Dynamics in Geometric Models of Matter

自从爱因斯坦在他的广义相对论中将引力解释为时空几何以来，物理学家们一直在寻找对所有已知力的统一和几何描述。在爱因斯坦发明广义相对论后不久，西奥多·卡鲁扎和奥斯卡·克莱因提出了引力和电磁的统一几何模型，电磁力被解释为曲率到第五维(除了时间和三个空间维度之外)。必须假定第五维度紧紧地卷曲成小圆圈。在五维几何的某些假设下，Kaluza-Klein模型再现了经典电磁学和广义相对论。在最近的一篇论文Atiyah中，Manton和Schroers提出了一个类似于Kalzua-Klein模型的物质几何模型，但交换了电场和磁场的作用。他们对卡卢扎和克莱因的假设进行了修正和放宽，得出了电子和质子等基本粒子的四维几何模型。这些模型是静态的(不包括时间)，并捕获了相关粒子的定性属性。这个项目的目标是通过澄清物质如何解释基本粒子的自旋以及如何用纯几何术语描述相互作用来发展物质的几何描述。这是一项雄心勃勃和冒险的研究，涉及数学和物理的许多不同研究领域。它可能为基本粒子、核物理和原子物理的一种全新的数学语言铺平道路。

Ever since Einstein's interpretation of gravity as spacetime geometry in his theory of general relativity, physicists have searched for a unified and geometric description of all known forces. Shortly after Einstein invented general relativity, Theodor Kaluza and Oscar Klein proposed a unified geometric model of gravity and electromagnetism, with electromagnetic forces being interpreted in terms of curvature into a fifth dimension (in addition to time and the three spatial dimensions). The fifth dimension had to be assumed to be tightly curled up into small circles. Under certain assumptions on the five-dimensional geometry, the Kaluza-Klein model reproduces classical electromagnetism together with general relativity. In a recent paper Atiyah, Manton and Schroers propose a geometric model of matter which is akin to the Kalzua-Klein model, but exchanges the roles of the electric and magnetic fields. Modifying and relaxing Kaluza and Klein's assumptions, they arrive models for elementary particles like the electron and the proton in terms of four-dimensional geometries. These models are static (time is not included) and capture qualitative properties of the particles in question. The goal of this project is develop the geometric description of matter by clarifying how it accounts for the spin of elementary particles and how one can describe interactions in purely geometric terms. This is ambitious and adventurous research which draws on many different research areas in mathematics and physics. It could pave the way for a radically new mathematical language for elementary particle, nuclear and atomic physics.