On the Quadratic Phase Quaternion Domain Fourier Transform and on the Clifford algebra of $R^{3,1}$

Abstract

We study application of the Clifford algebra and the Grassmann algebra to image recognitions in $(3+1)D$ using quaternions. Following S.L.Adler, we construct a quaternion-valued wave function model with fermions and bosons of equal degrees of freedom, similar to Cartan's supersymmetric model. The Clifford algebra ${\mathcal A}_{3,1}$ is compared with ${\mathcal A}_{2,1}$ and the model applied to the $(2+1)D$ non-destructive testing is extended. The fixed point lattice actions are calculated for 7 paths in $(3+1)D$ space with lengths less than or equal to 8 lattice units. Comparison with the quaternion time approach of Ariel, quaternion Fourier transform of Hitzer and the tensor renormalization group approach to classical lattice models are also discussed.

Figures (12)

• Figure 1: L19. The upper right corner ball is $e_2e_4$, the lower right corner one is $-e_2e_4$
• Figure 2: L20. The upper right corner ball is $e_2e_4$, the lower left corner one is $-e_2e_4$
• Figure 3: L21. The upper right corner ball is $e_1e_4/e_2e_4$, the lower left corner one is $-e_3e_4$.
• Figure 4: L22. The upper right corner ball is $e_1e_4/e_2e_4$, the lower left corner one is $-e_3e_4$.
• Figure 5: L23. The upper right corner ball is $e_1e_4$, the upper left corner one is $-e_2e_4$.