# Presenting A Realistic, Stochastic, And Local Model Of Quantum Mechanics

Quantum mechanics is certainly one the most puzzling branches of physics. While a mathematical formalism that describes quantum mechanics is well known since almost one century and has been thoroughly tested experimentally, the interpretation of such formalism and whether it hides more fundamental ‘laws’ that would govern quantum mechanics is still the subject of profound debate.

One of the most intriguing aspects of quantum mechanics is the so-called wave-particle duality: individual particles that do not interact with each other can exhibit wave-like patterns when statistical distributions of measured events are considered. Richard Feynman stated that the phenomenon of electron diffraction in the famous double-slit experiment is “impossible, absolutely impossible to explain in any classical way, and has in it the heart of quantum mechanics. In reality, it contains the only mystery”. Many other quantum phenomena that puzzle scientists and the general public, such as the emergence of discrete levels of energy and angular momentum, the uncertainty principle, tunneling of potential barriers, entanglement, etc. are indeed related with this wave-like behavior and wave superposition in particular.

Quantum theory postulates that it is fundamentally impossible to go beyond a description in terms of probability amplitudes, those abstract complex numbers that serve to calculate probability distributions. Any mechanism is provided to describe the individual events that perhaps contribute to the observed statistical distributions. Nevertheless, most physicists embrace the orthodox view that the description offered by quantum mechanics is complete and that, indeed, probability amplitudes are the only thing that can be described.

Now, imagine a totally different attitude, a model for quantum mechanics based on real physical states, objective and independent of the observer. In such perspective, at a given time, particles have definite values for any possible observable, regardless if a measurement is actually made or not. In particular, real particle trajectories are assumed to exist.

Such “realistic” interpretations have been sought since the early quantum history. Perhaps the best-known realistic theory, the De Broglie-Bohm mechanics, describes individual particles as following well-defined trajectories that are regulated by a guiding wave. However, this theory explicitly appeals to a “nonlocal” mechanism, since the guiding wave is supposed to be able to instantaneously influence the particle trajectory far away. Even so, it is generally accepted as a valid theory, as it passes the famous Bell’s inequality test. However, its nonlocality leaves unsatisfied those who are aiming at explaining the quantum behavior by assuming a local-realistic, event-based behavior, that is, at the same time, complying with the principle of realism and that of locality.

Antonio Sciarretta has recently presented a realistic, stochastic, and local model that reproduces nonrelativistic quantum mechanics results without using its abstract mathematical formulation. In the proposed model, at a given time, individual particles have definite values for position and momentum, among other observables, thus fulfilling the requirement of **realism**. The **stochastic** behavior that is manifested by the empirical evidence of quantum mechanics is explained by assuming a fundamental randomness in particles trajectories. The emergence of quantum behavior is a consequence of the particular rules of motion chosen.

The motion of individual particles and their interaction with external forces take place on a discrete space-time under the form of a lattice. Particle trajectories are asymmetric random walks, with transition probabilities being simple functions of a few quantities that are either randomly attributed to the particles during their preparation at the sources, or stored in the lattice nodes that the particle visits during the walk. The lattice-stored information is progressively built as the nodes are visited by successive emissions. This process, where particles leave a “footprint” in the lattice that is used by subsequent particles, is ultimately responsible for the quantum behavior.

Therefore the interactions between subsequent emissions fulfill the **locality** requirement, albeit through the mediation of the lattice. Being based on discrete motion, the proposed model is rather peculiar as it only involves integer values for time, space, velocity, energy, etc. of a given particle. In addition, some rational valued quantities are derived from these primary quantities. Mathematical operations on physical quantities are reduced to arithmetic operations that, in fact, ultimately reflect the fundamental probability rules.

With Antonio Sciarretta’s model, quantum predictions are retrieved as probability distributions of similarly-prepared ensembles of particles. In the paper, the model is assessed by reproducing several quantum scenarios, including free particle motion, constant external force, harmonic oscillator, particle in a box, the Delta potential, particle on a ring, particle on a sphere. In particular, momentum entanglement is reproduced with the model that, albeit local, passes the Bell’s test (violates Bell’s inequality).

These findings are described in the article entitled A Local-Realistic Model of Quantum Mechanics Based on a Discrete Spacetime, published in the journal Foundations of Physics. This work was led by Antonio Sciarretta.