Motion in 3D and polarization
In all previous posts we have considered a local-realistic model for quantum mechanics working in a one-dimensional scenario, where motion is only along dimension x. However, ordinary space is made of three dimensions, thus the lattice geometry should reflect this fact. The extension of the model to the 3D case is not completely trivial and will be described here.
A first element has been already introduced in this post, where the source preparation and the attribution of a source momentum to particles have been described. The key notion is the limiting condition that we recall here as
Simple equations of motion are trivially extended to three dimensions. Basically, in the absence of quantum forces, each dimension is independent from the others, except for the limiting condition above. Similarly, no special treatment is required for external forces, which are intended to work separately on each dimension.
Motion rules in the presence of quantum forces introduce a few new quantities. The rule discussed here for the quantum momentum is, in fact, generalized as
}[n] "v_{Qd}[n]=v_{0d}-\rho_d^2\sum_{\ell,\lambda}v_Q^{(\ell,\lambda)}[n]")
where the three (momentum) polarizations are random numbers attributed at the source to each particle emission, such that
This polarization vector "controls" the split of the PBM (vQ(ℓ,λ)) among the various dimensions. Note that the PBM themselves remain scalar quantities.
The other modifications needed by the 3D model apply to the Quantum Reset conditions. When a lattice boson (LB) transfers its momentum to a newly-created particle boson (PB), the PBM is initialized as
}&space;\Leftarrow&space;\frac{\omega_{xt}^{(\ell,\lambda)}}{\sum_{d=1}^2&space;\rho_d^2&space;\delta_d} "v_Q^{(\ell,\lambda)} \Leftarrow \frac{\omega_{xt}^{(\ell,\lambda)}}{\sum_{d=1}^2 \rho_d^2 \delta_d}")
where δd is the path difference in the d-direction. The new LB is initialized with a momentum
}&space;\Leftarrow&space;\sum_{d=1}^3&space;v_{Qd}&space;\delta_d^{(\ell,\lambda)}-\epsilon^{(\ell,\lambda)} "\omega_{xt}^{(\ell,\lambda)} \Leftarrow \sum_{d=1}^3 v_{Qd} \delta_d^{(\ell,\lambda)}-\epsilon^{(\ell,\lambda)}")
which generalizes the reset rule given and the aforementioned post and later in this one, where the role of the phase difference had been introduced.
The last generalization concerns the External Reset (ER), when external bosons are captured. The span reset is generalized as
Note that the 1D ER rule is retrived when the force is applied along one direction only.
How probability densities can be computed from these generalized rules will be discussed next. The role of the newly-introduce polarization vector in spin dynamics will be also the subject of a future post.
A first element has been already introduced in this post, where the source preparation and the attribution of a source momentum to particles have been described. The key notion is the limiting condition that we recall here as
Simple equations of motion are trivially extended to three dimensions. Basically, in the absence of quantum forces, each dimension is independent from the others, except for the limiting condition above. Similarly, no special treatment is required for external forces, which are intended to work separately on each dimension.
Motion rules in the presence of quantum forces introduce a few new quantities. The rule discussed here for the quantum momentum is, in fact, generalized as
This polarization vector "controls" the split of the PBM (vQ(ℓ,λ)) among the various dimensions. Note that the PBM themselves remain scalar quantities.
The other modifications needed by the 3D model apply to the Quantum Reset conditions. When a lattice boson (LB) transfers its momentum to a newly-created particle boson (PB), the PBM is initialized as
where δd is the path difference in the d-direction. The new LB is initialized with a momentum
which generalizes the reset rule given and the aforementioned post and later in this one, where the role of the phase difference had been introduced.
The last generalization concerns the External Reset (ER), when external bosons are captured. The span reset is generalized as
Note that the 1D ER rule is retrived when the force is applied along one direction only.
How probability densities can be computed from these generalized rules will be discussed next. The role of the newly-introduce polarization vector in spin dynamics will be also the subject of a future post.
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