On a Microscopic Representation of Space-Time V

Based on previous work on the Dirac algebra and su * (4) Lie algebra generators, using Lie transfer we've associated spin to line and Complex reps. Here, we discuss the construction of a Lagrangean in terms of invariant theory using lines or linear Complex reps like F µν , its dual F αβ , or even quadratic terms like e.g. F µν F µν , or F a µν F a µν with respect to regular linear Complexe. In this context, we sketch briefly the more general framework of quadratic Complexe and show how special relativistic coordinate transformations can be obtained from (invariances with respect to) line transformations. This comprises the action of the Dirac algebra on 4 × 2 'spinors', real as well as complex. We discuss a classical picture to relate photons to linear line Complexe so that special relativity emerges naturally from a special line (or line Complex) invariance, and compare to Minkowski's fundamental paper on special relativity. Finally, we give a brief outlook on how to generalize this approach to general relativity using advanced projective and (line) Complex geometry related to P 5 and the Plücker-Klein quadric as well as transfer principles.

Advances in Applied Clifford Algebras

Using the Dirac (Clifford) algebra γ µ as initial stage of our discussion, we summarize previous work with respect to the isomorphic 15dimensional 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 R 3. 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 p 2 ', 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.

Physics of Atomic Nuclei, 2012

We start from a noncompact Lie algebra isomorphic to the Dirac algebra and relate this Lie algebra in a brief review to low energy hadron physics described by the compact group SU(4). This step permits an overall physical identification of the operator actions. Then we discuss the geometrical origin of this noncompact Lie algebra and 'reduce' the geometry in order to introduce in each of these steps coordinate definitions which can be related to an algebraic representation in terms of the spontaneous symmetry breakdown along the Lie algebra chain su*(4) -→ usp(4) -→ su(2)×u(1). Standard techniques of Lie algebra decomposition(s) as well as the (physical) operator identification give rise to interesting physical aspects and lead to a rank-1 Riemannian space which provides an analytic representation and leads to a 5-dimensional hyperbolic space H 5 with SO(5,1) isometries. The action of the (compact) symplectic group decomposes this (globally) hyperbolic space into H 2 ⊕H 3 with SO(2,1) and SO(3,1) isometries, respectively, which we relate to electromagnetic (dynamically broken SU(2) isospin) and Lorentz transformations. Last not least, we attribute this symmetry pattern to the algebraic representation of a projective geometry over the division algebra H and subsequent coordinate restrictions.

We recall some basic aspects of line and line Complex representations, of symplectic symmetry emerging in bilinear point transformations as well as of Lie transfer of lines to spheres. Here, we identify SU(2) spin in terms of (classical) projective geometry and obtain spinorial representations from lines, i.e. we find a natural non-local geometrical description associated to spin. We discuss the construction of a Lagrangean in terms of line/Complex invariants. We discuss the edges of the fundamental tetrahedron which allows to associate the most real form SU(4) with its various related real forms covering SO(n,m), n + m = 6.

It is well known that any two arbitrary observers S and S moving relative to each other with a speed v < c in isotropic space see a 4-dimensional real spacetime. We demonstrate that the two observers should naturally see the spacetime as a complexified 4-dimensional manifold described by the Kähler manifold commonly studied in string theory. Such a complex spacetime has, on large scales, been demonstrated to be a natural consequence of special relativity when quantum effects are included in relativistic mechanics and are thus of much significance in quantum gravity, quantum super string theory, particle physics and cosmology

arXiv: History and Philosophy of Physics, 2023

In order to ask for future concepts of relativity, one has to build upon the original concepts instead of the nowadays common formalism only, and as such recall and reconsider some of its roots in geometry. So in order to discuss 3-space and dynamics, we recall briefly Minkowski's approach in 1910 implementing the nowadays commonly used 4-vector calculus and related tensorial representations as well as Klein's 1910 paper on the geometry of the Lorentz group. To include microscopic representations, we discuss few aspects of Wigner's and Weinberg's 'boost' approach to describe 'any spin' with respect to its reductive Lie algebra and coset theory, and we relate the physical identification to objects in $P^{5}$ based on the case $(1,0)\oplus(0,1)$ of the electromagnetic field. So instead of following this -- in some aspects -- special and misleading 'old' representation theory, based on 4-vector calculus and tensors, we provide and use an alternat...

2019

The article compiles the existing information on complex space-time, from which a sketch of its mathematical structure can be outlined. We divided space-time into geometric and physical layers. The elements of the geometric layer are any four-vectors because, although it is an orthogonal space, it does not have a metrics. The physical layer is created by objects having energy, whose state coordinates are proper or singular paravectors, therefore it is posible to define a metrics for these objects. However, this metrics has no classic properties and it is not a pseudometric of the Minkowski space-time either, so we called it para-metrics. In the physical complex space-time the triangle inequality and the Cauchy inequality have been proved but, however, they have the opposite directions as their equivalents known from Euclidean geometry. At the end, we give a few tips on how to simplify the idea of a complex space-time.

Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension ict, with the unit imaginary producing the correct spacetime distance x 2 {c 2 t 2 , and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary i~ffi ffiffiffiffiffiffi ffi {1 p , with the Clifford bivector i~e 1 e 2 for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis e 1 and e 2 . We find that with this model of planar spacetime, using a twodimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.

Journal of Physics A: Mathematical and General, 1989

Annalen der Physik, 2007

An alternative to the representation of complex relativity by self-dual complex 2-forms on the spacetime manifold is presented by assuming that the bundle of real 2-forms is given an almostcomplex structure. From this, one can define a complex orthogonal structure on the bundle of 2forms, which results in a more direct representation of the complex orthogonal group in three complex dimensions. The geometrical foundations of general relativity are then presented in terms of the bundle of oriented complex orthogonal 3-frames on the bundle of 2-forms in a manner that essentially parallels their construction in terms of self-dual complex 2-forms. It is shown that one can still discuss the Debever-Penrose classification of the Riemannian curvature tensor in terms of the representation presented here. Contents 1 Introduction…………………………………………………………………………. 1 2 The complex orthogonal group…………………………………………………….. 3 3 The geometry of bivectors………………………………………………………….. 4 3.1 Real vector space structure on A 2 ……………………………………………….. 5 3.2 Complex vector space structure on A 2 …………………………………………... 8 3.3 Relationship to the formalism of self-dual bivectors……………………………. 12 4 Associated principal bundles for the vector bundle of 2-forms………………….. 16 5 Levi-Civita connection on the bundle SO(3, 1)(M)………………………………… 19 6 Levi-Civita connection on the bundle SO(3; C)(M)……………………………….. 24