The lattice

Evolution of particles on a lattice

The proposed model assumes a discrete spacetime. Let us limit the discussion to one dimension for simplicity. The values of position are thus restricted to the integer multiples of a fundamental quantity X. Similarly, the values of time are restricted to the integer multiples of the fundamental quantity T. The spacetime may be therefore thought as if it is constituted by a grid, or lattice, whose nodes can be visitued by particles during their evolution.

Illustration of the lattice

The evolution of a particle is the particular succession of nodes x[n], t[n], where n is the integer counter that describe advance in history, called number of iterations.

Advance in time is unidirectional and unitary, that is

Advance in space is still unitary, but a particle can either advance in one of the two directions, or stay at rest,

where v is a random variable called momentum. The whole motion is regulated by this variable, which, at a given iteration, can take only three values: 1, 0, and -1.

The lattice dimensions and the Uncertainty principle

The fundamental quantities can be evaluated from Heisenberg's uncertainty principle. First consider a particle for which v is identically equal to 1. This situation is clearly the limit imposed by speed of light, so that it must be

On the other hand, consider the problem of observing a particle's momentum. The quantity v is a random variable and its expected value cannot be directly measured but only inferred from a (large) number N of samples. For N=1, the possible values of the average momentum are (in physical units): -c, 0, and c. Thus its uncertainty is equal to c. For N=2, the average momentum can take the values -c, -c/2, 0, c/2, and c. So its uncertainty is equal to c/2. Generalizing these results, after N observations, the uncertainty on the average momentum is equal to c/N.

Now consider the observation of the particle's position. Taking a measurement that lasts N iterations implies that the position could vary between -NX and +NX, thus its uncertainty is equal to 2NX (in physical units). Multiplying the momentum uncertainty with the position uncertainty, one obtains

by virtue of Heisenberg's principle (in one dimension, see below for three dimensions), where m is the mass (see also future posts on this point). Combining the two latter equations, one obtains for the lattice dimensions

that is, half the Compton length and time.

The same result is obtained if the average energy is to be observed. Take the energy as v². Its possible values are 0 and 1 (in lattice units). With a N=1 iteration observation, its uncertainty is thus equal to c² (in physical units). With N=2, the possible outcomes for the average energy are 0, c²/2, and c², thus the uncertainty is c²/2. After N observations, the energy uncertainty is c²/N. Multiplying this value by the mass, the factor 1/2 and the time uncertainty NT, one obatins the Planck constant h as it should be.

In three dimensions, the quantity h should be replaced by the reduced Planck constant h/(2π). In fact, the factor 2π is half the solid angle of a sphere with unit radius: in one dimension it is replaced by half the measure of the unit 1-sphere, which is the factor 1.

These relations are the same that have been used by the proponents of random walks derivation of the Schrödinger equation, particularly, G. Ord. Other authors have proposed - on totally different grounds - a fundamental quantization of spacetime based on Compton wavelength, including seminal work by H.S. Snyder.