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WIGRIS is a unification theory which is extended in scientific articles like and also a tool chest for high school students. A book will appear and a video demonstration for inner WIGRIS motions of a nucleon. For nucleon (standard model plus gravity) calculations with Moebius transformations, special relativity and difference equations (replacing differential equations) are a suitable theory exposed since energy exchanges are done by bosons with a fixed energy content. The building blocks are structures, geometrical-algebraical, technical, constructible for observations, motions. The students can then add further considerations on paper or as computer work. Additional MINT-WIGRIS examples are given in the MINT journal.
Contents
Example WIGRIS
Draw on paper or construct 3-dimensional with WIGRIS building blocks (from engineers tool boxes for instance) a hedgehog having six vectors, pointing in or out and attached to its surface. The vectors shall represent (first figure at right - on x,y,z coordinate axes) energy vectors with their initial point on the boundary B of an energy system P. Often B is geometrical similar to a 2-dimensional ball surface, a complex Riemannian sphere S². It is the boundary of a complex, inner 4-dimensional spacetime of P in motion.
In the figure at left is drawn a 3-dimensional spin coordinate system x,y,z for the location of P and its barycenter O to P. Below the same is drawn using spherical coordinates, radius r with center O and two angles phi and theta. The figure at right is a parametrization of B with six polar caps for the six color charges CC of quarks in a nucleon. The vectors CC point outwards in case energy is deleted from P, if energy is absorbed they point inwards. The polar caps represent geometrical whirls which have other mathematical descriptions than waves. If in the WIGRIS model three quarks with neutral (but exchangeable) r,g,b CC are in a nucleon P, the high school students can show to the observing class their construction in a video WIGRIS rotor S, using spherical coordinates, as a discrete, conical motion. In an explanation by a teacher, mathematical energy integrations (also differentiations) from complex numbers can be added. The gluon exchange in a nucleon between pairs of quarks uses the symmetry of a triangle in WIGRIS S. It acts mostly like a spring, but in WIGRIS S as an inner dynamic for integrating (use the complex residuation theory) energy vectors. The notion of pole (of order 1 for speed and potential, of order 2 for forces as differentiated derivations) or spacetime singularity is appropriate, boundary value problems also. Integrated are two electrical or gravity potentials, two linear or angular momentum, a harmonic oscillation and in addition, using the lever law, the setting of barycenters and a (Schwarzschild) radius of P. The new symmetry used is that of an equilateral triangle.
A similarly constructed model is shown and explained in the video WIGRIS wheel W,. A special relativistic coupling of the two S, W coordinate systems is also demonstrated as a video where S produces for the barycentrical GR gravity coordinates - a radial contraction/expansion in three sizes while the relativistic speed turns GR radii in W at the same time on equilateral spirals. This video is like three dogs on a triangles vertices - head to tail - pursuing or running away from one another on spirals. An older version from 2001 is found in www.uni-ulm.de/~gkalmbac/
Moebius transformation MT's as symmetry on S² are useful to demonstrate dipoles and fields.
Use them to map a complex plane into a second one. The poles a,b are a source and a sink which can also be identical. They are drawn in the figure right with a surrounding field, found for instance for electromagnetism. The Riemannian sphere S² is mapped through the geographical projection from a pole at infinity onto such a complex plane. For dipoles, 0 zero is a second pole. The two points can be mapped to arbitrary points a,b in the complex plane. Concerning WIGRIS W, a 3-dimensional field structure for the 3-dimensional Pauli spin of P was described by Hopf. The three Pauli matrices act as metrical, quadratic forms and map a 3-dimensional sphere onto S² such that the first S² coordinate is the complex dot product, the second the complex cross product, both suitably scaled, and the third coordinate can be used for a torus description of leptons. The inverse Hopf map combined with the 3-dimensional stereographic projection, and projected into a 3-dimensional space, represents an electromagnetic field. The dynamical, discrete, constructible WIGRIS Geometry as tool chest is new for young MINT engineers to be.
MINT Journal
In the journal MINT (Mathematik, Informatik, Naturwissenschaften, Technik) are printed in its first part many articles concerning a MINT instruction, in its second part scientific articles form different areas. For the local use also parts are found which are of historical interest:
In the MINT Journal are articles, written for and from highschool students 12th grade of the Gymnasium. They add as enrichment to school curricula. Scientific articles are published and themes of general or local interest. Reports from the Archives KHE are given.
Example Hedgehog Energies
Energy systems in the universe may get unstable. The big bang postulated in physics is an example. Other examples are colliding galaxies G, stars or comets, nuclear decay or the last year discovered Higgs Boson or an abstract system HB, as nucleon decay insufficiently treated in science, the internet and world press. For HB I postulate a 6fold biological energies bifurcation, and before a stochastical heat chaos occurs, the generation of eight gluons. More details on the qualities of the six generated energies are found in and as download on the webside www.rgnpublications.com.
