Bell's inequality in a regular discrete topology

"I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case nothing remains of my entire castle in the air gravitation theory included. [and of] the rest of modern physics." from a letter of Einstein's quoted in Abraham Pais's. Subtle is the Lord.

The simplest approach to a fully discrete physical model is a regular topological connection of points each with a finite state. The state at each point changes in discrete time steps based on the state of the point and the states of a specified set of near neighbors. This is a finite difference equation. It is often used to approximate continuous differential equations when an exact solution is unknown or impractical. A discrete model for physics must approximate the continuous wave equation to very high accuracy. This is both the electromagnetic wave equation and the relativistic Schödinger equation for particles with zero mass.

The topology of spatial points is not, by itself, a geometry. Special relativity must be true to very high accuracy and thus the same object will appear to differ in length in different inertial frames. Geometry is a function of both the spatial topological grid and the laws of state transformation over time.

In finite difference equations, the velocity of wave propagation depends on the state change laws and may be much faster than the rate of one spatial step in one time step even though physical effects can propagate up to that rate. This opens the possibility of superluminal but still local effects limited by a velocity faster than light.

These nonlinearities can introduce chaotic like behavior that appears to be random. This suggests the speculative possibility that photons are distributed structures that usually accurately model the wave equation, but can transform chaotically into other particles under the right circumstances. The point like particles we observe are the focal point of these transformations whose size is determined by the uncertainty principle. If these particles exist they would be somewhat like attractors in chaos theory. Structures that model the existing fundamental particles of physics must be enormously large and complex if the discretization occurs in the range of the Planck time and distance scales.

It is easy to construct simulations of difference equations but impractical to do simulations on the scale required to model such particles. If something like this approach is correct it will probably take decades of experimentalists and theorists feeding on each others work to develop a detailed theory. Engineers may be involved as well as they use the new physics along the way to build more powerful computers. If this approach is close to the truth than there probably are currently feasible experiments testing Bell's inequality that could start this process.

Conservation laws that characterize the continuous equation will, to high accuracy, be obeyed by the discretized model. These laws will impose constraints on what particle transformations can occur. Many transformations can be expected to start to occur that cannot be completed because of the conservation laws or other structural constraints. These could be a source of the virtual particles of quantum mechanics. The process of particle transformation is one of chaotic like convergence to a stable but dynamic state as one set of attractor-like structures (particles) converge to a new set of attractor-like structures

It is plausible that this class of models could account for the existing experimental tests of Bell's inequality. Early in this series of experiments it was observed that quantum mechanics has a property called delayed determinism, meaning that the outcome of an observation may not be determined until some time after the event occurred[3]. We see this explicitly in the creation of virtual particles that start to come into existence before it is determined whether the transformation that has started will be able to be completed. Quantum mechanics does not have an objective definition of "event". That is why there are competing philosophical interpretations to fill this gap. Delayed determinism is an inherent part of the process of converging to a stable state. There is no absolute definition of "stable". It is a matter of probabilities. Any transformation might ultimately be reversed although that probability may become negligibly small rapidly. Delayed determinism and the lack of an objective definition of event makes it tricky to say just when an experiment has conclusively violated Bell's inequality.

There are significant weaknesses in existing experiments[4]. The process of converging to a stable state is a bit like water seeking its level. All the available mechanisms (paths of water flow) are used in creating the final state. The process of converging to a stable state may exploit every available loophole as well as superluminal effects possible with this class of models. If some finite difference approximation to the wave equation is a correct, then eventually quantum mechanics will disagree with experimental results in tests of Bell's inequality. One can think of these experiments as trying to determine iff there is a physical quantum collapse process with a local (but not relativistic) space-time structure that can be observed experimentally. If experiments were conducted with this possibility in mind, it might facilitate detecting an experimental violation of quantum mechanics.

A more complete and technical description is in [1]. For a discussion of the wider implications of discreteness for mathematics and philosophy as well as physics see [2].

[1] P. Budnik, "Emergent Properties of Discretized Wave Equations," Complex Systems 19(2) 2010.

[2] P. Budnik, What Is and What Will Be: Integrating Spirituality and Science, Los Gatos, CA: Mountain Math Software, 2006.

[3] J. D. Franson, "Bell's Theorem and Delayed Determinism," Physical Review D, 31(10), 1985 pp. 2529-2532.

[4] P. G. Kwiat, P. H. Eberhard, A. M. Steinberg, and R. Y. Chiao, "Proposal for a Loophole-Free Bell Inequality Experiment," Physical Review A, 49(5), 1994 pp. 3209-3220.