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From MINT: as I was asked for a revised version, read this, copied form my word file:

A unification, concerning particles, gravity and mass

Gudrun Kalmbach H.E.

In CERN experimental physics and their computer simulations you find discussions like: Higgs and the weak W- and Z-bosons get their mass from swallowed ordinary masses by the known mass-radius inversion at a Schwarzschild radius. This may be actually not wrong. What is missing is gravity. Why Higgs sets particles mass is not geometrically realized by CERN. Such a geometry and its symmetry is described now, extending the known one for the three basic forces in physics from the standard model. It uses spacetime singularities, vectorial poles, an Einstein mass-frequency energy plane, dynamical integration and differentiation and whirls - as third energy character, -in addition to the wave, particle characters of physics.

Higgs fields HF without gravity GR are only one part concerning mass. Fields consist in the quantum geometry of finite, discrete vectors as eigenvectors of the relevant matrices. They can use linear or curved arrangements for the field presentation. If HF sets discrete mass points as weight, they are mostly on end points of vectors, for instance a GR potential vector, a Newtonian force vector or an angular momentum vector. Since CERN excludes with Higgs fields gravitational fields, the reference for a projective 5-dimensional GR field is [1].

For Gleason and Einstein measures (metrics), I remind to the Copenhagen interpretation: If one system P is observed by a second system Q, not only (linear or non-linear) scalings of coordinate units or a scalar as output value of this measurement occur, but also P can get into another state (see 1.-6. below). These are boundary or measurement problems, concerning also particles mass for systems in interaction. Einstein’s special relativity [2] sets the Minkowski metric SRm for spacetime. SRm rescales in measurements mass.

Beside Einstein‘s two metrics, Gleason measures T of weight W, not used in physics today, are responsible for mass as weight and allow for another mass-setting (not Higgs) of combined systems like nucleons, atoms, stars or galaxies. In the case of three quarks in a nucleon, the different mass a, b  of two quarks set a maximum M and minimum value m for the frame function f of  T and the nucleon weight and inertial mass W is computed through a third value (W - M – m) where also membran oscillations of the nucleon/quarks triangle count für W. The f  weights are  related to 2a + b minus the binding energy of quarks. The mass defect of neutrons and electrons in atoms are also using T’s.

For coordinates, recall that the three not commutative Pauli matrices as geometric reflections set Euclidean metric for space (x, y, z) coordinates, The orthogonal coordinate triple for f  has vectorial located its weights on a Riemannian ball surface S2. Energy systems P, have a 2-dimensional spherical boundary S2 on which MT’s interact with the environment of P for energy exchanges.

The use of Moebius transformations MTs as symmetries of S2 for Einstein’s two metrics can be found in the references below (invariant are areas - in differences form (length x time-interval) in spacetime or in differential form (dr x dt) in its tangential space). The new symmetry group suggested to unify GR with the standard model of physics are the MTs. They have one or two poles, also applied to mass. Their geometrical invariants are cross ratios, applying here to invariant areas. Relevant matrices with components 0, +1 or –1 are of order 1,2,3 and infinity. The order 3 matrices are complex cubic root of unity uj, considered as rotations, and combine with the first Pauli matrix (for instance) to non-commutative symmetry groups of order 6. This is a new symmetry group for the generation of particle series and can be extended as one factor a group of order 12 by a second factor, containing the identiy and a reflection.

For the six color charges of quarks CERN has no suggestions. Here are geometrical, dynamical axioms and the orientations for the screwed or conic 3-dimensional case: The color charge pairing is noted as x or xbar, x from the set {r, g, b} (r = red, not radius, g green, b blue). 3-dimensional cross products set right hand screws L = r × p (L angular momentum, r radius), Fl = Q(v × B) (Fl Lorentz force, Q EM-charge, v speed, B magnetic field) for the well known GR and electromagnetic EM triples. Here the multiplication of the non-real (u1u2) = 1 = u3 is used. Pauli spin S = (Sx, Sy, Sz) has left hand screws for its space coordinates, using for instance the quaternionic multiplication of Pauli matrices. Color charges are conic whirls with a cyclic (in time) cw clockwise or mpo counterclockwise orientation on its boundary circle, having a crosscut quark triangle. (Spherical and Einstein energy coordinates are used instead of the four Pauli spacetime coordinates [6] .)

1. red r (mpo, EM(pot) potential of EM) integration is about a complex pole rotation (about a circle) for bag radii r of particles, nucleons and larger systems such as earth, galaxies. - This gives 2-dimensional boundaries S2. The nucleon quark triangle is of large size, also for the following.

2. g (cw, stochastic E(heat) energy) has a complex polar angle coordinate; radial integrated is the second cosmic speed (with weight, also the Schwarzschild radius factor for the general relativity Einstein metric is set). The Gleason weight operators allow the stochastic energy transfer in form of mass defects to speed/frequency in order to generate stable nucleons and atoms. The Boltzmann constant is set. Gravitons with no weight attached are rotating whirls as superposition of r,g,b vectorial whirls.

3. gbar (mpo; E(rot) rotational, whirl energy) has a space angle and time integrates angular momentum. The whirl character of energies is generated.

