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Idempotents and Nilpotents in the Clifford Algebra of Euclidean 3-Space and Their Interconnections in Physics
2009jbr | O. Mornev  // mornev@mail.ru

Within the framework of Space Algebra, the Clifford algebra $Cl_{3}$ generated by the three-dimensional Euclidean space $E_{3}$ over ${\rm {\mathbb R}}$, a structure of idempotents and nilpotents of index 2 is investigated. The general view of these elements is derived \textit{ab init}, and their algebraic and geometric properties are revealed. The equivalence of action of the groups of phase transformations $(U_{1} )$ and rotations $(SO_{3} )$ on the nilpotents of index 2 is discovered: the phase transformations of the nilpotent, which are realized by its multiplications on the complex exponents, lead to spatial rotations of the nilpotent in $E_{3}$, and vice versa. It is proved that the nilpotents of index 2 are the unique elements of $Cl_{3}$, for which the equivalence of action of the groups $U_{1}$ and $SO_{3}$ takes place; thus, this property of nilpotents is a characteristic one. The results obtained are applied to analyzing geometry of vacuum solutions to the Maxwell equations without sources, which describe plane harmonic electromagnetic waves, the photons, with two types of helicity. On the basis of the analysis performed, the non-formal hypothesis is formulated that the real physical space is at least a six-dimensional one: in the minimal case its basis consists of six linearly independent elements, three vectors and three bivectors generated by these base vectors.

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