Polarization in action: 3D probability densities
In the latest post, we have discussed how quantum mechanical behavior in a three dimensional space can be reproduced by a local-realistic model, which has been partly published here, here, and here. In 3D, we have introduced "polarization" as a collection of three rational numbers attributed to particles at their source. The role of such polarization will be clearer in this post, where we shall discuss how QM probability densities are retrieved.
The procedure is quite similar to that already discussed here (without external forces) and here (with external forces) for 1D motion. Following the same reasoning, after lattice training, the particle boson momenta (PBM) for span-trace pairs (ℓ, λ) tend to
}=\frac{\sqrt{P_0^{(\ell\lambda)}}}{\sum_{d=1}^3&space;\rho_d\delta_d}\omega_{xt}^{(\ell\lambda)} "v_Q^{(\ell\lambda)}=\frac{\sqrt{P_0^{(\ell\lambda)}}}{\sum_{d=1}^3 \rho_d\delta_d}\omega_{xt}^{(\ell\lambda)}")
where all quantities have been introduced in previous posts (P0 is the source probability dunction, ω is the lattice boson momentum (LBM) δ is the path difference, ρ is polarization, and the subscript d denotes the three dimensions.
After lattice training, the LBM are
}=\frac{\sin\left(\pi\sum_{d=1}^3&space;\delta_d^{(\ell\lambda)}\frac{x_d-\left(x_0^{(\ell)}+x_0^{(\lambda)}\right)/2}{t}-\epsilon^{(\ell\lambda)}\right)}{\pi} "\omega_{xt}^{(\ell\lambda)}=\frac{\sin\left(\pi\sum_{d=1}^3 \delta_d^{(\ell\lambda)}\frac{x_d-\left(x_0^{(\ell)}+x_0^{(\lambda)}\right)/2}{t}-\epsilon^{(\ell\lambda)}\right)}{\pi}")
where ε is the phase difference and x0 denotes the source position.
Now, in the generalized motion rule (the second in this post) polarization weight the various PBM when combined to form the quantum momentum. The latter is eventually evaluated as
}}\sin\left(\pi\sum_{d=1}^3&space;\delta_d^{(\ell\lambda)}\frac{x_d-\left(x_0^{(\ell)}+x_0^{(\lambda)}\right)/2}{t}-\epsilon^{(\ell\lambda)}\right)}{\pi&space;\sum_{d=1}^3&space;\rho_d\delta_d^{(\ell\lambda)}} "v_{Qd}=v_{0d}-\rho_d\sum_{\ell,\lambda} \frac{\sqrt{P_0^{(\ell\lambda)}}\sin\left(\pi\sum_{d=1}^3 \delta_d^{(\ell\lambda)}\frac{x_d-\left(x_0^{(\ell)}+x_0^{(\lambda)}\right)/2}{t}-\epsilon^{(\ell\lambda)}\right)}{\pi \sum_{d=1}^3 \rho_d\delta_d^{(\ell\lambda)}}")
We can now compute the joint probability density function of the three average spatial coordinates ρ(〈x〉;t) from the latter equation and the average motion equation. The calculation is summarized in the 2017 paper and reproduced here below (averages are in bold).
Note that, when evaluating the Jacobian, the correct expression is obtained regardless of the specific values of the polarizations that, in fact, do not appear in the result. However, their role is to weight the path differences (which are integers) in the sum at the denominator of the first equation above, in order to avoid that positive and negative path differences cancel out, provoking a division by zero. Since polarizations are random rational numbers, this possibility is virtually ruled out.
where all quantities have been introduced in previous posts (P0 is the source probability dunction, ω is the lattice boson momentum (LBM) δ is the path difference, ρ is polarization, and the subscript d denotes the three dimensions.
After lattice training, the LBM are
where ε is the phase difference and x0 denotes the source position.
Now, in the generalized motion rule (the second in this post) polarization weight the various PBM when combined to form the quantum momentum. The latter is eventually evaluated as
We can now compute the joint probability density function of the three average spatial coordinates ρ(〈x〉;t) from the latter equation and the average motion equation. The calculation is summarized in the 2017 paper and reproduced here below (averages are in bold).
Note that, when evaluating the Jacobian, the correct expression is obtained regardless of the specific values of the polarizations that, in fact, do not appear in the result. However, their role is to weight the path differences (which are integers) in the sum at the denominator of the first equation above, in order to avoid that positive and negative path differences cancel out, provoking a division by zero. Since polarizations are random rational numbers, this possibility is virtually ruled out.
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