4. bbar has a complex, scaled ict time coordinate (cw, E(magn) magnetic energy, c speed of light) is for the wave character of energies. Time or/and space integrated are waves or (harmonic) oscillations. The nucleon triangle is of middle size, also for kinetic energy. Magnetic momentum of particles are on the time coordinate and linearly coupled with Pauli spin vectors S.

5. b, iu frequency f coordinate (mpo, E(kin) kinetic energy, for frequency and Newton’s momentum p). Time integrated is the Newtonian force vector to p; - dark energy is projected into spacetime (1.-4. above) in form of finite energy strings as dynamical conic or cylindrical helix lines or vectors, moving with p. The Planck h constant is set in E = hf (E energy).

6. rbar, iw mass coordinate (cw, E(pot) GR potential and mass) - the GR contraction of the nucleon triangle is small, also for 3.; vectorial it is a black hole SL spacetime or a singular pole (dark matter), using the first Pauli matrix; radial integrated is the first cosmic speed; setting barycenters for masses, using the lever law of physics. The gravitational constant is set.

This represents a 5-dimensional spin, two flat orientations cw, mpo and three rotating vectors, replacing the 3-dimensional Pauli Sx,Sy,Sz-vectors. The physics differentation is here replaced by a difference equation with the characteristic equation of the sixth‘ roots of unity which generates these six states in nucleons. The 6-cycle is for the strong interaction SI (see [3] for this), while a 4-cycle for EM and the weak interaction WI uses a difference equation with the characteristic polynomial of the fourth‘ roots of unity (the Pauli EM/WI space is 1.-4; alternatively, set in this space the Maxwell equations for EM.). The barycentrical GR stretching-squeezing of Einstein‘s relativity is noted as size of the nucleon triangle; its radial version is in SI coordinates; a spiralic, dynamic version is found in the internet, observed in WI coordinates in relative motion towards the SI coordinates.

The quantum space [5] for SI is the complex operator space [7] C3 which is projected projectively by the SI gluon Gell-Mann matrices into 4-dimensional spaces with boundaries S2 as follows (z1 is 3.,4.; z2 is 1.,2.; z3 is 5.,6.):

the projection of energies from the (z1, a, z3), a = 0, 1, projective space into the vacuum is into spacetime (z1, z2, b), b = 0, 1, of physics, for instance as localized particles (1.), waves (4.) or whirls (3.). Black holes SLs with (d, z2, z3). d = 0, 1, coordinates can use physics laws in their outer environment. Their energetic, decay, normed projective structure is partly described by the old WI Hopf S3 geometry and the new WIGRIS (1.-6.) S5 geometry. Sn are scaled n-dimensional unit spheres in a real (n+1)-dimensional space. Beside stereographic projections, matrix generated and projective normings, there are complex maps over C3 with subspace ranges and domains, using the S1 (circle) fibre bundle structure of S3,5. The projective norming of S5 by S1 is a projective complex space PC2 as inner 4-dimensional spacetime location for nucleons with an S2 boundary, arising for instance from a decaying Higgs S5. The complex function values are set on a fourth Cantor coordinate z4, added to C3. An infinite dimensional Hilbert space (see quantum mechanics) is not needed.

The hedgehog [8]  is for an  educational kit MINT-WIGRIS, a mathematical-technical tool chest to learn this geometry and symmetries. The physical laws, symmetries and metrics are set when they first appear. Further informations on potentials are found in [9], on particle physics, matter in [10] and on S3, S5 in [11].

References

[1] E. Schmutzer: Projektive einheitliche Feldtheorie. Verlag Harri Deutsch, Frankfurt, 2004

[2] E. Schmutzer: Relativistische Physik. Akad. Verlagsges., Leipzig, 1968

[3] Editorial Board and G.Kalmbach: Articles in the Journal MINT (Mathematik, Informatik, Naturwissenschaften, Technik), vol. 1-28, 1997-2013, MINT Verlag, Bad Woerishofen

[4] Gudrun Kalmbach: A Complex 3-dimensional World. A Universe with Moebius Transformations. Evolution of Nucleons and Hydrogen. Intern. J. of Pure and Applied Math. 20, 2005, 539-540; 52, 2009, 289-300; 57, 2009, 111-120

[5] Gudrun Kalmbach: Quantum Measures and Spaces. Kluwer, Dordrecht, 1998, ISBN 0-7923-5288-2

[6] Gudrun Kalmbach: A Graviton Group D3. In: K.-E. Hellwig et al.: Quantum Structures. Kluwer, Dordrecht, 1996.

[7] Gudrun Kalmbach: A Conception of the World. 2001 Animation with Help,

www.uni-ulm.de/~gkalmbac/

[8] Gudrun Kalmbach: Hedgehog balls for Nucleons. PJAAM 7, 2013, 1-5

[9] T. Poston and 1. Stewart: Catastrophe Theory. Pitman, London, 1978. Note: Look up the elliptic umbilic for potentials, the gravity wheel and the 6roll mill.

[10] K. Stierstadt: Physik der Materie. VCH, Weinheim, 1989.

[11] H. Samelson: Ueber die Sphaeren, welche als Gruppenraeume auftreten. Comm. Math. Helv. 13, 1940, S. 144-155. Note: Look up the 3- and 5-dimensional spheres, arising in the WI and SI geometries.