The Hyperbolic Spinor Representation of Transformations in $$\mathbb {R}_1^3$$ R 1 3 by Means of Split Quaternions

Abstract

In this study, firstly, we give a different approach to the relationship between the split quaternions and rotations in Minkowski space $$\mathbb {R}_1^3$$ . In addition, we obtain an automorphism of the split quaternion algebra $$H'$$ corresponding to a rotation in $$\mathbb {R}_1^3$$ . Then, we give the relationship between the hyperbolic spinors and rotations in $$\mathbb {R}_1^3$$ . Finally, we associate to a split quaternion with a hyperbolic spinor by means of a transformation. In this way, we show that the rotation of a rigid body in the Minkowski 3-space $$\mathbb {R}_1^3$$ expressed the split quaternions can be written by means of the hyperbolic spinors with two hyperbolic components. So, we obtain a new and short representation (hyperbolic spinor representation) of transformation in the 3-dimensional Minkowski space $$\mathbb {R}_1^3$$ expressed by means of split quaternions.