The Conformal Image of the Electron

The construction of a transformation image is pivotal to the transformation design method for steering waves around an object. Constructing the conformal images of an electron, however, remains a challenge, primarily due to the complexities involved in realizing the spin structure within the image space. This study proposes a novel solution in this regard of the electron's spin structure on a $2d$ surface, leveraging on the interactions between the electron and a vector potential. By introducing a scalar potential to facilitate the generation of a conformal metric structure and differential operator in the Dirac equation, it is demonstrated that the conformal image of the electron can indeed be constructed within an appropriate electromagnetic field. Furthermore, an equivalence between the geometric effect of a conformal deformation on the electron and the action of an electromagnetic force is established. Electromagnetic fields are used to generate form-invariant conformal images of the electron within a plane and on a sphere. These examples help showcase the effectiveness of the currently presented method in realizing the conformal images of the electron in various settings.