By Patrick R. Girard

ISBN-10: 3764377909

ISBN-13: 9783764377908

Using Clifford algebras in mathematical physics and engineering has grown speedily lately. while different advancements have privileged a geometrical strategy, this booklet makes use of an algebraic method that may be brought as a tensor made from quaternion algebras and offers a unified calculus for far of physics. It proposes a pedagogical advent to this new calculus, in line with quaternions, with purposes normally in targeted relativity, classical electromagnetism, and basic relativity.

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**Additional info for Quaternions, Clifford Algebras and Relativistic Physics**

**Example text**

Cliﬀord algebras are directly related to the quaternion algebra via the following theorem ([13],[14]). 1 (Cliﬀord, 1878). If n = 2m (m integer), the Cliﬀord algebra C2m is the tensor product of m quaternion algebras. If n = 2m − 1, the Cliﬀord algebra C2m−1 is the tensor product of m − 1 quaternion algebras and the algebra (1, ω) where ω is the product of the 2m generators (ω = e0 e1 · · · e2m−1 ) of the algebra C2m . The tensor product of the algebras A and B is deﬁned as follows [8, p. 57]. Consider two algebras A and B with x, y ∈ A and u, v ∈ B; the tensor product A ⊗ B is deﬁned by the relation (x ⊗ u) (y ⊗ v) = (xy) ⊗ (uv) .

Group of conformal transformations 53 the inverse transformation results from −1 − ac −1 [1 − x ac ] x−1 = (x ) = (x ) and thus −1 x = (1 − x ac ) x. The composition of two transformations gives −1 (x ) −1 (x ) = x−1 + ac , −1 = (x ) −1 =x −1 =x + bc + ac + b c + cc with c = a + b and thus belongs indeed to the group; if one permutes the two transformations, one obtains the same resulting transformation. 14) |dx| , |x| 2 |1 + xac | −1 dx = (1 + xac ) dx (1 + ac x) −1 . 12) results from x = (1 + xac )−1 x = x(1 + ac x)−1 , (1 + xac ) x = x = x (1 + ac x) which entails the relation.

Using real matrices, ⎡ ⎤ x0 = ct ⎢ ⎥ x1 ⎥, X=⎢ 2 ⎣ ⎦ x 3 x ⎡ ⎤ x 0 = ct ⎢ ⎥ x1 ⎥ X =⎢ 2 ⎣ ⎦ x 3 x one can write dX = X − X = dϕv i Ki X 44 with Chapter 3. 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 ⎥ ⎥.

### Quaternions, Clifford Algebras and Relativistic Physics by Patrick R. Girard

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