牛顿子量子引力、WIMP与天体力学--结果综述
Newtonion Quantum Gravity, WIMPs and Celestial Mechanics -- Summary of Results
摘要：
讨论了开普勒问题中牛顿量子引力半经典渐近性中的首项。对于暗物质，冰或尘埃粒子在引力场的恒星或大质量行星这解释了如何迅速行星或环系统可以形成考虑半经典方程的渐近的相关薛定谔波函数。我们扩展这种治疗各向同性谐振子潜力在二维和三维寻找明确的解决方案，特洛伊小行星的重要应用。关于限制性三体问题的更详细的结果将在即将发表的论文中介绍。进一步在二维空间中，我们找到了牛顿相应的旋转轨道的显式解，解释了行星近日点的推进。利用柯西特征曲线给出了方程解存在唯一的充分必要条件，并给出了方程的一般隐式解法。使用这种方法，我们解决了二维幂律问题在我们的设置。我们的方法给出了一个洞察的行为，半经典轨道的经典轨道附近的运动的复常数如何出现从SO（4）对称群和粒子密度必须为不同的中心势的圆形螺旋。这些常数在旋转轨道中具有重要作用。此外，它们和上述密度可能会在行星环系统的早期历史中产生可观察到的影响。我们相信这里的量子螺旋与经典开普勒椭圆轨道相关联，解释了星系的演化，在它们的中心有一个中子星或黑洞，从螺旋到椭圆。在这方面，我们包括一些最近的结果，量子曲率，扭转以及反重力和螺旋分裂，酸测试我们的想法。
英文摘要：
We discuss the leading term in the semi-classical asymptotics of Newtonian quantum gravity for the Kepler problem. For dark matter, ice or dust particles in the gravitational field of a star or massive planet this explains how rapidly planets or ring systems can be formed by considering semi-classical equations for the asymptotic of the relevant Schrodinger wave function. We extend this treatment to isotropic harmonic oscillator potentials in two and three dimensions finding explicit solutions, with important applications to the Trojan asteroids. More detailed results on the restricted 3-body problem are in a forthcoming paper. Further in two dimensions, we find the explicit solutions for Newton's corresponding revolving orbits, explaining planetary perihelion advance in these terms. A general implicit approach to solving our equations is given using Cauchy characteristic curves giving necessary and sufficient conditions for existence and uniqueness of our solutions. Using this method we solve the two-dimensional Power Law Problem in our setting. Our methods give an insight to the behavior of semi-classical orbits in the neighborhood of classical orbits showing how complex constants of the motion emerge from the SO(4) symmetry group and what the particle densities have to be for circular spirals for different central potentials. Such constants have an important role in revolving orbits. Moreover, they and the aforementioned densities could give rise to observable effects in the early history of planetary ring systems. We believe the quantum spirals here associated with classical Keplerian elliptical orbits explain the evolution of galaxies, with a neutron star or black hole at their centre, from spiral to elliptical. In this connection we include a brief account of some recent results on quantum curvature, torsion as well as anti-gravity and spiral splitting, acid tests of our ideas.
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