On a Microscopic Representation of Spacetime

Rolf Dahm, Beratung für IS, Germany

Using the Dirac algebra $\Gamma^{\mu}$ as an initial stage of our discussion, we find the isomorphic 15-dimensional Lie algebra su*(4) as complex embedding of sl(2,H). Based on previous work, we can identify spin-isospin hadron states as representations of the compact group SU(4), and it is very interesting to discuss various group chains based on SU*(4) or SU(4) resp. their maximal compact subgroup USp(4). Within this context, we discuss some exponential representations emerging in coset de-compositions, some of their properties and their relation to ’wellknown’ procedures of standard QFT. On the other hand, these technical procedures hide a lot of the geometrical (and physical) background. By reverting to very old (and at that time basic) knowledge of physics and mathematics, at first we are led to line geometry and some transfer principles which technically reflect in the group chains found above by choosing Lie’s (differential) approach. The geometrical description, however, leads to the beautiful framework of line complexes which comprises Dirac’s ’square root of $p^2$’, the discussion of ’equations of motion’ mostly in terms of ’moving points’, electromagnetism, etc. This old framework not only covers these actual approaches but it also restores simple access to physical properties in terms of polar and null polar systems after having introduced homogeneous coordinates and projective geometry.