Energy systems tend to decay through intermediate energy transfer into stable systems in case their old localization allows no equilibrium. In some stable energy systems Q, their inner energies is distributed in grids which I introduce for nuclear, vectorial (HB decays) spactime grids. Stars in newly generated G are macroscopic examples. Now astrophysics research studies the possibility of dark matter DM or dark energies DE in G. Adding up the observable G mass shows that G should decay, it is too low. On a cover page for downloads a method is described to solve this problem on nucleon base. Research in astrophysics goes in terms of nuclear QCD grids, color charge-gluon particles bound. Color charges of quarks in nucleons, however, are not treated sufficiently by QCD.
A warning is necessary, since for instance gravitons, postulated as particle carriers for gravity GR, are not experimentally verified today. I introduced for WIGRIS as new nuclear theory a color charge 3-fold whirl graviton grid for stable, color charge neutral nucleons and the group of Moebius transformations MT (acting on every a nucleon ball bounding surface) sphere S2.
MT’s are responsible for the Pauli spin group, already found about 100 years ago and experimentally well documented. The Pauli symmetry group SU(2) grids are for spacetime with a quaternionic non-commutative matrix presentation and with the non-commutative special relativistic SR metric, measuring other inertial systems with +v or -v.
The Sn are unit spheres in a higher (n + 1)-dimensional space, for n = 2 as ball surfaces. For quantum mechanics QM I suggest a complex 3-, finite dimensional Hilbert space as a matrix operator space C3 in which energy systems can be formed or decay. For HB localizations as projected energy grid, a 5-dimensional unit sphere S5 in C3 as energy carrier is suitable; its coordinates get scaled by an angle, reminding to the Einstein SR measuring angle towards other energy systems P.
In the HB case the S5 fibre bundle with fibre S1 becomes, geometrically projected, a complex 2-dimensional space, projectively closed by a sphere S2. The sphere S2 is a boundary of a newly generated system Q, arising from decaying HB’s. Its energy is sitting inside S2, can be a nucleon, a star or a galaxy. According to Einstein, it can also be observed by other stable P, having another grids measuring coordinate system.
Coordinates and measures are bound to local energy systems, not universal and not commutative as the Pauli spin. Using a scaling of relevant MT’s to coordinates 0,±1, I observe (see the literature below) that there are three groups of MT’s. Some MT count with their powers natural numbers or integers Z and are of infinite order like Z. Other ones have finite order 2, 3 or 6. Spin matrices are of order 2, reflections. The ones of order 3 can be used for instance as turning angles 0,±120 degree of an equilateral triangle. They are presented by the 3 cubic roots of unity which I attribute to the cubic GR behaviour. As symmetry group, like the Pauli spin group of order 4 or also QCD of order 8, they generate, together with the first Pauli spin matrix of order 2 (or a similar reflection of order 2), symmetry groups of order 6 or 12 (not used in physics today) for the 12fold fermionic series.
These symmetry groups are for stable nucleons as energy systems bound by an S2 with 3 quarks inside, arising from HB decays, having 6 color charges and gluon (QCD) bound integrating states CST. Single quarks don’t exist in the universe, but decay; also two quark mesons decay. CST is a spacetime grid, only possible for 3 or more quarks: For their paired color charges, hidding orthogonal as gluon frequencies, a circle as Lissajous figure is generated as boundary of a conic rotation where the quark not involved is kept fixed as the cones tip. The nucleons state changes in a 6 cycle with the triangles symmetry group, replacing in this case the Pauli spin group of quantum mechanics.
Example cross product and 7 dimensions
A cross product in seven dimensions is like the 3-dimensional one. This is in detail used in a book and article for a Quantum Mathematics 7-dimensional space of nucleons where the seven cross products realize six technical possible integrations of force-vectorial energies. The seventh cross product is for the Pauli spin in the 3-dimensional space with the Euclidean metric. Since Einstein's affine Minkowski metric is not Euclidean, this new 7 dimensional space is not Euclidean. It has for physics many until now unpublished Gleason operators for metrical and probabilistic energy distributions in spacetime. The tools are beside Gleason, norming a complex 5 dimensional sphere, belonging to the strong interaction, to a local complex projective 2 dimensional space E with a bounding 2 sphere and its actions, the Moebius transformations (which allow polar singularities of order 1 and 2 as derivatives in spacetime or its tangent bundle). E has for local energy systems with bounding 2 sphere and Higgs bosonic mass local 7 dimensional coordinates as vectorial tuple.
