We consider a mass-less manifestly covariant {\it linear} Schrödinger equation. First, we show that it possesses a class of non-dispersive soliton solution with finite-size spatio-temporal support inside which the quantum amplitude satisfies the Klein-Gordon equation with finite {\it emergent} mass. We then proceed to interpret the soliton wave function as describing a particle with finite mass, energy and momentum. Inside the spatio-temporal support, the wave function shows spatio-temporal internal vibration with angular frequency and wave number that are determined by the energy-momentum of the particle as firstly conjectured by de Broglie. Imposing resonance of the internal vibration inside the spatio-temporal support leads to Planck-Einstein quantization of energy-momentum. The first resonance mode is shown to recover the classical energy-momentum relation developed in special relativity. We further show that the linearity of the Schrödinger equation allows one to construct many solitons solution through superposition, each describing a particle with various masses, energies and momenta.
Testing de Broglie's double solution in the mesoscopic regime
本文给出了任意线性外势作用下非线性薛定谔方程的解。非线性表示自聚焦相互作用。这些解是导频波与峰值孤子的乘积，峰值孤子的速度服从Louis de Broglie在1926年导出的导引方程。我们的近似的有效性程度增加时，孤子的大小减小，变得可以忽略不计的典型尺寸相比，在导频波的变化。我们讨论的可能性，以揭示他们的存在，通过实施一个humpty-dumpty Stern-Gerlach干涉仪在介观制度。
Whence Nonlocality? Removing spooky action at a distance from the de Broglie Bohm pilot-wave theory using a time-symmetric version of de Broglie double solution
Fermion self-trapping in the optical geometry of Einstein-Dirac solitons
我们用Finster，Smoller和Yau [Phys Rev. D 59，104020（1999）]介绍的形式分析了具有高角动量的多费米子的引力局域态。我们表明，由此产生的孤子一样的波函数可以自然地解释的一种形式的自陷，费米子成为本地化的外壳上的位置对应于那些'凸起'在光学几何形状创建自己的能量密度。
