Unified Representation of 3D Multivectors with Pauli Algebra in Rectangular, Cylindrical and Spherical Coordinate Systems

Abstract

In practical engineering, the use of Pauli algebra can provide a computational advantage, transforming conventional vector algebra to straightforward matrix manipulations. In this work, the Pauli matrices in cylindrical and spherical coordinates are reported for the first time and their use for representing a three-dimensional vector is discussed. This method leads to a unified representation for 3D multivectors with Pauli algebra. A significant advantage is that this approach provides a representation independent of the coordinate system, which does not exist in the conventional vector perspective. Additionally, the Pauli matrix representations of the nabla operator in the different coordinate systems are derived and discussed. Finally, an example on the radiation from a dipole is given to illustrate the advantages of the methodology.

Keywords

• geometric algebra
• Clifford algebra
• Pauli matrices
• coordinate systems
• multivectors