Digital Quantum Reality

In a recent post , we have discussed how the local-realistic model of quantum mechanics includes external forces. Probability densities of position, momentum, etc., can be derived from those rules of motion, which will be shown in this post. We limit our attention to force fields that standard physics describes as derived from quadratic potentials . In our terminology, the effective momentum transfer ('force' in the standard terminology) generally depends on the lattice node and has the form For what discussed in the aforementioned post, we treat this force as occurring at each node. We start considering ensemble of particles emitted by a single source. The general rules of motion prescribe that the span changes its sign at each external reset, that is, by virtue of the "equivalence" assumption above, at each iteration according to the expression Therefore, there are no possible differences between the span and the trace found. Consequently...

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 . Figure by Borb, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=3816716 A force field is t...

In Quantum Mechanics, position and momentum are conjugate variables . The probability density function in momentum space, φ(p;t), is the Fourier transform  of the wave function in position space, ψ(x;t). A consequence of that is the Kennard inequality that describes the Uncertainty principle . A related consequence is that position and momentum operators are noncommuting, thus these observables are said to be incompatible and cannot in general be measured simultaneously. In our  local-realistic model , there is no need of the abstract mathematical formulation of standard quantum mechanics. In the latest post we have seen that the Schrödinger equation and the Born rule can be retrieved without appealing to complex wave functions. Indeed, we have already evaluated the position probability density function (in the case of a free particle with any type of source preparation) from the model's  elementary rules of motion . The key role in this evaluation is played by ex...

Quantum Mechanics theory describes systems with wave functions  and their evolution with the Schrödinger equation . Probabilities of the outcomes of measurements on a quantum system are given by the Born rule . In particular, the probability of a single particle to be in a certain position is obtained by squaring the modulus of the wave function (which is a function of position). In our local-realistic model of quantum mechanics, there is no Hilbert space, no complex numbers, no wave functions. As we have illustrated in previous posts, probability density functions (themselves, large-time approximations of probability mass functions) are obtained directly  from the stochastic rules of motion assumed. For example, we have seen in this post that, for a free particle, the pdf of position is obtained as where the indexes l and λ represent any two of the possible source, x 0 their locations, P 0 their probability, δ their relative distance (reflected by the differenc...

In the proposed local-realistic model for quantum mechanics, when the rules of motion for single particle instances are applied to an ensemble of similarly prepared particles, give rise to probability densities . In a previous post , we have discussed probability distribution for position in the case of free particles without quantum forces, that is, particles that are emitted from a single source. Illustration by user Geek3, Wikimedia Commons, https://commons.wikimedia.org/wiki/File:Hydrogen_eigenstate_n5_l2_m1.png Here we shall discuss how this probability density function changes in the presence of quantum forces. The derivation is rather straightforward. We shall use the already discussed large-time approximation and replace the position x with its average value as the argument of the probability density, with ⟨x⟩ = Vt, where we take the "trained" (i.e., steady-state) value for V. We also know from the elementary rules of motion that the momentum propensity i...

In standard Quantum Mechanics, wave packets  (where the represented variable is a probability amplitude) are an essential tool to describe particles in many scenarios. In the local-realistic model that we present in this blog, we have discussed in earlier posts how to describe ensembles of particles emitted from a discrete set of posssible sources. Particles emitted from different sources will generally have a different span when they reach the same lattice node at the same lifetime, which results in quantum forces . Yet another piece of iformation is missing so far, to describe phase that plays such a big role, notably, in wave packets. The model assumes that, at sources, particles are attributed a further quantity, denoted as phase source ε, that is generally a rational number, together with the momentum propensity v 0 . We will see in a future post that the list of source-attributed quantites is not over! The phase source plays a role in the boson dynamics. In parti...

The local-realistic model for quantum mechanics that we are presenting in this blog has the feature that quantum behavior (superposition) emerges after a large number of reproductions of the same system (e.g., a double-slit preparation). A Matlab code to simulate simple scenarios has been presented in this post , where the lattice is considered as already "trained". In a previous post , we have discussed how particle boson momenta also converge to a steady-state average value. Here below is a further accelerated code that takes advantage of such property and considers particles already "trained" and their momenta already converged. In practice, the whole mechanism of quantum reset is ignored and replaced by the first line in bold. The probabilities of QR are precalculated. Note also that a lag of several iterations has been introduced to avoid numerical oscillations and reproduce somehow the time scale of the training process. %%% Simulate an ensemble of Np p...