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Using the Dirac (Clifford) algebra γμ as initial stage of our discussion, we summarize previous work with respect to the isomorphic 15 dimensional Lie algebra su*(4) as complex embedding of sl(2,H), the relation to the compact group SU(4) as well as subgroups and group chains. The main subject, however, is to relate these technical procedures to the geometrical (and physical) background which we see in projective and especially in line geometry of R3. This line geometrical description, however, leads to applications and identifications of line Complexe and the discussion of technicalities versus identifications of classical line geometrical concepts, Dirac’s ‘square root of p2’, the discussion of dynamics and the association of physical concepts like electromagnetism and relativity. We outline a generalizable framework and concept, and we close with a short summary and outlook.

The spinors of fermions are derived as polarized isotropic multivectors within the Clifford algebra of the Minkowski spacetime. It is assumed that each spinor Clifford multiplied with its gradeinverse constitutes that primitive idempotent in the minimal left ideal of which it is located, that is, we have. Each Cartan spinor in Minkowski algebra is decomposable into a product of extension and torsion. The list of fermions is complete. The spinors presented obey the symmetries of the forces of nature. They do not rely on the addition of auxiliary bundles.

In previous parts of this publication series, starting from the Dirac algebra and SU*(4), the 'dual' compact rank-3 group SU(4) and Lie theory, we have developed some arguments and the reasoning to use (real) projective and (line) Complex geometry directly. Here, we want to extend this approach further in terms of line and Complex geometry and give some analytical examples. As such, we start from quadratic Complexe which we've identified in parts III and IV already as yielding naturally the 'light cone' when being related to (homogeneous) point coordinates and infinitesimal dynamics by tetrahedral Complexe (or line elements). This introduces naturally projective transformations by preserving anharmonic ratios. We summarize some old work of Plücker relating quadratic Complexe to optics and discuss briefly their relation to spherical (and Schrödinger-type) equations as well as an obvious interpretation based on homogeneous coordinates and relations to conics and second order surfaces. Discussing (linear) symplectic symmetry and line coordinates, the main purpose and thread within this paper, however, is the identification and discussion of special relativity as direct invariance properties of line/Complex coordinates as well as their relation to 'quantum field theory' by complexification of point coordinates or Complexe. This can be established by the Lie mapping¹ which relates lines/Complexe to sphere geometry so that SU(2), SU(2)×U(1), SU(2)×SU(2) and the Dirac spinor description emerge without additional assumptions. We give a short outlook in that quadratic Complexe are related to dynamics e.g. power expressions in terms of six-vector products of Complexe, and action principles may be applied. (Quadratic) products like are natural quadratic Complex expressions which may be extended by line constraints λk = 0 with respect to an 'action principle' so that we identify 'quantum field theory' with projective or line/Complex geometry having applied the Lie mapping.

In order to extend our approach based on SU$*$(4), we were led to (real) projective and (line) Complex geometry. So here we start from quadratic Complexe which yield naturally the 'light cone' $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{0}^{2}=0$ when being related to (homogeneous) point coordinates $x_{\alpha}^{2}$ and infinitesimal dynamics by tetrahedral Complexe (or line elements). This introduces naturally projective transformations by preserving anharmonic ratios. Referring to old work of Pl{\"u}cker relating quadratic Complexe to optics, we discuss (linear) symplectic symmetry and line coordinates, the main purpose and thread within this paper, however, is the identification and discussion of special relativity as direct invariance properties of line/Complex coordinates as well as their relation to 'quantum field theory' by complexification of point coordinates or Complexe. This can be established by the Lie mapping which relates lines/Complexe to sphere geometry so that SU(2), SU(2)$\times$U(1), SU(2)$\times$SU(2) and the Dirac spinor description emerge without additional assumptions. We give a short outlook in that quadratic Complexe are related to dynamics e.g.~power expressions in terms of six-vector products of Complexe, and action principles may be applied. (Quadratic) products like $F^{\mu\nu}F_{\mu\nu}$ or $F^{a\,\mu\nu}F^{a}_{\mu\nu}$, $1\leq a\leq 3$ are natural quadratic Complex expressions ('invariants') which may be extended by line constraints $\lambda k\cdot\epsilon=0$ with respect to an 'action principle' so that we identify 'quantum field theory' with projective or line/Complex geometry having applied the Lie mapping.

Recursive distinctioning (RD) is a name coined by Joel Isaacson in his original patent document describing how fundamental patterns of process arise from the systematic application of operations of distinction and description upon themselves. Recursive distinctioning means just what it says. A pattern of distinctions is given in a space based on a graphical structure (such as a line of print or a planar lattice or given graph). Each node of the graph is occupied by a letter from some arbitrary alphabet. A specialized alphabet is given that can indicate distinctions about neighbors of a given node. The neighbors of a node are all nodes that are connected to the given node by edges in the graph. The letters in the specialized alphabet (call it SA) are used to describe the states of the letters in the given graph and at each stage in the recursion, letters in SA are written at all nodes in the graph, describing its previous state. The recursive structure that results from the iteration of descriptions is called recursive distinctioning.

Article was abstracted by the editors of the journal.
No need to remind me from above!
Thanks nevertheless!

Since Immanuel Kant’s Inaugural Dissertation of 1770 we assume that the concepts of space and time are not abstracted from sensations of external things. But outer experience is considered possible at all only through an inner representation of space and time within the cognitive system. In this work we describe a representation which is both inner and outer. We add to the Kantian imagination that “forms of nature, matter, space and time are intelligible, perceivable and comprehensible”, the idea that these four are indeed intelligent, perceiving, grasping and clear. They are active systems with their own intelligence. In this paper on the mind-matter interface we create the mathematical prerequisites for an appropriate system representation. We show that there is an oriented logic core within the space–time algebra. This logic core is a commutative subspace from which not only binary logic, but syntax with arbitrary real and complex truth classifiers can be derived. Space–time algebra too is obtained from this inner grammar by two rearrangements of four basic forms of connectives. When we conceive the existence of a few features like polarity between two appearances, identification and rearrangement of the latter as basic and primordial to human cognition and construction, the intelligence of space–time is prior to cognition, as it contains within its representation the basic self-reference necessary for the intelligible de-convolution of space–time. Thus the process of nature extends into the inner space.

This is a book that the author wishes had been available to him when he was student. It reflects his interest in knowing (like expert mathematicians) the most relevant mathematics for theoretical physics, but in the style of physicists. This means that one is not facing the study of a collection of definitions, remarks, theorems, corollaries, lemmas, etc. but a narrative-almost like a story being told-that does not impede sophistication and deep results. It covers differential geometry far beyond what general relativists perceive they need to know. And it introduces readers to other areas of mathematics that are of interest to physicists and mathematicians, but are largely overlooked. Among these is Clifford Algebra and its uses in conjunction with differential forms and moving frames. It opens new research vistas that expand the subject matter. In an appendix on the classical theory of curves and surfaces, the author slashes not only the main proofs of the traditional approach, which uses vector calculus, but even existing treatments that also use differential forms for the same purpose. © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.