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

The Hyperbolic Spinor Representation of Transformations in $$\mathbb {R}_1^3$$ R 1 3 by... In this study, firstly, we give a different approach to the relationship between the split quaternions and rotations in Minkowski space $$\mathbb {R}_1^3$$ R 1 3 . In addition, we obtain an automorphism of the split quaternion algebra $$H'$$ H ′ corresponding to a rotation in $$\mathbb {R}_1^3$$ R 1 3 . Then, we give the relationship between the hyperbolic spinors and rotations in $$\mathbb {R}_1^3$$ 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 $$\mathbb {R}_1^3$$ 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 $$\mathbb {R}_1^3$$ R 1 3 expressed by means of split quaternions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Applied Clifford Algebras Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Physics; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Applications of Mathematics; Physics, general
ISSN
0188-7009
eISSN
1661-4909
D.O.I.
10.1007/s00006-018-0844-0
Publisher site
See Article on Publisher Site

Abstract

In this study, firstly, we give a different approach to the relationship between the split quaternions and rotations in Minkowski space $$\mathbb {R}_1^3$$ R 1 3 . In addition, we obtain an automorphism of the split quaternion algebra $$H'$$ H ′ corresponding to a rotation in $$\mathbb {R}_1^3$$ R 1 3 . Then, we give the relationship between the hyperbolic spinors and rotations in $$\mathbb {R}_1^3$$ 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 $$\mathbb {R}_1^3$$ 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 $$\mathbb {R}_1^3$$ R 1 3 expressed by means of split quaternions.

Journal

Advances in Applied Clifford AlgebrasSpringer Journals

Published: Feb 24, 2018

References

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