If one examines the mathematical tools used in physics, one finds essentially three calculi: the classical vector calculus, the tensor calculus and the spinor calculus. The three-dimensional vector calculus is used in nonrelativistic physics and also in classical electromagnetism which is a relativistic theory. This calculus, however, cannot describe the unity of the electromagnetic field and its relativistic features. As an example, a phenomenon as simple as the creation of a magnetic induction by a wire with a current is in fact a purely relativistic effect. A satisfactory treatment of classical electromagnetism, special relativity and general relativity is given by the tensor calculus. Yet, the tensor calculus does not allow a double representation of the Lorentz group and thus seems incompatible with relativistic quantum mechanics. A third calculus is then introduced, the spinor calculus, to formulate relativistic quantum mechanics. The set of mathematical tools used in physics thus appears as a succession of more or less coherent formalisms. Is it possible to introduce more coherence and unity in this set? The answer seems to reside in the use of Clifford algebra. One of the major benefits of Clifford algebras is that they yield a simple representation of the main covariance groups of physics: the rotation group SO(3), the Lorentz group, the unitary and symplectic unitary groups. Concerning SO(3), this is well known, since the quaternion algebra ℍ which is a Clifford algebra (with two generators) allows an excellent representation of that group. The Clifford algebra ℍ ⊗ ℍ, the elements of which are simply quaternions having quaternions as coefficients, allows us to do the same for the Lorentz group.
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