A survey of octonian generators, extending the Pauli spin generators of spacetime to 8 dimensions, shows at right the identity e0 for projecting vecors and vector fields into spacetime. The Feigenbaum bifuration adds to the vector splitting of energy attributes: first into the electromagnetic 1 and mass (Higgs) 5 potentials. 1 splits into heat 2 and magnetism 4; 5 splits for cosmic speeds into rotational 3 and kinetic 6 energy. The last four energies split off the strong interaction from gravity with 8 gluons generated. Then heat chaos occurs. Wavelength 1 and frequency 6 generate light 7. The projective Pascal geometry of SI is for the complex 3-dimensional space C3. Its Pascal line contains three 4-dimensional subspaces: 1234 of spin (weak interaction and electromagnetism) – the spacetime of physics, the light cone 1456 and the nuclear rotor 2356 of WIGRIS (figure 6 roll mill).
WIGRIS is illustrated now by additional examples: Australian geologists AG publish that the earth is rotational not as stable as assumed earlier. They measured that the hot kernel K, and also fluid parts between K and the surface ES we live on, have different rotational speeds than our day and night; in addition these speeds change in time. Einstein introduced for such, only partly related, energies SR, quoted for this research from AG. The finding is that parts like K have a SR speed v towards ES. In different coordinate systems EISK,ES belonging for instance to K and ES, the SR measure scales all energies and spacetime coordinates. Observe, that EISK,ES can reach only a turning angle of a sinus value 1. Then decays of matter occur, - as known for light energies, the electromagnetic interaction, with this maximal observable speed in the universe when atomic electrons release or absorbe it.
For the two nuclear forces strong SI and weak WI interaction I postulated synchronized two CST with an integrating 4 state spin grid for WI. Synchronized are for instance parallel spins and magnetic momenta of quarks which allow a generated protons electrical + charge to attach an electron in a hydrogen shell with its - charge rotate (notice: also in neutrons the quarks electrical charges are added to a neutral charge). As mentioned earlier, in the two SI, WI coordinate systems in SR motion against one another, a 3-fold radial graviton contraction and expansion is barycentrical for the 3 quark masses barycenters as a geometrical triangle in SI, while the triangle in WI coordinates is spiralic contracting and expanding with 60 degree angles between radii.
Another example is speed of light: Einstein SR measures need an observer as measuring apparatus Q and another measured system P. From physics without such P,Q I quote the not observable waves of QM, living in an infinite dimensional, complex Hilbert space. If energies have not such a couple Q, P, they cannot be observed. The complex couple for a wave is in QM its complex conjugate and they can be observed, but only as a real probability distribution in some spacetime. If complex speeds higher than the speed of light exist, they are not observable.
Barycentrical GR coordinates generate many 3-dimensional vectorial, energies carrying subspaces U in C3 for a suitable probabilistic space location; also Gleason operators for measures are generated for them. I remind to the physical claim that mass, now in form of Higgs bosons, interact only with charges and produce this way the fermion series and the three WI bosons. The quarks color charges are responsible for the quark series, the electrical charges and the neutral electrical charge for leptons and the WI bosons. Gluons, gravitons, photons, phonons don`t get a mass attached.
I suggest that no direct proofs for non-observable energies are possible. SR is only one restriction for them, spacetime grids have as lower bounds the Planck constant h. If DM, DE are considered, one should take into account the matrix operator bound geometry in C3 and the above-mentioned grids. Neither local R4 spacetime coordinates for one of many possible C3 subspaces are sufficient to explain them, nor an infinite dimensional Hilbert space of QM. The old theories need a revision on the grounds of modern findings.
As new research I suggest my new symmetry groups like the MT with boundary value problems through S2, which have not been known to Pauli, Einstein 100 years ago. I present this together with boundaries S2 (snail shells for inner energies, slowing down their speeds below speed of light) as a new possibility for physics and I developed on this base a dynamical WIGRIS energy distribution for nucleons. The search for 5 more Higgs bosons according to my triangle symmetry group of order 6 or the cyclic symmetry group of order 6 may confirm the 12-fold mass series of fermions with the added conjugation operator C of physics as group of order 2. I request the search for whirl motions of gravity, as proposed by WIGRIS for nucleons. In the book the reader can learn that Einstein's general relativistic gravity is also graviton whirls bound. 9 figures from this book are added:
The first five files are for macroscopic running machines for flows, demonstrating that fields can be generated by flows. The next figure shows that periodic functions generate curved geometries for energies locations. The projective view, extending Einstein's affine Minkowski space is adopted in the following figure. The figure pulsation shows that also for nucleons Bohr radii for their equilibrium states occur, not only for electrons in atomic shells.