arXiv subjects

Alexandre Gondran

Publications and source records attributed to Alexandre Gondran.

At least 19 recordsLinked to original sources

arXiv2025

An experiment to improve understanding wave-particle duality

This article presents an experiment that can be conducted today and that could provide a deeper understanding of the interaction between the wave and particle aspects of an atom. The wave-particle duality is often presented as mutually exclusive: one considers either the wave aspect or the particle aspect. Our proposed experiment involves both aspects simultaneously and raises new questions. It is a slightly modified version of Young's double-slit interference experiment (a grid of narrow slits is added between the two wide slits) and is carried out using Rydberg atoms. Young-type interference experiments typically involve only the de Broglie wave $\psi$, which depends solely on the mass and velocity of the atoms. However, with Rydberg atoms having a large principal quantum number, the ``size'' of the atom-particle also becomes significant. The two large slits are wide enough to allow the Rydberg atoms to pass through, whereas the grid of narrow slits prevents them from passing through. We numerically simulate the possible outcomes based on different hypotheses regarding wave-particle interaction. Conducting the experiment in practice would allow us to distinguish between these hypotheses and deepen our understanding of wave-particle interaction. The conceptual framework of Louis de Broglie's double solution theory is well-suited to this experiment because it distinguishes between two types of waves: an external or statistical wave (de Broglie's wave) and an internal or physical wave (corresponding to the physical particle). We will examine the relevance of this approach.

physics.atom-ph↗

arXiv2025

A pseudo-random and non-point Nelson-style process

We take up the idea of Nelson's stochastic processes, the aim of which was to deduce Schr\"odinger's equation. We make two major changes here. The first one is to consider deterministic processes which are pseudo-random but which have the same characteristics as Nelson's stochastic processes. The second is to consider an extended particle and to represent it by a set of interacting vibrating points. In a first step, we represent the particle and its evolution by four points that define the structure of a small elastic string that vibrates, alternating at each period a creative process followed by a process of annihilation. We then show how Heisenberg's spin and relations of uncertainty emerge from this extended particle. In a second step, we show how a complex action associated with this extended particle verifies, from a generalized least action principle, a complex second-order Hamilton-Jacobi equation. We then deduce that the wave function, accepting this complex action as a phase, is the solution to a Schr\"odinger equation and that the center of gravity of this extended particle follows the trajectories of de Broglie-Bohm's interpretation. This extended particle model is built on two new mathematical concepts that we have introduced: complex analytical mechanics on functions with complex values [8, 7, 12] and periodic deterministic processes [8, 9]. In conclusion, we show that this particle model and its associated wave function are compatible with the quantum mechanical interpretation of the double-scale theory we recently proposed [11].

quant-ph↗

arXiv2023

Double-scale theory

We present a new interpretation of quantum mechanics, called the double-scale theory, which expends on the de Broglie-Bohm (dBB) theory. It is based, for any quantum system, on the simultaneous existence of two wave functions in the laboratory reference frame : an external wavefunction and an internal one. The external wave function corresponds to a field that pilots the center-of-mass of the quantum system. The external wave spreads out in space over time. Mathematically, the Schr\"odinger equation converges to the Hamilton-Jacobi statistical equations when the Planck constant tends towards zero and the Newton trajectories are therefore approximations of the dBB trajectories. The internal wave function corresponds to the interpretation proposed by Edwin Schr\"odinger for whom the particle is extended. Then, the internal wave remains confined in space. Its converges, when h -> 0, to a Dirac distribution. Furthermore, we show that non-stationary solutions can exist such that the 3N-dimensional configuration space of the internal wave function can be rewritten as the product of N individual 3-dimensional internal wave functions.

quant-ph↗

arXiv2023

Semiclassical gravity in the de Broglie-Bohm theory and the double scale theory

We present a semiclassical gravity in the framework of the double scale theory, a new interpretation of quantum mechanics which expands on the de Broglie-Bohm (dBB) theory. In this interpretation, any quantum system is associated with two wave functions, one external and one internal. The external wave function drives the center-of-mass, as in the dBB theory, and spreads with time. The internal wave function corresponds to the density of an extended particle. It remains confined in space and its average position corresponds to the center of mass of the particle. We define, for an N-body system, new semi-classical gravity equations called Einstein-de Broglie equations. The external wave function of each body depends only on the internal wave functions of the N--1 others. They are deduced from Einstein's semi-classical equation by using the internal wave functions instead of the usual wave function. If we restrict to the dBB interpretation, we find the Newton-de Broglie equations that Struyve and Lalo{\"e} have recently proposed independently. We show that the Newton-de Broglie and Einstein-de Broglie equations converge to Newton's gravity when h $\rightarrow$ 0, which gives a theoretical validation.

physics.gen-ph↗

arXiv2020

External and internal wave functions: de Broglie's double-solution theory?

