Capturing the external-force bosons
Although we have denoted as "quantum forces" the interactions between a particle and the previous instances in an ensemble (interactions mediated by the lattice) that are ultimately rensponsible of quantum behavior, in the previous posts we have considered free particles that do not interact with the external world and are not subject to external "true" forces.
However, the local-realistic model of quantum mechanics that we promote in this blog describes external forces as well. The principle is similar to that governing quantum forces: during their walk on the fundamental lattice, particles m ay capture external bosons, which mediate external forces, and incorporate their momentum. The event of encountering and capturing an external boson is denoted as "External Reset (ER)", in analogy to the Quantum Reset (QR) discussed in this post.
A force field is thus characterized by two numbers: the probability for a particle to capture a boson at any particular site, and the momentum transferred by the boson to the particle momentum propensity V. Both these quantities depend on the characteristics of the force field. Let us abandon for a moment the 1D case we have been treating in all recent posts, and let us take the example of a spherically-symmetric potential (a force that depends only on the distance from a certain lattice node O). If the source O emits a certain number N of bosons (e.g., true photons) at each iteration, all having the same intensity of the source momentum, albeit random direction (that point will be clarified in a future post where 3D motion rules will be discussed), the probability ρER to find one of such bosons at a node distant r lattice units from O is given (for sufficiently large r) by the ratio of an element of the spherical surface around the node and the surface area of the sphere with radius r, which is
=\frac{X^2}{4\pi&space;r^{'2}}=\frac{1}{4\pi&space;r^2} "\rho_{ER}(r)=\frac{X^2}{4\pi r^{'2}}=\frac{1}{4\pi r^2}")
where X is the lattice fundamental size and r' is the distance in physical units (r' = X r). Of course, the quantity ρER is always less than 1, as a probability should be.
The momentum transferred at each ER will generally depend both on the boson and the particle features. We can denote this momentum as
}&space;\rho_{F} "\Delta v_{ER} = v_Q^{(F)} \rho_{F}")
where vQ(F) is the momentum carried by the boson (we use the same nomenclature as for the PBMs) and ρF is the probability of transfer, which generally depends on the particle's feature. Overall, the expected value of the transferred momentum can be easily evaluated as
}&space;\rho_{F}}{4\pi&space;r^2} "f=\frac{v_Q^{(F)} \rho_{F}}{4\pi r^2}")
For example, Coulomb's law can be parametrized with ρF = ke q1 and vQ(F) = q2/(4π), where ke is Coulomb's constant, while q1 and q2 are the electric charges of the flying and the source particle supposedly residing at O, respectively. Other, non-spherically-symmetric potentials, can be treated with the same approach and parametrized with an average transferred momentum f, that is generally a function of the node x and the iteration n.
The latter modifies the momentum propensity according to the rule
 "V[n]=v_Q[n]+v_F[n]=v_Q[n]+\sum_{n'=n_0+1}^{n} f(x[n'],n')")
where vQ is the quantum momentum (as discussed in this post, the sum of source momentum and all of the PB momenta carried on by the particle). The formula above generalizes the rule V = vQ that is valid for free particles. It is clear that the effective quantity f mimics the classical concept of force.
A correction to this rule to cope with special relativity will be discussed in a future post.
The last effect of external bosons is on the particle's span that we have introduced in this post and denoted with ℓ. At an external reset, the span is just inversed,
However, the local-realistic model of quantum mechanics that we promote in this blog describes external forces as well. The principle is similar to that governing quantum forces: during their walk on the fundamental lattice, particles m ay capture external bosons, which mediate external forces, and incorporate their momentum. The event of encountering and capturing an external boson is denoted as "External Reset (ER)", in analogy to the Quantum Reset (QR) discussed in this post.
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| Figure by Borb, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=3816716 |
A force field is thus characterized by two numbers: the probability for a particle to capture a boson at any particular site, and the momentum transferred by the boson to the particle momentum propensity V. Both these quantities depend on the characteristics of the force field. Let us abandon for a moment the 1D case we have been treating in all recent posts, and let us take the example of a spherically-symmetric potential (a force that depends only on the distance from a certain lattice node O). If the source O emits a certain number N of bosons (e.g., true photons) at each iteration, all having the same intensity of the source momentum, albeit random direction (that point will be clarified in a future post where 3D motion rules will be discussed), the probability ρER to find one of such bosons at a node distant r lattice units from O is given (for sufficiently large r) by the ratio of an element of the spherical surface around the node and the surface area of the sphere with radius r, which is
where X is the lattice fundamental size and r' is the distance in physical units (r' = X r). Of course, the quantity ρER is always less than 1, as a probability should be.
The momentum transferred at each ER will generally depend both on the boson and the particle features. We can denote this momentum as
where vQ(F) is the momentum carried by the boson (we use the same nomenclature as for the PBMs) and ρF is the probability of transfer, which generally depends on the particle's feature. Overall, the expected value of the transferred momentum can be easily evaluated as
For example, Coulomb's law can be parametrized with ρF = ke q1 and vQ(F) = q2/(4π), where ke is Coulomb's constant, while q1 and q2 are the electric charges of the flying and the source particle supposedly residing at O, respectively. Other, non-spherically-symmetric potentials, can be treated with the same approach and parametrized with an average transferred momentum f, that is generally a function of the node x and the iteration n.
The latter modifies the momentum propensity according to the rule
A correction to this rule to cope with special relativity will be discussed in a future post.
The last effect of external bosons is on the particle's span that we have introduced in this post and denoted with ℓ. At an external reset, the span is just inversed,
This rule plays a role within non-homogeneous force scenarios, like potential walls.
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