Prepare and move two entangled particles
In this blog we are presenting and supporting a local and realist model for quantum mechanics, based on simple rules of (stochastic) motion on a discrete spatiotemporal lattice. The model is realist in the sense that at each instant particles have definite properties such as position, momentum, energy, etc. and these are independent of any possible measurement. Locality means that particles interact only with other beables that are resident in the lattice nodes actually visited. In particular, quantum behavior is reproduced for an ensemble of similarly-prepared particles and thanks to a footprint mechanism where particles leave some information in the node they visit which influece the behavior of subsequent particles.
One of the most common objections to the fact that one even tries to build a local and realist model, is that these models are simply impossible, as they are supposedly ruled out by Bell 's theorem. In this post and in future ones we shall demonstrate that the opposite is true, and that our model is indeed capable of violating Bell's inequalities.
A key element of Bell tests is the preparation of pairs of entangled particles. Many-particle systems where each particle evolve independently of the others are naturally represented in the proposed model: it is sufficient to attribute independent source quantities to the two emission sets, in addition to some flag that distinguishes the two sets. The Reset Condition is activated only for particles and lattice traces having the same flag.
We shall now consider here particles that are entangled in momentum (in a future post we shall consider spin entanglement). Such particles are emitted at sources as pairs. The two entangled particles are denoted with superscripts R = {I,II}. The source preparation attributes "entangled" source momenta, according to the rule
}+v_0^{(II)}=0 "v_0^{(I)}+v_0^{(II)}=0")
Without loss of generality, we shall denote v0 the source momentum of particle I.
In addition to source momentum and possible position-dependent source phase, each particle is attributed a random source phase 𝜑 = U[−1,1], which is the same for both particles,
}=\varphi^{(II)}=\varphi "\varphi^{(I)}=\varphi^{(II)}=\varphi")
With such simple prepration, we are ready now to discuss how the entanglement of with nR = 2 particles modifies the rules of motion described in previous posts.
The first modification concerns the lattice boson momentum (LBM) generated at Quantum Reset, that is now twice the one that would be created by non-entangled particles,
}\Leftarrow&space;n_R(|\ell-\lambda|v_Q^{(R)}-(\epsilon^{(R)}-\epsilon_{xt})) "\omega_{xt}^{(\ell\lambda,R)}\Leftarrow n_R(|\ell-\lambda|v_Q^{(R)}-(\epsilon^{(R)}-\epsilon_{xt}))")
In addition, the particle boson momentum (PBM) is one half the value for non-entangled particles, that is
Finally, the quantum momentum propensity is modified as follows:
}&space;-&space;\sum_{\ell,\lambda}&space;\left(&space;v_Q^{(\ell\lambda,R)}&space;+&space;\frac{\epsilon^{(\ell\lambda,R)}-\phi}{|\ell-\lambda|}&space;\right&space;) "v_Q^{R} = v_0^{(R)} - \sum_{\ell,\lambda} \left( v_Q^{(\ell\lambda,R)} + \frac{\epsilon^{(\ell\lambda,R)}-\phi}{|\ell-\lambda|} \right )")
where the new term between parantheses represents a relativistic correction and will be back-introduced in the non-entangled case in a future post.
With these rules at hand, we are now ready to evaluate the probability densities of position and momentum for entangled pairs, which will lead retrieving the standard Bell test statistics.
One of the most common objections to the fact that one even tries to build a local and realist model, is that these models are simply impossible, as they are supposedly ruled out by Bell 's theorem. In this post and in future ones we shall demonstrate that the opposite is true, and that our model is indeed capable of violating Bell's inequalities.
A key element of Bell tests is the preparation of pairs of entangled particles. Many-particle systems where each particle evolve independently of the others are naturally represented in the proposed model: it is sufficient to attribute independent source quantities to the two emission sets, in addition to some flag that distinguishes the two sets. The Reset Condition is activated only for particles and lattice traces having the same flag.
We shall now consider here particles that are entangled in momentum (in a future post we shall consider spin entanglement). Such particles are emitted at sources as pairs. The two entangled particles are denoted with superscripts R = {I,II}. The source preparation attributes "entangled" source momenta, according to the rule
Without loss of generality, we shall denote v0 the source momentum of particle I.
In addition to source momentum and possible position-dependent source phase, each particle is attributed a random source phase 𝜑 = U[−1,1], which is the same for both particles,
With such simple prepration, we are ready now to discuss how the entanglement of with nR = 2 particles modifies the rules of motion described in previous posts.
The first modification concerns the lattice boson momentum (LBM) generated at Quantum Reset, that is now twice the one that would be created by non-entangled particles,
In addition, the particle boson momentum (PBM) is one half the value for non-entangled particles, that is
Finally, the quantum momentum propensity is modified as follows:
where the new term between parantheses represents a relativistic correction and will be back-introduced in the non-entangled case in a future post.
With these rules at hand, we are now ready to evaluate the probability densities of position and momentum for entangled pairs, which will lead retrieving the standard Bell test statistics.
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