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.