arXiv 2103.170742021-03-29gr-qcmath-phmath.MP

自旋与引力相互作用的几何研究

Geometric studies of the interplay between spin and gravity

Loïc Marsot

备注：: PhD thesis, 141 pages, 12 figures

摘要：

这篇博士论文是我在Serge Lazzarini的监督下在理论物理中心进行的一些工作的结论。手稿中提出了两个主题，都旨在研究基本粒子的自旋对其他已知理论的影响。首先是研究Lévy-Leblond-Newton（LLN）方程，该方程用于描述自旋为1/2的量子系统的演化，该系统与其自身的引力势耦合。在回顾了非相对论性量子力学的一些对称性以及如何借助Bargmann结构将它们几何化之后，我们回忆起什么是Lévy-Leblond方程。在这一章中，LLN方程在Bargmann结构上以完全协变的方式描述。这个协变公式有助于导出方程的动力学对称性和守恒量。这个方程的对称群就是已知的薛定谔-牛顿群。第二部分讨论了广义相对论中带自旋粒子的运动轨迹。我们首先强调Souriau的几何方法来推导Mathisson-Papapetrou-Dixon（MPD）方程，然后讨论不同的可能的自旋补充条件（SSC）存在关闭MPD方程组。在本章中，我们假设Tulczyjew SSC来描述光子的轨迹，从而得到Souriau-Saturnini方程。然后，我们提出了一些工作，我们已经应用这些方程，在史瓦西时空，并在时空变形的引力波。在第一项工作中，我们发现了引力双折射：轨道偏离测地线平面，这取决于光子的螺旋度及其波长。第二项工作也在引力干涉测量实验中发现了双折射，但这种效应比我们能检测到的要低很多个数量级。

英文摘要：

This PhD thesis is the conclusion of some of my works carried out at the Centre de Physique Théorique, under the supervision of Serge Lazzarini. Two subjects are presented in the manuscript, both aiming at studying the effect of the spin of elementary particles on otherwise known theories. First is a study of the Lévy-Leblond-Newton (LLN) equation, which is used to describe the evolution of a quantum system with spin 1/2 that is coupled to its own gravitational potential. After reviewing some symmetries of non relativistic Quantum Mechanics, and how to geometrize them with the help of Bargmann structures, we recall what is the Lévy-Leblond equation. In this chapter, the LLN equation is described in a fully covariant way on Bargmann structures. This covariant formulation then helps to derive the dynamical symmetries of the equation, and conserved quantities. The symmetry group of this equation turns out to be the already known Schrödinger-Newton group. The second part deals with the trajectory of particles with spin in General Relativity. We first highlight Souriau's geometric method to derive the Mathisson-Papapetrou-Dixon (MPD) equations, and then discuss the different possible Spin Supplementary Conditions (SSC) that exist to close the system of MPD equations. In this chapter, we assume the Tulczyjew SSC to describe the trajectory of photons, leading to the Souriau-Saturnini equations. We then present some works where we have applied these equations, in a Schwarzschild spacetime, and in a spacetime deformed by a gravitational wave. In the first work, we found gravitational birefringence: the trajectory deviates from the geodesic plane, depending on the helicity of the photon and its wavelength. The second work presented also found birefringence in gravitational interferometry experiments, however the effect is many orders of magnitude lower than what we can detect.

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摘要：

英文摘要：

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