The Surabale of Space-Time and Its Algebra Signature Split

Abstract

A “surable” is a category given by a special manifold of geometric algebra frames. It is a bale brought on by a surjective map the equivalence classes of which can constitute base elements of the associative algebra. It is also a stranded braid of idempotents based on a sheaf of base unipotents. The stranded braid of idempotents which are thought to represent particles or fields consists of fibers strictly separated by mutual annihilation throughout the bundle. The surabale can be defined on the Clifford algebra of space-time. Then it constitutes a bundle of frames which – though covering all dimensions of the geometric algebra – turn out as isomorphic to the ground space generating the algebra. Because of this, the mass shell and the Dirac-Hestenes equation can be defined on the whole surabale. As a result the equation is preserved when acted on by the symmetries of the transformation group of the standard model. The Kahler-equation simply turns into a Dirac-Hestenes equation on the inhomogenous surabale, yet with the same simple differential 1-forms of the linear equation. This shows very beautifully that the equation of motion as well as the invariance of the surabale under the standard model symmetry can be formulated base free. The Clifford bases – instead of Grasmann – just brings in the riches of representation, that is, the emergence of the standard model. But its grading, in a way, is an illusion. Studying the dimension of the space-time-split in quadratic Clifford algebras, it turns out that the dimension of the positive space-like component reproduces their period-8 properties. Considering as an example the Minkowski space-time in the Lorentz metric rather than in (+ − − − ) we can see that physicists found the electroweak symmetries in the negative part of the geometry, here denoted as $$ {\user1{\mathcal{N}}}^{ - } $$ , but did not realize the strong force symmetries in the positive part since those depend on the graded motion in the graded subspace $$ {\user1{\mathcal{P}}}^{ + } $$ . It is comparatively difficult to find the generators of the group capable to represent the classic SU(3) with its root space A2. Though the approach put forward gives satisfying answers to some classical problems of relativistic quantum mechanics, it does not solve the most important riddle which has been variously pointed out by Professor Oziewicz, namely, mechanics is not governed by the Lorentz- or Poincare-group. The simplest argument to be held against it, is that the Lorentz/Poincare group by definition is the symmetry group of the metric tensor in the empty space-time without bodies and radiation. How can such a Bewegungsgruppe of the empty space-time be related to the physics and mechanics of material bodies? [1], [2] May be this first argument is not convincing enough. But Oziewicz has listed a considerable number of arguments concerning the whole observation process against the present day unquestioned but incorrect application of the full Lorentz group. To clarify this will still need some more fundamental efforts which do not concern the main subject of this paper.