We propose an interpretative framework for quantum mechanics corresponding to the specifications of Louis de Broglie's double-solution theory. The principle is to decompose the evolution of a quantum system into two wave functions: an external wave function corresponding to the evolution of its center of mass and an internal wave function corresponding to the evolution of its internal variables in the center-of-mass system. Mathematical decomposition is only possible in certain cases because there are many interactions linking these two parts. In addition, these two wave functions will have different meanings and interpretations. The external wave function "pilots" the center of mass of the quantum system: it corresponds to the Broglie pilot wave. When the Planck constant tends to zero, it results mathematically from the convergence of the square of the module and the phase of the external wave function to a density and a classical action verifying the Hamilton-Jacobi statistical equations. This interpretation explains all the measurement results, namely those yielded by interference, spin measurement (Stern and Gerlach) and non-locality (EPR-B) experiments. For the internal wave function, several interpretations are possible : the one of the pilot wave can be applied in cascade to the internal wave function. However, the interpretation proposed by Erwin Schr{\"o}dinger at the Solvay Congress in 1927 and restricted to the internal wave function is also possible. For Schr{\"o}dinger, the particles are extended and the square of the module of the (internal) wave function of an electron corresponds to the density of its charge in space. We present many arguments in favour of this interpretation, which like the pilot wave interpretation is realistic and deterministic. Finally, we will see that this double interpretation serves as a frame of reference by which to better understand the debates on the interpretation of quantum mechanics and to review the relationships between gravity and quantum mechanics.

quant-ph↗

arXiv2019

Lower Bound for (Sum) Coloring Problem

The Minimum Sum Coloring Problem is a variant of the Graph Vertex Coloring Problem, for which each color has a weight. This paper presents a new way to find a lower bound of this problem, based on a relaxation into an integer partition problem with additional constraints. We improve the lower bound for 18 graphs of standard benchmark DIMACS, and prove the optimal value for 4 graphs by reaching their known upper bound.

cs.DM↗

arXiv2018

Optimality Clue for Graph Coloring Problem

In this paper, we present a new approach which qualifies or not a solution found by a heuristic as a potential optimal solution. Our approach is based on the following observation: for a minimization problem, the number of admissible solutions decreases with the value of the objective function. For the Graph Coloring Problem (GCP), we confirm this observation and present a new way to prove optimality. This proof is based on the counting of the number of different k-colorings and the number of independent sets of a given graph G. Exact solutions counting problems are difficult problems (\#P-complete). However, we show that, using only randomized heuristics, it is possible to define an estimation of the upper bound of the number of k-colorings. This estimate has been calibrated on a large benchmark of graph instances for which the exact number of optimal k-colorings is known. Our approach, called optimality clue, build a sample of k-colorings of a given graph by running many times one randomized heuristic on the same graph instance. We use the evolutionary algorithm HEAD [Moalic et Gondran, 2018], which is one of the most efficient heuristic for GCP. Optimality clue matches with the standard definition of optimality on a wide number of instances of DIMACS and RBCII benchmarks where the optimality is known. Then, we show the clue of optimality for another set of graph instances. Optimality Metaheuristics Near-optimal.

cs.DM↗

arXiv2017

The de Broglie-Bohm weak interpretation

We define the de Broglie-Bohm (dBB) weak interpretation as the dBB interpretation restricted to particles in unbound states whose wave function is defined in the three-dimensional physical space, and the dBB strong interpretation as the usual dBB interpretation applied to all wave functions, in particular to particles in bound states whose wave function is defined in a 3N-dimensional configuration space in which N is the number of particules. We show that the current criticisms of the dBB interpretation do not apply to this weak interpretation and that, furthermore, there are theoritical and experimental reasons to justify the weak dBB interpretation. Theoretically, the main reason concern the continuity existing for such particles between quantum mechanics and classical mechanics: we demonstrate in fact that the density and the phase of the wave function of a single-particle (or a set of identical particles without interaction), when the Planck constant tends to 0, converges to the density and the action of a set of unrecognizable prepared classical particles that satisfy the statistical Hamilton-Jacobi equations. As the Hamilton-Jacobi action pilots the particle in classical mechanics, this continuity naturally concurs with the weak dBB interpretation. Experimentally, we show that the measurement results of the main quantum experiments (Young's slits experiment, Stern and Gerlach, EPR-B) are compatible with the de Broglie-Bohm weak interpretation and everything takes place as if these unbounded particles had trajectories. In addition, we propose two potential solutions to complete the dBB weak interpretation.

quant-ph↗

arXiv2017

Heads or tails in zero gravity: an example of a classical contextual "measurement"

