Source preparations and momentum

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In an earlier post, we have seen that two of the building blocks of a local and realistic model of Quantum Mechanics are the assumption of a discrete spacetime (the size of Compton wavelenght) and a three-valued momentum as a random variable, whose probability distribution is completely determined by its expected value, or momentum propensity.

At a given iteration, quantum propensity V is the sum of two contributions,

where vQ is the momentum contribution of quantum forces, and vF that of standard, "external" forces.

Quantum momentum

The quantum momentum is a quantity carried by the particle, which is first assigned at a certain value during the particle's preparation and may subsequently vary as a function of quantum forces. When a particle is prepared at a source, vQ is initialized to a value that we denote as v0 , the source momentum. This quantity is itself a random variable: it can take rational values and is uniformly distributed between -1 and +1.

If the possible source is unique, then there will be no quantum forces acting (see future posts on this point), and vQ will be always equal to v0 all along the particle's evolution. Thus, in absence of both quantum and external forces (single-source free particle), V will be always equal to v0. The particle will randomly fluctuate along the trajectory



where n0 is the iteration at which the particle has been emitted and x0 is the source location.

Three-dimensional source momentum

So far, we have considered one-dimensional motion. In 3D, position is not described by a single integer x but by a triple {xd}, with d spanning from 1 to 3. Correspondingly, momentum and momentum propensity actually read v={vd} and V={Vd}, respectively.

The source momentum is also a triple v0={v0d}. Each of the three components is randomly attributed at the source a value that is uniformly distributed between -1 and +1. The choice of each component is made independently from the others. However, and here special relativity plays a first role, only those combinations for which

 
are valid and actually attributed to the particle.

One might imagine that the process of attributing the source momentum is a 3D (pseudo-)random uniform number generation process where numbers can be selected only inside a unit sphere.

In future posts, we shall see which other quantites are attributed at a particle's preparation, and how free particles evolve in a more detail.




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