Real Spinorial Groups

Exponential factorization of multivectors in Cl(p,q), p+q<3

“…Possibly, the factorizations that we present here may also find future applications in a number of sciences, including many branches of physics, signal and image processing, neural network computations, computer algebra, encryption, robotics, and computer vision. The present work can further be applied in the study of Lipschitz versors; see, for example, E.4.2 in Xambó‐Descamps, 12 and Lounesto, 3 and pinor and spinor groups. The factors we expose are expected to be related to the factors of polar decomposition of multivectors, 13 but we will study this in a later work.…”

“…12 e 12 h(e 12 )e −𝛼 1 z e −𝛼 0 = e −𝛼 0 e −𝛼 1 z ′ e −𝛼 12 e 12 h(e 12 ),z ′ = e −𝛼12 e 12 h(e 12 ) z e 𝛼 12 e 12 h(e 12 ), z ′ 2 = 0. (6.19) 6.1.2 Assuming |m 0 | = |m 12 | ≠ 0…”

“…12 e 12 h(e 12 )(m 1 e 1 + m 2 e 2 )e −𝛼 ′ 0 e −𝛼1 2 h(e 12 ) = (m 1 e 1 + m 2 e 2 )e −2𝛼 ′ 0 e −𝛼 12 e 12 e 𝛼 12 e 12 h(e 12 )(m 1 e 1 + m 2 e 2 )h(e 12 ) = (m 1 e 1 + m 2 e 2 )h(e 12 )(m 1 e 1 + m 2 e 2 )h(e 12 )Depending on the sign of the square of V in (6.39), the square of v will be ±1.For v 2 = −1, which occurs for |m 1 | < |m 2 | and h(e 12 ) = ±1 or for |m 1 | > |m 2 | and h(e 12 ) = ±e 12 , we can factorize m as m = e 𝛼 0 e 𝛼 1 v e 𝛼 12 e 12 h(e 12 ),…”

On factorization of multivectors in Cl(2,1)$$ Cl\left(2,1\right) $$, by exponentials and idempotents

“…5,35 Furthermore, a complete factorization study of Cl (1,3) and Cl (3,1) that are both of great importance in special relativity and relativistic physics 4,22,23,36 may be of considerable interest. The present work can, for example, be applied in the study of Lipschitz versors; see, for example, E.4.2 in Xambó-Descamps, 8 pinor and spinor groups, and in the development of Clifford Fourier and wavelet transformations, 9,36 compare also the motivation for this research in Section 1.…”

“…Here, we endeavor to extend this approach to the higher dimensional associative Clifford geometric algebra Cl (2,1), which plays an important role in geometry, physics, and computer science. [1][2][3][4][5][6][7][8] Namely, it is the physical algrebra of 2 + 1 space-time and the conformal geometric algebra Cl(1 + 1, 1) of one-dimensional Euclidean space R 1 . Our results may therefore be of special interest in the special theory of relativity and for the conformal geometry of a Euclidean line.…”