set (we'll call it "Spacetime"). Then define S(<w) to be the set of all possible finite sequences of elements of S (where w stands for omega because I don't have a math font). Now if s is an element of S, define the "length" of s as |s|, and |empty| = 0 and if x is in S(w) then |x| = w.
2. Now put a metric topology on S by defining t such that if t is in S(<w) and s is in S(<=w) and |t| < |s| and t(i) = s(i) then t is an initial segment of s. Define a
tree as a subset T of S(<=w) closed under initial segments. Put the discrete topology on S and the product topology on S(w). Define the metric as:
d(x,y) = 2 ^ (-n-1) if (x != y) ^ n = { min(i in w: x(i) != y(i) }
and 0 otherwise
3. The standard basis for this topology is the clooen sets N(s) : { x in S(w) such that s is a subset of x }.
This topology is
zero dimensional.
It is isomorphic to the Cantor space.
I could go on, but you get the idea. You can build a Cantor space from "just about anything". Once you have it, you can map it to any space built from the open unit interval, which includes the Hilbert space, and then it naturally follows there is a 1-1 metric correspondence, which in our case becomes a probability measure
