^{1, 2}

^{1}

^{1}

^{2}

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 group

Many properties of physical systems can be revealed from the analysis of proper mathematical structures used in descriptions of these systems. The geometry of space-time, one of the main physical characteristics of nature, can be understood as a reflection of symmetries of physical signals we receive and of the algebra used in the measurement process. Since all observable quantities we extract from single measurements are real, in geometrical applications it is possible to restrict ourselves to the field of real numbers. To have a transition from a manifold of the results of measurements to geometry, one must be able to introduce a distance between some objects and an etalon (unit element) for their comparison. In algebraic language all these physical requirements mean that to describe geometry we need a composition algebra with the unit element over the field of real numbers.

Besides usual real numbers, according to the Hurwitz theorem, there are three unique normed division algebras—complex numbers, quaternions, and octonions [

The essential feature of all normed composition algebras is the existence of a real unit element and a different number of hypercomplex units. The square of the unit element is always positive, while the squares of the hypercomplex basis units can be negative as well. In physical applications one mainly uses division algebras with Euclidean norms, whose hypercomplex basis elements have negative squares, similar to the ordinary complex unit

In this paper we propose parameterizing world-lines (paths) of physical objects by the elements of real split octonions [

In geometric application four of the eight real parameters in (

The eight basis units in (

Conjugations of octonionic basis units, which can be understood as the reflection of vector-like elements,

Using (

As for the case of ordinary Minkowski space-time, we assume that for physical events the corresponding “intervals” given by (

To find the geometry associated with the signals (

To represent the active rotations in the space of

It is known that associative transformations of split octonions can be done by the specific simultaneously rotations in two (and not in three as for

Infinitesimal transformations of coordinates which accompany the group of active transformations of octonionic basis units,

The Lorentz-type transformations (

We notice that if we consider rotations by the angles

For completeness note that there exists a second well-known representation of

In the algebra of split octonions there exist only three space-like parameters,

Euclidean rotations around one of the space-like axes,

Analysis of boosts of

When

From (

For the reference frame (

Consider photons moving in the

Now consider the last class of automorphisms (Appendix

It is known that in split algebras there can be constructed special elements with zero norms, which are called zero divisors [

The zero-norm condition,

For photons,

For massive particles the time coordinate,

Applying the condition (

To conclude, in this paper, it was analyzed consequences of describing physical signals in terms of split octonions over the field of real numbers. Eight real parameters of split octonions were related to space-time coordinates, the phase function (classical action), and wavelengths characterizing physical signals. The

The algebra of split octonions is a noncommutative, nonassociative, nondivision ring. Any element (

Split octonions with nonzero norms,

One can construct also the polar form of a split octonion with nonzero norm:

Depending on the values of (

(i) Every split octonion with negative norm,

(ii) Every split octonion with positive norm,

(iii) Every split octonion with positive norm,

The norm of a split octonion (

The 8-dimensional octonionic space (

(i) The three pseudovector-like basis elements,

(ii) For the three vector-like basis elements,

(iii) The last vector-like basis element,

So the one-side products (

The decomposition of a split octonion in the form (

From the expressions of a hypercomplex unit of split octonions by two other basis elements, which form associative triplets with the selected unit, one can find orthogonal to it planes of rotations. An automorphism (

Similar to (

One can define also hyperbolic rotations (automorphisms) around the three vector-like units

Finally for the rotations around the time direction,

The group

By transition to our coordinates,

In the algebra of split octonions two types of zero divisors, idempotent elements (projection operators) and nilpotent elements (Grassmann numbers), can be constructed [

There exist four noncommuting (totally eight) primitive idempotents:

We have also the four noncommuting classes (totally twelve) of primitive nilpotents:

For the completeness we note that the idempotents and nilpotents obey the following algebra:

Using commuting zero divisors any octonion (

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the Shota Rustaveli National Science Foundation Grant ST09-798-4-100. Otari Sakhelashvili acknowledges also the scholarship of World Federation of Scientists.

_{8}

_{2}and the rolling ball

_{2}. I. Complex and real forms of g

_{2}and their maximal subalgebras