Mediating quantum forces: dynamics of "boson"
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-x0. The span is left as a footprint to the nodes visited. At a certain iteration, each node (x,t) thus possesses a trace λxt.
A Quantum Reset occurs when 𝜆 is different from ℓ. In this case, the span is exchanged with the trace, that is, the particle takes the trace and the node takes the span,
Moreover, the QR creates a new momentum-carrying boson, that is labeled with the particular mark (ℓ𝜆). This boson is left to the lattice node, becoming a footprint or Lattice Boson (LB) and replacing the previously existing LB with the same label, if it existed, so that at one time there is at most one pair of (ℓ𝜆) bosons. The old LB is transferred to the particle and becomes a Particle Boson (PB).
This boson exchange goes along with an exchange of momentum. The old LB transferred to the particle (PB) brings a momentum
}\Leftarrow\frac{\omega_{xt}^{(\ell\lambda)}}{|\ell-\lambda|} "v_Q^{(\ell\lambda)}\Leftarrow\frac{\omega_{xt}^{(\ell\lambda)}}{|\ell-\lambda|}")
Conversely, the new LB is created with a momentum
}\Leftarrow&space;|\ell-\lambda|&space;v_Q "\omega_{xt}^{(\ell\lambda)}\Leftarrow |\ell-\lambda| v_Q")
where the particle's quantum momentum, vQ, is the sum of the source momentum and of all the PB momenta carried on by the particle,
} "v_Q=v_0-\sum_{\ell,\lambda}v_Q^{(\ell\lambda)}")
At iterations where no quantum reset occurs, both the PB momenta and the LB momenta undergo a decay (tend to progressively vanish). The iterative rule for the PB momenta is
}[n]=v_Q^{(\ell\lambda)}[n-1]\left(1-\frac{1}{2(n-n_{QR}^{(\ell\lambda)})}&space;\right&space;) "v_Q^{(\ell\lambda)}[n]=v_Q^{(\ell\lambda)}[n-1]\left(1-\frac{1}{2(n-n_{QR}^{(\ell\lambda)})} \right )")
where nQR is the iteration of the last Quantum Reset, so that the decay is inversely proportional to the lifetime of the boson. Clearly, in the absence of further QR, the PB momenta would converge to zero. This decay rule can be regarded as a discrete counterpart of a continuous-time decay that is inversely proportional to the square root of the lifetime, vQ ~ 1/√t.
The iterative rule for the LB momenta is slightly more complex, being inversely proportional to the squared lifetime of the boson,
}[n]=\omega_{xt}^{(\ell\lambda)}[n-1]\left(1-\left(&space;\frac{\omega_{xt}^{(\ell\lambda)}[n_{QR}^{(\ell\lambda)}]}{n-n_{QR}^{(\ell\lambda)}}&space;\right&space;)^2\right&space;) "\omega_{xt}^{(\ell\lambda)}[n]=\omega_{xt}^{(\ell\lambda)}[n-1]\left(1-\left( \frac{\omega_{xt}^{(\ell\lambda)}[n_{QR}^{(\ell\lambda)}]}{n-n_{QR}^{(\ell\lambda)}} \right )^2\right )")
The continuous-time counterpart of this decay is something like ω ~ e(const./t). Differently from the PB momentum, in the absence of further Quantum Resets the LB momentum tends to a steady-state value that is different from zero.
On the basis of these rules, all possible complex interference patterns are built, as it will be shown 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-x0. The span is left as a footprint to the nodes visited. At a certain iteration, each node (x,t) thus possesses a trace λxt.
 |
| An exemple of Quantum Reset process with exchange of trace and momentum between a particle and a footprint |
A Quantum Reset occurs when 𝜆 is different from ℓ. In this case, the span is exchanged with the trace, that is, the particle takes the trace and the node takes the span,
Moreover, the QR creates a new momentum-carrying boson, that is labeled with the particular mark (ℓ𝜆). This boson is left to the lattice node, becoming a footprint or Lattice Boson (LB) and replacing the previously existing LB with the same label, if it existed, so that at one time there is at most one pair of (ℓ𝜆) bosons. The old LB is transferred to the particle and becomes a Particle Boson (PB).
This boson exchange goes along with an exchange of momentum. The old LB transferred to the particle (PB) brings a momentum
Conversely, the new LB is created with a momentum
where the particle's quantum momentum, vQ, is the sum of the source momentum and of all the PB momenta carried on by the particle,
At iterations where no quantum reset occurs, both the PB momenta and the LB momenta undergo a decay (tend to progressively vanish). The iterative rule for the PB momenta is
where nQR is the iteration of the last Quantum Reset, so that the decay is inversely proportional to the lifetime of the boson. Clearly, in the absence of further QR, the PB momenta would converge to zero. This decay rule can be regarded as a discrete counterpart of a continuous-time decay that is inversely proportional to the square root of the lifetime, vQ ~ 1/√t.
The iterative rule for the LB momenta is slightly more complex, being inversely proportional to the squared lifetime of the boson,
The continuous-time counterpart of this decay is something like ω ~ e(const./t). Differently from the PB momentum, in the absence of further Quantum Resets the LB momentum tends to a steady-state value that is different from zero.
On the basis of these rules, all possible complex interference patterns are built, as it will be shown in a future post.
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