Playing the game of heads or tails in zero gravity demonstrates that there exists a contextual "measurement" in classical mechanics. When the coin is flipped, its orientation is a continuous variable. However, the "measurement" that occurs when the coin is caught by clapping two hands together gives a discrete value (heads or tails) that depends on the context (orientation of the hands). It is then shown that there is a strong analogy with the spin measurement of the Stern-Gerlach experiment, and in particular with Stern and Gerlach's sequential measurements. Finally, we clarify the analogy by recalling how the de Broglie-Bohm interpretation simply explains the spin "measurement".

quant-ph↗

arXiv2015

A heuristic extended particle 2D-model compatible with quantum mechanics

In this paper we propose an extended particle model whose evolution is deterministic. In dimension 2, the extended particle is represented by four points that define a small elastic string that vibrates, alternating between a creation process and an annihilation process. First we show how the spin and the Heisenberg uncertainty relations emerge from this extended particle. We then show how the complex action associated with this extended particle satisfies, from a generalized principle of least action, a second order complex Hamilton-Jacobi equation. Third, we show that the wave function, which admits this action as a complex phase, satisfies the Schr{\"o}dinger equation. Finally, we show that the gravity center of this extended particle follows the trajectories proposed by the de Broglie-Bohm interpretation well as the Schr{\"o}dinger interpretation. This model is built on two new mathematical concepts we have introduced: complex analytical mechanics on complex-valued functions and a periodic deterministic process.

math-ph↗

arXiv2015

Replacing the Singlet Spinor of the EPR-B Experiment in the Configuration Space with two Single-Particle Spinors in Physical Space

Recently, for spinless non-relativistic particles, Norsen, Marian and Oriols show that in the de Broglie-Bohm interpretation it is possible to replace the wave function in the configuration space by single-particle wave functions in physical space. In this paper, we show that this replacment of the wave function in the configuration space by single-particle functions in the 3D-space is also possible for particles with spin, in particular for the particles of the EPR-B experiment, the Bohm version of the Einstein-Podolsky-Rosen experiment.

quant-ph↗

arXiv2014

Measurement in the de Broglie-Bohm interpretation: Double-slit, Stern-Gerlach and EPR-B

We propose a pedagogical presentation of measurement in the de Broglie-Bohm interpretation. In this heterodox interpretation, the position of a quantum particle exists and is piloted by the phase of the wave function. We show how this position explains determinism and realism in the three most important experiments of quantum measurement: double-slit, Stern-Gerlach and EPR-B. First, we demonstrate the conditions in which the de Broglie-Bohm interpretation can be assumed to be valid through continuity with classical mechanics. Second, we present a numerical simulation of the double-slit experiment performed by Jönsson in 1961 with electrons. It demonstrates the continuity between classical mechanics and quantum mechanics: evolution of the probability density at various distances and convergence of the quantum trajectories to the classical trajectories when h tends to 0. Third, we present an analytic expression of the wave function in the Stern-Gerlach experiment. This explicit solution requires the calculation of a Pauli spinor with a spatial extension. This solution enables to demonstrate the decoherence of the wave function and the three postulates of quantum measurement: quantization, the Born interpretation and wave function reduction. The spinor spatial extension also enables the introduction of the de Broglie-Bohm trajectories, which gives a very simple explanation of the particles' impact and of the measurement process. Finally, we study the EPR-B experiment, the Bohm version of the Einstein-Podolsky-Rosen experiment. Its theoretical resolution in space and time shows that a causal interpretation exists where each atom has a position and a spin. This interpretation avoids the flaw of the previous causal interpretation. We recall that a physical explanation of non-local influences is possible.

quant-ph↗

arXiv2014

Variations on Memetic Algorithms for Graph Coloring Problems

Graph vertex coloring with a given number of colors is a well-known and much-studied NP-complete problem.The most effective methods to solve this problem are proved to be hybrid algorithms such as memetic algorithms or quantum annealing. Those hybrid algorithms use a powerful local search inside a population-based algorithm.This paper presents a new memetic algorithm based on one of the most effective algorithms: the Hybrid Evolutionary Algorithm HEA from Galinier and Hao (1999).The proposed algorithm, denoted HEAD - for HEA in Duet - works with a population of only two individuals.Moreover, a new way of managing diversity is brought by HEAD.These two main differences greatly improve the results, both in terms of solution quality and computational time.HEAD has produced several good results for the popular DIMACS benchmark graphs, such as 222-colorings for \textless{}dsjc1000.9\textgreater{}, 81-colorings for \textless{}flat1000\_76\_0\textgreater{} and even 47-colorings for \textless{}dsjc500.5\textgreater{} and 82-colorings for \textless{}dsjc1000.5\textgreater{}.

