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Quantum-phase-field: from the Broglie–Bohm double-solution program to doublon networks
Different forms of linear and non-linear field equations, so-called ‘phase-field’ equations, are studied in relation to the de Broglie–Bohm double-solution program. This defines a framework in which elementary particles are described by localised non-linear wave solutions moving by the guidance of a pilot wave, defined by the solution of a Schrödinger-type equation. First, we consider the phase-field order parameter as the phase for the linear pilot wave, second as the pilot wave itself, and third as a moving soliton interpreted as a massive particle. In the last case, we introduce the equation for a superwave, the amplitude of which can be considered as a particle moving in accordance to the de Broglie–Bohm theory. Lax pairs for the coupled problems are constructed in order to discover possible non-linear equations that can describe the moving particle and to propose a framework for investigating coupled solutions. Finally, doublons in 1 + 1 dimensions are constructed as self-similar solutions of a non-linear phase-field equation forming a finite space object. Vacuum quantum oscillations within the doublon determine the evolution of the coupled system. Applying a conservation constraint and using general symmetry considerations, the doublons are arranged as a network in 1 + 1 + 2 dimensions, where nodes are interpreted as elementary particles. A canonical procedure is proposed to treat charge and electromagnetic exchange.