Abstract In the Scale Relativity Theory, wave-corpuscle duality is analyzed both in the Schrödinger and in the Madelung multifractal scenarios. Therefore, in the Schrödinger scenario, this duality implies the functionality of a multifractal sinus – Gordon differential equation based on 2×2 matrices of real variable elements, while in the Madelung scenario, the same duality implies the solving of the multifractal hydrodynamic equation system.
Abstract In the Scale Relativity Theory, wave-corpuscle duality is analyzed both in the Schrödinger and in the Madelung multifractal scenarios. Therefore, in the Schrödinger scenario, this duality implies the functionality of a multifractal sinus – Gordon differential equation based on 2×2 matrices of real variable elements, while in the Madelung scenario, the same duality implies the solving of the multifractal hydrodynamic equation system.
By assimilating any complex system into a multifractal, a new approach for describing the dynamics of such systems is proposed by means of the Multifractal Theory of Motion. In such context, the description of these dynamics is accomplished through continuous and non-differentiable curves (multifractal curves), giving rise to two scenarios. The first scenario is a Schrödinger-type multifractal scenario, a situation in which the motion laws can be related to the SL(2R) algebra invariant functions. The second scenario is a Madelung-type multifractal scenario, a situation in which if the differentiable and non-differentiable components of the velocity field satisfy a particular restriction, an SL(2R) symmetry can also be highlighted. Moreover, correlative dynamics in either of the two scenarios, based on the same SL(2R) symmetry, can be obtained by Riccati-type gauges, which imply Stoler coherent states. Several cases induced by the SL(2R) symmetry are also analyzed.
Considering that the chaotic effect of associated wave packet on the particle itself results in movements on the fractal (continuous and non-differentiable) curves of fractal dimension DF, wave–particle duality through an extension of the scale relativity theory is given. It results through an equation of motion for the complex speed field, that in a fractal fluid, the convection, dissipation and dispersion are reciprocally compensating at any scale (differentiable or non-differentiable). From here, for an irrotational movement, a generalized Schrodinger equation is obtained. The absence of dispersion implies a generalized Navier–Stokes type equation, whereas, for the irrotational movement and the fractal dimension, DF = 2, the usual Schrodinger equation results. The absence of dissipation implies a generalized Korteweg–de Vries type equation. In such conjecture, at the differentiable scale, the duality is achieved through the flowing regimes of the fractal fluid, i.e. the wave character by means of the non-quasi-autonomous flowing regime and the particle character by means of the quasi-autonomous flowing regime. These flowing regimes are separated by '0.7 structure'. At the non-differentiable scale, a fractal potential acts as an energy accumulator and controls through the coherence the duality. The correspondence between the differentiable and non-differentiable scales implies a Cantor space-time. Moreover, the wave–particle duality implies at any scale a fractal.
原有量子力学的由Schrodinger方程不能很好地描述微观粒子的局域特性和波-粒二象性，因此，量子力学必须改进和发展。在非线性系统中我们考虑了粒子之间或它与背景场之间的非线性相互作用，于是必须用非线性Schrodinger方程去描述的微观粒子的状态。至此，微观粒子的状态和特征相对于线性系统发生了根本性的变化。在这种情况下不论外势场如何变化，但非线性Schrodinger方程都能给出具有波-粒二象特性的解，它由包络孤子和一个载波组成，有一个具有确定的位置的质心；同时，微观粒子此时也具有一个确定的质量，能量和动量，和它们遵守的质量，能量和动量守恒定律，在外场作用时粒子是稳定的，并满足经典运动方程和遵守哈密顿方程和拉格朗日方程。于是用非线性Schrodinger方程去描述的微观粒子具有一个明显的的局域特性和波-粒二象性， 从而彻底消除了原有量子力学存在近一个世纪的困难和问题。基于这些新的结果我们可建立起完整和系统的非线性量子力学理论体系，推动量子力学的发展。因此这一研究具有重要意义。 In original quantum mechanics the Schrodinger equation is, in essence, a wave equation, wherein the microscopic particles depicted have only a wave feature, and not a corpuscle nature. These descriptors do not agree only with the Broglie relation of wave-corpuscle duality, but also with experimental results and the traditional knowledge of particle concept. Meanwhile, the theory gives only some approximate solutions. Thus a series of contradictory representations and problems occur in quantum mechanics, which have re-sulted in durative disputations focused on the area of physics and have not led to any united conclusions until now. The only way to solve these problems and difficulties is to develop the quantum mechanics. We inves-tigate in detail wave-corpuscle duality of microscopic particles described by a nonlinear Schrodinger equa-tion in nonlinear quantum systems. Concretely speaking, we here study the properties of the solution of the nonlinear Schrodinger equation, the stability of microscopic particles, invariance and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, and so on. Studied results show that the solution of the nonlinear Schrodinger equation depicting microscopic particles is a soliton and have a wave-corpuscle duality, microscopic particles have always a mass center and possess determinant positions, sizes, mass, mo-mentum, energy and form, their mass, momentum and energy satisfy corresponding conservation laws, their dynamic states can be described by both nonlinear Schrodinger equation and classical Lagrangian and Ham-ilton equations, their motions obey the classical Newton-type law of motion. These properties indicate that the microscopic particles described by a nonlinear Schrodinger equation have a corpuscle nature. However, the solutions of dynamic equation are some solitary waves which are accompanied by a carrier wave to propagate in space-time, and possess certain frequency and wave speed. Thus microscopic particles described by nonlinear quantum mechanics have also wave feature. Thus we can affirm that the microscopic particles depicted by a nonlinear Schrodinger equation have a perfect wave-corpuscle duality, which are in essence different from those in linear quantum mechanics. Based on these results we can establish a new nonlinear quantum mechanics, which can solve these difficulties and problems disputed for about a century in linear quantum mechanics.
We show that the Schr\"odinger equation can be derived assuming the Galilean covariance of a generic wave equation and the validity of the de Broglie's wave-particle duality hypothesis. We also obtain from this set of assumptions the transformation law for the wave function under a Galilean boost and prove that complex wave functions are unavoidable for a consistent description of a physical system. The extension to the relativistic domain of the above analysis is also provided. We show that Lorentz covariance and wave-particle duality are consistent with two different transformation laws for the wave function under a Lorentz boost. This leads to two different wave equations, namely, the Klein-Gordon equation and the Lorentz covariant Schr\"odinger equation.
We perform a critical analysis of some quantum mechanical models such as the hydrodynamic model (Madelung's model), de Broglie's theory of double solution etc., specifying both mathematical and physical inconsistencies that occur in their construction.These inconsistencies are eliminated by means of the fractal approximation of motion (physical objects moving on continuous and non-differentiable curves, i.e. fractal curves) developed in the framework of Scale Relativity (SR) (Nottalle, L., 1993;Chaline, J. et al, 2009;