Conformal Geometric Algebra and Galilean Spacetime

Abstract

This paper explores the application of geometric algebra to Galilean spacetime and its physical implications. We introduce the Galilean Spacetime Algebra (GSTA), a five-dimensional conformal geometric algebra (CGA) generated by a specific metric, and demonstrate its utility in representing special Galilean transformations, rotations, and boosts. The general form of special Galilean transformations within the GSTA is derived, demonstrating their preservation. While the tensor formulation of Galilean electromagnetism is well-established, our work offers a fresh insight by deriving it from a geometric algebra perspective, utilizing the GSTA, and demonstrates how it seamlessly reduces to the familiar Maxwell equations in the non-relativistic limit. A significant aspect of this research is the introduction of Galilean spinors as elements of the minimal left ideals of the GSTA. We illustrate how these spinors can be utilized to construct the L\'evy-Leblond equation for a free electron, along with its corresponding matrix representation. Furthermore, we establish a connection between the GSTA and the four-component dual numbers introduced by Majernik, suggesting pathways for developing a covariant formulation of Newtonian gravity. This work not only clarifies the geometric interpretation of Galilean symmetries but also opens avenues for future research in non-relativistic physics, highlighting the advantages of using CGA in this context.