By Patrick R. Girard
Using Clifford algebras in mathematical physics and engineering has grown quickly in recent times. while different advancements have privileged a geometrical technique, this publication makes use of an algebraic process that may be brought as a tensor manufactured from quaternion algebras and gives a unified calculus for far of physics. It proposes a pedagogical advent to this new calculus, in accordance with quaternions, with purposes ordinarily in targeted relativity, classical electromagnetism, and normal relativity.
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Extra resources for Quaternions, Clifford Algebras and Relativistic Physics
Example text
Complex quaternions ⎡ 0 ⎢1 K1 = ⎢ ⎣0 0 1 0 0 0 0 0 0 0 ⎤ 0 0⎥ ⎥, 0⎦ 0 ⎡ 0 ⎢0 K2 = ⎢ ⎣1 0 0 0 0 0 1 0 0 0 ⎤ 0 0⎥ ⎥, 0⎦ 0 ⎡ 0 ⎢0 K3 = ⎢ ⎣0 1 0 0 0 0 0 0 0 0 ⎤ 1 0⎥ ⎥. 2) the matrices defined for the infinitesimal transformations of SO(3) 0 0 0 0 ⎤ 0 0 0 0 ⎥ ⎥, 0 −1 ⎦ 1 0 ⎡ 0 0 ⎢0 0 M2 = ⎢ ⎣0 0 0 −1 0 0 0 0 ⎤ 0 1⎥ ⎥, 0⎦ 0 ⎡ 0 ⎢0 M3 = ⎢ ⎣0 0 ⎤ 0 0 0 0 −1 0 ⎥ ⎥. 2) are those of the unbound Kepler problem, which identifies the corresponding symmetry group as being SO(1, 3). 4 Four-vectors and multivectors in H(C) Let x = (x0 + i x), y = (y0 + i y) be two four-vectors and their conjugates xc and yc ; one can define the exterior product x∧y = 1 (xyc − yxc ) 2 ⎡ ⎢ ⎢ =⎢ ⎣ 0, (x2 y3 − x3 y2 ) + i (x1 y0 − x0 y1 ) , (x3 y1 − x1 y3 ) + i (x2 y0 − x0 y2 ) , (x1 y2 − x2 y1 ) + i (x3 y0 − x0 y3 ) ⎤ ⎥ ⎥ ⎥ ⎦ = [0, x × y + i (y0 x − x0 y)] with x ∧ y = −y ∧ x.
A third rotation of angle γ (proper rotation angle) around the vector k transforms the basis i , j , k into the basis e1 , e2 , e3 = k . Give the quaternion r of the rotation X = rX rc . Determine ω = 2rc dr , dt ω=2 dr rc . dt Give the components of the basis vectors ei . 1: Euler’s angles: α is the angle of precession, β the angle of nutation and γ the angle of proper rotation. Chapter 3 Complex quaternions From the very beginning of special relativity, complex quaternions have been used to formulate that theory [45].
In combining two symmetries, one obtains x = −axc a, x = −bxc b, hence x = (bac )x(ac b) = rxrc with r (= bac ) ∈ H, rc = abc = ac b, rrc = 1. The unitary quaternion r can be expressed in the form θ θ r = cos + u sin 2 2 where the unit vector u (u2 = −1) is the axis of rotation (going through the origin) and θ the angle of rotation of the vector x around u (θ is taken algebraically given the direction of u and using the right-handed screw rule). The conservation of the norm of x results from x xc = rxrc rxc rc = xxc .