cs.AI↗

arXiv2013

The theory of the double preparation: discerned and indiscerned particles

In this paper we propose a deterministic and realistic quantum mechanics interpretation which may correspond to Louis de Broglie's "double solution theory". Louis de Broglie considers two solutions to the Schrödinger equation, a singular and physical wave u representing the particle (soliton wave) and a regular wave representing probability (statistical wave). We return to the idea of two solutions, but in the form of an interpretation of the wave function based on two different preparations of the quantum system. We demonstrate the necessity of this double interpretation when the particles are subjected to a semi-classical field by studying the convergence of the Schrödinger equation when the Planck constant tends to 0. For this convergence, we reexamine not only the foundations of quantum mechanics but also those of classical mechanics, and in particular two important paradox of classical mechanics: the interpretation of the principle of least action and the the Gibbs paradox. We find two very different convergences which depend on the preparation of the quantum particles: particles called indiscerned (prepared in the same way and whose initial density is regular, such as atomic beams) and particles called discerned (whose density is singular, such as coherent states). These results are based on the Minplus analysis, a new branch of mathematics that we have developed following Maslov, and on the Minplus path integral which is the analog in classical mechanics of the Feynman path integral in quantum mechanics. The indiscerned (or discerned) quantum particles converge to indiscerned (or discerned) classical particles and we deduce that the de Broglie-Bohm pilot wave is the correct interpretation for the indiscerned quantum particles (wave statistics) and the Schrödinger interpretation is the correct interpretation for discerned quantum particles (wave soliton). Finally, we show that this double interpretation can be extended to the non semi-classical case.

quant-ph↗

arXiv2012

From interpretation of the three classical mechanics actions to the wave function in quantum mechanics

First, we show that there exists in classical mechanics three actions corresponding to different boundary conditions: two well-known actions, the Euler-Lagrange classical action S_cl(x,t;x_0), which links the initial position x_0 and its position x at time t, the Hamilton-Jacobi action S(x,t), which links a family of particles of initial action S_0(x) to their various positions x at time t, and a new action, the deterministic action S(x,t;x_0,v_0), which links a particle in initial position x_0 and initial velocity v_0 to its position x at time t. We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend on the initial density. In the first case, where the initial quantum density is a classical density, the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the initial density. In the second case where initial density converges to a Dirac density, the density converges to the Dirac function and the quantum action converges to a deterministic action. Therefore we introduce into classical mechanics non-discerned particles, which satisfy the statistical Hamilton-Jacobi-equations and explain the Gibbs paradox, and discerned particles, which satisfy the deterministic Hamilton-Jacobi equations. Finally, we propose an interpretation of the Schrodinger wave function that depends on the initial conditions (preparation). This double interpretation seems to be the interpretation of Louis de Broglie's "double solution" idea.

quant-ph↗

arXiv2012

The Principle of Least Action as Interpreted by Nature and by the Observer

In this paper, we show that the difficulties of interpretation of the principle of least action concerning "final causes" or "efficient causes" are due to the existence of two different actions, the "Euler-Lagrange action" (or classical action) and the "Hamilton-Jacobi action". These two actions, which are not clearly differentiated in the texbooks, are solutions to the same Hamilton-Jacobi equation, but with very different initial conditions: smooth conditions for the Hamilton-Jacobi action, singular conditions for the Euler-Lagrange action. There are related by the Minplus Path Integral which is the analog in classical mechanics of the Feynmann Path Integral in quantum mechanics. Finally, we propose a clear-cut interpretation of the principle of least action: the Hamilton-Jacobi action does not use "final causes" and seems to be the action used by Nature; the Euler-Lagrange action uses "final causes" and is the action used by an observer to retrospectively determine the trajectory of the particle.

quant-ph↗

arXiv2011

The two limits of the Schr\"odinger equation in the semi-classical approximation: discerned and non-discerned particles in classical mechanics

We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend to the initial density . In the first case where the initial quantum density is a classical density rho_0(x), the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the density rho_0(x). In the second case where initial density

quant-ph↗

arXiv2010

Discerned and Non-Discerned Particles in Classical Mechanics and Quantum Mechanics Interpretation

We introduce into classical mechanics the concept of non-discerned particles for particles that are identical, non-interacting and prepared in the same way. The non-discerned particles correspond to an action and a density which satisfy the statistical Hamilton-Jacobi equations and allow to explain the Gibbs paradox in a simple manner. On the other hand, a discerned particle corresponds to a particular action that satisfies the local Hamilton-Jacobi equations. We then study the convergence of quantum mechanics to classical mechanics when hbar -> 0 by considering the convergence for the two cases. These results provide an argument for a renewed interpretation of quantum mechanics.

quant-ph↗