Abstract
It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3 + 1)-theory (e.g., number of dimensions, existence of maximal velocities, Heisenberg uncertainty, and particle generations). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie groupG2. This group generates specific rotations of (3 + 4)-vector parts of split octonions with three extra time-like coordinates and in infinitesimal limit imitates standard Poincare transformations. In this picture translations are represented by noncompact Lorentz-type rotations towards the extra time-like coordinates. It is shown how theG2algebra’s chirality yields an intrinsic left-right asymmetry of a certain 3-vector (spin), as well as a parity violating effect on light emitted by a moving quantum system. Elementary particles are connected with the special elements of the algebra which nullify octonionic intervals. Then the zero-norm conditions lead to free particle Lagrangians, which allow virtual trajectories also and exhibit the appearance of spatial horizons governing by mass parameters.
Highlights
Many properties of physical systems can be revealed from the analysis of proper mathematical structures used in descriptions of these systems
The (4 + 4)-norms of the real split octonions are invariant under the group SO(4, 4) of tensorial transformations of parameters, while the representation of rotations by octonions themselves corresponds to the real noncompact form of Cartan’s exceptional Lie group G2NC (the subgroup of SO(4, 4)), which under certain conditions imitate the Poincaretransformations in ordinary space-time
We examined the G2NC-effects on a quantum system moving at constant speed with respect to the observer, which emits light in various planes relative to the direction of motion, and new parity-violating effect on aberration and Doppler shift was derived
Summary
Many properties of physical systems can be revealed from the analysis of proper mathematical structures used in descriptions of these systems. The essential feature of all normed composition algebras is the existence of a real unit element and a different number of hypercomplex units. In physical applications one mainly uses division algebras with Euclidean norms, whose hypercomplex basis elements have negative squares, similar to the ordinary complex unit i. The introduction of vectorlike basis elements (with positive squares) leads to the split algebras with pseudo-Euclidean norms. The squares (inner products) of seven of the hypercomplex basis elements of split octonions give the unit element, 1, with the different signs: Jn2 = 1, jn2 = −1,. JnI = −IJn = jn, jnI = −Ijn = Jn. Conjugations of octonionic basis units, which can be understood as the reflection of vector-like elements, Jn† = −Jn,. A second condition is that for physical signals the vector part of split octonions (1) should be time-like: c2t2 + λnλn > xnxn. The classification of split octonions by the values of their norms is presented in Appendix A
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
