On Universal SO(4,4) Particle Theory

In elementary particle physics, today's models are built on the foundation of a high dimensional space time, for instance String Theory in a 10,1 space-time continuum. Standard GUT models, for instance, may be constructed over a SU(5) x SO(3,1) representation. However, Maxwell's theory is built on the famous SO(4,2) conformal group. In physics, one finds other De-Sitter representations, meaning noncompact real forms of D(3) and their subgroups, such as SO(4,1).
The well known fifteen gamma matrices used in Dirac theory span SO(4,2), both elementary representations are possible.

Here we will present a new model based on pseudo-octonions, very similar to Gürsey's quark model, constructed over compact octonions.

Pseudo-octonions keep the form (-,-,-,-,+,+,+,+) invariant, meaning all these De-Sitter (sub)groups, including the Lorentz group, are subgroups of such a SO(4,4) model. Once again, in contrast to all these models, this means position space is a four-dimensional projective space and is isomorphic to timespace (four-dimensional projective (compact), also). Therefore, rotations in Euclidean three-dimensional position space described by the usual SO(3) spin group (Spin(3)) and find their analogue in rotations in timespace. It can easily be shown from extended Maxwell theory that these rotations in timespace exist and they have been well known for over sixty years. Heisenberg's ISO-spin is time-spin. It should be noted that the one-dimensional timeline may of course easily be embedded in any higher dimensional timespace. In this fashion, an ISO-spin flip would find its geometrical analogue in an actual three-dimensional timespace (Euclidian version). This, of course, is the most important result of this paper.

Therefore, these pseudo-Octonions will be named space-time octonions. Two sub structures are identified. A SO(3,3) sector, where the eight 8-tuples (eigen vectors of the specific Cartan algebra) are identified as a spin up, spin down proton neutron pair, and they are antiparticles (including the correct sign of the magnetic moment). A SO(4,2) decomposition is also possible. In this case, some new particles may appear. For instance, "hidden" particles with some attributes of this so-called dark matter. These two sectors, SO(4,2) and SO(3,3), are clearly Lorentz covariant (SO(3,1,)). Following Gürsey's ansatz, the Lorentz group is then exchanged with the exceptional Lie group G(2) . Here, however, we of course use the unique non-compact version of G(2). Evidently, these particle systems are no longer Lorentz covariant, thus they do not exist in the real world. But here, we identify with the correct quantum numbers four generations of quark pairs: The well known up-down, strange-charm, top-bottom, and now adding, fire-ice.

Going back to standard Dirac SO(4,2) theory, this physical system acts on three dimensional projective position space and a one-dimensional projective time line.
Finally we remark that Dirac theory can easily be represented in a SO(3,3) version, too.


Please contact me for further details: anton.schober@alumni.tu-berlin.de