运动电荷的运动学结构与惯性几何

中文摘要 AI 辅助

我们追问：带电粒子在其中运动的几何，有多少由其运动本身固定，又有多少必须由粒子自身携带。我们仅要求辛结构像哈密顿方程那样将速度与动量联系起来，并且同时要求它对每个能量都成立。特别地，我们证明满足这一要求的结构是典范辛结构及其被底空间上的闭二形式扭转后的结构。场提供二形式，粒子提供其前的乘子，我们将该乘子在构成上识别为粒子的电荷。因此，单一能量支配一族结构，每个粒子取用其电荷所固定的那一个。随后我们追问：粒子必须携带什么才能被赋予动量，并证明该映射的度决定了场以何种阶次进入牛顿第二定律。反对称双线性形式不产生洛伦兹力，而Randers度量则产生一个力——该长度与黎曼长度的差在速度上是线性的。此外，我们发现在扭转结构中该度量已经存在，作为自由能某个水平上的原像，并且其输运定律是非线性的，目前尚无已知仿射联络可为之服务。在不定号差下，长度与动力学分离，极值曲线从最短变为最长。在标准球面上，磁单极通量不留下这样的度量，而输运仍然存在，并且预量子化在给定作用量单位时将电荷限制为格点。如此，我们得出结论：每个荷质比都获得其自身的几何，因此根据功能主义标准，它们中没有一个是带电粒子的时空。

英文摘要

We ask how much of the geometry a charged particle moves in is fixed by its motion, and how much a particle must bring. We ask of a symplectic structure only that it relate velocity to momentum as Hamilton's equations do, and we ask it of every energy at once. In particular, we show that the structures meeting that demand are the canonical one and its twists by a closed two-form of the base. A field provides the two-form, a particle the multiplier before it, which we identify constitutively with its charge. Thus, a single energy governs a family of structures, and each particle takes the one its charge fixes. We then ask what a particle must bring to be given a momentum, and we show that the degree of that map settles the degree at which a field enters Newton's Second Law. An antisymmetric bilinear form returns no Lorentz force, whilst a Randers metric returns one --- a length whose difference from a Riemannian one is linear in the velocity. Moreover, we find that metric already within the twisted structure, as its primitive over a level of the free energy, and its law of transport to be nonlinear, no affine connection being known to serve. Under an indefinite signature the length parts from the dynamics, and the extremals turn from shortest to longest. On the round sphere a monopole flux leaves no such metric, whilst the transport remains and prequantisation, given a unit of action, restricts the charge to a lattice. In this manner, we conclude that each charge-to-mass ratio receives a geometry of its own, so that by a functionalist criterion none of them is the spacetime of a charged particle.

发表机构

• Universidad Autónoma Metropolitana – Azcapotzalco（墨西哥自治大学阿兹卡波察尔科分校）

机构由 AI 辅助整理，请以论文原文为准。