É. Cartan’s Supersymmetry, Noncommutative Geometry and Propagation of Time in the Kaluza-Klein-Like Universe ^†

É. Cartan [1] has shown that in a space $E_{2\nu }$ or $E_{2\nu +1}$ in which the fundamental form of vector fields $F=x^1{x}^{1^{\prime }}+x^2{x}^{2^{\prime }}+\cdots +x^{\nu }{x}^{{\nu }^{\prime }}$ is defined, semi-spinors $\varphi$, $\psi$ and vectors or bivectors that satisfy specific symmetry can be introduced. When $\nu =4$, he showed that there exists a group G which leaves invariant the trilinear form $\mathcal{F}=\varphi {}^{T}CX\psi$, where $X=(x^1,x^2,x^3,x^4,x^{1^{\prime }},x^{2^{\prime }},x^{3^{\prime }},x^{4^{\prime }})$ and three quadratic forms $F={\displaystyle \sum _{i=1}^4}{x}^i{x}^i$, $\Phi =\varphi {}^{T}C\varphi$ and $\Psi =\psi {}^{T}C\psi$. The transformation group G of vectors and semi-spinors consists of five types $G_{23},G_{12},G_{13},G_{123}$ and $G_{132}$. When one adopts non-commutative geometry like Connes [2], one can pull back at each bundle point on the ${\mathbf{S}}^3$ model, two fibre points corresponding to $x^4\in U\left(1\right)$ and $x^{4^{\prime }}\in U{\left(1\right)}^{\prime }$. We allow $x^4$ and $x^{4^{\prime }}$ to run in different directions of time. The transformations $G_{23},G_{12},G_{13},G_{123}$ and $G_{132}$ contain the triality symmetry of octonions which can appear as the colour degrees of freedom of quark gluon systems. The transformation properties of vectors and semi-spinors which have the transformation property similar to $G_2$ symmetry could be the origin of different properties of baryonic systems, from those of leptonic systems which are defined by the quaternion symmetry of the Dirac equations. A specific superposition of time-reversal symmetry violating wave and original wave can enhance signals and can be used in nonlinear elastic wave technology of memristor [3,4].