Dual Hyperquaternion Poincare Groups
Abstract
A new representation of the Poincare groups in n dimensions via dual hyperquaternions is developed, hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof). This formalism yields a uniquely defined product and simple expressions of the Poincare generators, with immediate physical meaning, revealing the algebraic structure independently of matrices or operators. An extended multivector calculus is introduced (allowing a possible sign change of the metric or of the exterior product). The Poincare groups are formulated as a dual extension of hyperquaternion pseudo-orthogonal groups. The canonical decomposition and the invariants are discussed. As concrete example, the 4D Poincare group is examined together with a numerical application. Finally, the hyperquaternion representation is compared to the quantum mechanical one. Potential applications include in particular, moving reference frames and computer graphics.
