From the abstract quaternion group, we have defined the quaternion algebra ℍ, then the complex quaternion algebra ℍ(ℂ) and the Clifford algebra ℍ ⊗ ℍ. The quaternion algebra gives a representation of the rotation group SO(3) well-known for its simplicity and its immediate physical significance. The Clifford algebra ℍ⊗ℍ yields a double representation of the Lorentz group containing the SO(3) group as a particular case and having also an immediate physical meaning. Furthermore, the algebra ℍ⊗ℍ constitutes the framework of a relativistic multivector calculus, equipped with an associative exterior product and interior products generalizing the classical vector and scalar products. This calculus remains relatively close to the classical vector calculus which it contains as a particular case. The Clifford algebra ℍ⊗ℍ allows us to easily formulate special relativity, classical electromagnetism and general relativity. In complexifying ℍ⊗ℍ, one obtains the Dirac algebra, Dirac’s equation, relativistic quantum mechanics and a simple formulation of the unitary group SU(4) and the symplectic unitary group USp(2,ℍ). Algebraic or numerical computations within the Clifford algebra ℍ ⊗ ℍ have become straight-forward with software such as Mathematica. We hope to have shown that the Clifford algebras ℍ⊗ℍ over ℝ and ℂ constitute a coherent, unified, framework of mathematical tools for special relativity, classical electromagnetism, general relativity and relativistic quantum mechanics. The quaternion group consequently appears via the Clifford algebra as a fundamental structure of physics revealing its deep harmony.
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