“Do Carroll particles move?” The answer depends on the characteristics of the particle such as its mass, spin, electric charge, and magnetic moment. A massive Carroll particle (closely related to fractons) does not move; its immobility follows from Carroll boost symmetry which implies dipole conservation, but not conversely. A massless Carroll particle may propagate by following the Hall law, consistently with the partial breaking of the Carroll boost symmetry. The framework is extended to Carroll field theory. In $d=2$ space dimensions, the Carroll group has a two-fold central extension which allows us to generalize the dynamics to massive and massless particles, including anyons. The anyonic spin and magnetic moment combine with the doubly-extended structure parametrized by two Casimir invariants interpreted as intrinsic magnetization and non-commutativity parameter. The extended Carroll particle subjected to an electromagnetic background field moves following a generalized Hall law which includes a Zeeman force. This theory is illustrated by massless, uncharged anyons with doubly-centrally extended structure we call exotic photons, which move on the horizon of a Black Hole, giving rise to an anyonic spin-Hall Effect.

“卡罗尔粒子会移动吗？” 答案取决于粒子的特性，例如质量、自旋、电荷和磁矩。一个巨大的卡罗尔粒子（与分形密切相关）不会移动；它的固定性遵循卡罗尔增强对称性，这意味着偶极子守恒，但反之则不然。无质量的卡罗尔粒子可以按照霍尔定律传播，与卡罗尔增强对称性的部分破缺一致。该框架被扩展到卡罗尔场论。在$d=2$在空间维度上，卡罗尔群有两个中心延伸，这使我们能够将动力学推广到有质量和无质量的粒子，包括任意子。任意子自旋和磁矩与由两个卡西米尔不变量参数化的双延伸结构相结合，解释为固有磁化强度和非交换性参数。受到电磁背景场影响的扩展卡罗尔粒子遵循广义霍尔定律（包括塞曼力）移动。这一理论通过具有双中心延伸结构的无质量、不带电的任意子来说明，我们称之为奇异光子，它们在黑洞的视界上移动，产生任意子自旋霍尔效应。