By Patrick R. Girard
The use of Clifford algebras in mathematical physics and engineering has grown speedily in recent times. while different advancements have privileged a geometrical technique, this e-book makes use of an algebraic technique that may be brought as a tensor fabricated from quaternion algebras and gives a unified calculus for far of physics. It proposes a pedagogical advent to this new calculus, in keeping with quaternions, with purposes mostly in precise relativity, classical electromagnetism, and normal relativity.
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Extra info for Quaternions, Clifford Algebras and Relativistic Physics
1 0 0 −1 ⎥ ⎥, 0 0 ⎦ 0 0 32 Chapter 2. 5 Symmetries and invariants: Kepler’s problem Let us consider the motion of a particle of mass m, of momentum p and gravitating at the distance r of a mass M . The Hamiltonian is given by H= p2 k − , 2m r k = GM m and the equations of motion are q˙i = ∂H , ∂pi p˙i = −∂H . ∂qi Let F (qi , pi , t) be a physical quantity of the motion; one has ∂F ∂F ∂qi ∂F ∂pi dF = + + dt ∂t ∂qi ∂t ∂pi ∂t ∂F ∂Hi ∂F ∂H ∂F + . − = ∂t ∂qi ∂pi ∂pi ∂qi Introducing Poisson’s bracket of two functions u and v [u, v] = one obtains ∂u ∂v ∂u ∂v − ∂qi ∂pi ∂pi ∂qi dF ∂F = + [F, H] .
Hence, one sees that H(C) already allows us to develop a few notions of a multivector calculus. These notions will be developed later on in the more satisfying framework of the Cliﬀord algebra. 5. 1). 1: Special Lorentz transformation (pure): the axes remain parallel to themselves and the reference frame moves along the Ox axis. 8) cosh ϕ2 + i i sinh ϕ2 , bbc = 1, with X = (x0 + i x) , X = (x 0 + i x ), b = tanh ϕ = uc ; write γ = cosh ϕ, hence γ 2 = 1 + sinh2 ϕ = 1 + 1 γ= 1− u2 c2 u2 2 γ , c2 .
4 (N = 48). Example. 3. Crystallographic groups 27 for the middle points of the 12 sides j±k ± √ , 2 ± i±j ± √ . 2 (±i + k) √ , 2 The octaeder transforms into itself under the rotation x = rxrc , r being taken in the double octahedral group. The same property applies to the cube (dual of the octaeder) the 8 vertices of which are the centers of the faces of the above octaeder. 5 Double icosahedral group (N = 120) The 120 elements of this group are iα , jα, kα , i + m j + mk 2 i − m j + mk 2 i + m j − mk 2 i − m j − mk 2 α α , α , α , α , 1+i+j+k 2 β 1+i−j−k 2 β mi + j + m k , 2 α −mi + j − m k , 2 α mi + j − m k , 2 α −mi + j + m k , 2 1−i−j+k 2 β , 1−i+j−k 2 β , 1 + mj + m k 2 β 1 + mj − m k 2 m+m j+k 2 m+m j−k 2 β , , γ , , , β , 1 + mi + m j 2 β , γ 1 + m i + mk 2 α m i + mj + k , 2 α m i − mj + k , 2 α −m i − mj + k , 2 α −m i + mj + k , 2 1 − m i + mk 1 + mi − m j , 2 2 γ γ m+i+mk m+mi+j , , 2 2 γ γ m−i+mk m+mi−j , , 2 2 with α = 1, .
Quaternions, Clifford Algebras and Relativistic Physics by Patrick R. Girard