Digital Quantum Reality

In recent posts , I have described the creation and dynamics of "boson" in the proposed local-realistic model of quantum mechanics. In this post, we shall present a simulation code that implements these rules, allowing the local-realistic emergence of the QM outcome of a double slit experiment. The scenario is that introduced here . Written in the Matlab language, the code would like like that: %%% Simulate an ensemble of Np particles emitted at intervals Nti  %%% from either of Ns distinct sources xs(1,...,Ns) having  %%% probability Ps(1,...,Ns).  Evaluate the frequency of arrivals at %%% a 'screen' after Nt  iterations. %%% %%% Parameters: Np,Ns,Nt,Nti,xs,Ps. %%% %%% Evaluate the number of possible bosons for this scenario. B = Ns*(Ns-1)/2; %%% Attribute a label to each possible boson. %%% Bosons generated from the same pair of sources  %%% will have the same label. lab = NaN(Ns,Ns); for i = 1:Ns     lab(i,i+1:Ns) = [(i-1)*(Ns-i...

In a previous post , I have introduced the mechanism with which the proposed model treats the states superposition and ultimately represents intereference. The key roles are played by the footprints that a particle leaves on each lattice node visited, and the exchange of " bosons " between the particle and the footprints. These bosons possess a momentum, that is summed up to the particle-carried momentum to form the total momentum propensity (in the absence of external forces, which will be introduced in a future post). It is now time to describe in detail the rules concerning the formation and the evolution of such bosons. We have seen that each particle carries on a counter, called span , which sums up its position jumps since the emission. At a certain iteration, the span, ℓ, is thus equal to the distance covered so far, x - x 0 . The span is left as a footprint to the nodes visited. At a certain iteration, each node ( x,t ) thus possesses a trace λ xt . An exemple ...

After having described in the previous posts the motion of an ensemble of particles emitted at the same, perfectly localized in space, source, it is time to turn our attention to the emergence of quantum forces , which ultimately leads to the description of phenomena such as quantum superposition and the related interference. In this post, we shall discuss about the famous double-slit experiment and see how it is treated in the proposed model . In future posts, we will generalize the concepts introduced here. Consider a simplified, 1D version of a double-slit apparatus. where similar particles are emitted at a certain rate from either of two sources placed at positions x 0 = ±δ. Emitted particles have a constant speed along the perpendicular y direction, but a random initial speed along the x direction. At a certain distance from the sources along the y axis, a backstop is placed and particle arrivals are detected. Given the constancy of the y -axis velocity, will assume th...

The main results of this post about the stochastic motion of single particles can be obtained by considering such motion - for large times - as a Brownian motion with drift , where W(t) is a Wiener process  with zero mean and unit variance. The momentum propensity V here plays the role of the drift, while the variance is given by the diffusivity coefficient D  = 1- e , where e is the energy propensity (1+ V 2 )/2. For instance, we can derive the probability density from the Fokker-Planck equation of the Brownian motion, whose solution is clearly the Gaussian distribution shown in the aforementioned post, with the initial condition w = δ( x - x 0 ). Or, we can use the properties of stochastic processes to find the probability density of the  first return time to the source, i.e., the time at which the particle comes back for the first time at x =  x 0 . For some reason, I could not find a formula for that in the case of a Wiener proce...

In the latest post , we have discussed the substantial equivalence between the QM prediction of the position probability density for a free particle in 1D emitted by a perfectly localized source, via the Schrödinger equation and the Born rule (though the result is considered as non-normalizable) and the ensemble pdf of the position calculated with the proposed rules of motion and the proposed source preparation . The particle position can be regarded as a stochastic variable defined as where v is the momentum. We can define another stochastic variable as that is, the sum of momentum squared. Clearly, this variable is related to the accumulation of energy. It can take only positive values comprised between 0 and t . Its pmf is thus evaluated as where e =  (1+ V 2 )/2 is the energy propensity, a function of the momentum propensity. This binomial distribution is approximated for large times by a Normal distribution, with the mean at e∙t . Indeed, energy × tim...

The rules of particles' motion in the proposed model have been described as a random walk in this post , while in this post we have evaluated the probability mass function of position for one single particle emission. Each emission is characterized by one specific value of the source momentum v 0 that, in the absence of quantum or external forces, coincides with the momentum propensity V . However, for an  ensemble  of similarly-prepared particles, the source momentum is a random variable with uniform distribution in the range from -1 to 1 (see this post ). We are now interested in evaluating the position probability mass function (pmf) for the ensemble of particles. To do so, we just need to multiply the pmf w ( x ,  V ;  t ,  x 0 ) by the source momentum probability density and integrate over all possible values, The probability density function (pdf) of the source momentum is simply 1/2 (uniform distribution). Thus the ensemble pmf can be r...

We have described the simplest rules of motion (in the absence of any - quantum or external - forces) of the proposed local-realistic model for quantum mechanics in an earlier post . We have seen that these rules are inherently stochastic. Moreover, the probability distribution of the momentum depends on the source momentum that is, in turn, randomly attributed to particles at the source. It is therefore time now to derive the probability mass function of the position x reached after t iterations. Consider first a single particle with source at  x 0 , with a source momentum that coincides with the momentum propensity in this case (no external forces, no quantum forces),  v 0  =  V . The position pmf is given by the formula: for any x  comprised between  x 0 - t and  x 0 + t (the pmf is zero otherwise). Clearly, particles cannot be faster than light and thus cannot reach lattice nodes that are outside of the light cone from the source. Fo...

In this blog I am presenting a local and realistic model that simulates ensembles of similarly-prepared particles evolving on a discrete spacetime, with the aim of retrieving the results of Quantum Mechanics. With the basic rules of motion described in this post  and the source preparation in terms of momentum described  in this post , we can already simulate an ensemble of particles for a very basic scenario. We assume that there are no external forces (including potential barriers), nor " quantum forces " acting. What the latter condition means, will be clarified in future posts. For the moment, it might be regarded as the scenario where there is only one location (one lattice node) possible as the source of the emitted particles. In this scenario, a simple pseudocode implementing the 1D model can be written as follows: %%% Simulate an ensemble of Np particles for i = 1 to Np    %%% Attribute a random source momentum    v0 = rand[-1,1] ...

In an earlier post , we have seen that the spacetime lattice that forms the backbone of the proposed local-realistic QM model , has dimensions that correspond to the Compton length and time. These quantities are claerly variable with the mass and thus particle-dependent. How can different particle ensembles coexist and evolve on the same lattice, then? That is possible if all particles have a mass that is an integer fraction of a common multiple m 0 . If, say, the electron mass is m e =m 0 /k e , that of the top quark mass is m t =m 0 /k t , that of the tau particle is m τ =m 0 /k τ , etc., with all the k 's integers. Now, what is this common multiple  m 0 ? Particles with different mass have different lattice steps that are nevertheless integer multiples of the same minimal length Scientists have worked on the hypothesis of a discrete spacetime since long time. Most often, people prefer the concept of minimal length scale , that is the fundamental limit to the resolution ...