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

Abstract

In order to combine internal symmetries and spacetime that has Poincaré symmetry, it is necesary to introduce supersymmetry, Supersymmetry of Connes is based on involution, and that of Cartan is based on triality. Cartan’s supersymmetry allows violation of Lorentz symmetry and time reversal violation can occur.

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].