Physical Geometry

© Gustavo R. González Martín 2007

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...A unification through geometry... The book was born from a series of lectures over unified theories given at Universidad Simón Bolívar. Really, it is a coherent recollection of scattered publications on the geometric unification of physics, including unpublished works. The objective is to establish the foundations for this unification in order to give an answer to the following question: Is there a Physical Geometry? The fundamental ideas and some results are published in the references.

It is recognized that the action of matter defines certain concepts and their relations, all of them capable of geometrical representation. The main aspects of the theory are the following:

  1. The acceptance of non linear geometrical equations for the description of the universe.
  2. The group of relativity, the group of automorphisms of flat space time, is generalized to the group of automorphisms of the geometric algebra of space time.
  3. The classical field equations and the equations of motion are represented in terms of a geometric connection and matter frames that determine the geometry.
  4. Microscopic physics is seen as the study of linear geometric excitations, which are representations of the group, characterized by a set of discrete numbers.
  5. These excitation are also characterized by background parameters, calculable from the self energy in terms of quotients associated to the group.

The results indicate that gravitation and electromagnetism are unified in a non trivial manner. Multipole aproximations determine the geodesic motion with the Lorentz force. If we restrict to the gravitational part, we obtain the Einstein equation with cosmological constant and a geometric energy momentum tensor that indicates a geometric internal solution. In vacuum, the known gravitational solutions are obtained. The constant curvature parameter (geometric energy density) of a hyperbolic symmetric solution may be related, in the newtonian limit, to the gavitational constant. In general the gravitational parameter G is variable under nonriemannian fields. This effect may be interpreted as the presence of dark matter. Electromagnetism is related to an SU(2)Q subgroup. If we restrict to a U(1) subgroup we obtain Maxwell's field equations. The geometry has a canonical bracket operation for generalized Jacobi vector fields and determines fermionic and bosonic operator fields end their rules of quantization (QED). In fact, it appears that this geometry is the germ of quantum physics including its probabilistic aspects. The geometric nature of Planck's constant h and light speed c is determined by their respective relations to the connection and the metric. The mass may be defined in an invariant manner in terms of energy, depending on the connection and matter frames. The geometry shows a triple structure that determines various physical triple structures in the classification of particles. The geometric excitations have fundamental quanta of charge e, action h/2 and flux h/2e that may be used to explain the fractional quantum Hall effect. The quotient of bare masses of three stable particles may be calculated and leads us to a surprising geometric expression, previously known but physically unexplained, that gives the value 1836.1181 for the proton to electron mass ratio. There are connection excitations whoose masses correspond to the weak WZ boson masses and allow a geometric interpretation of Weinberg's angle and represent a weak interaction . The geometric equation of motion (a generalized Dirac equation) determines the anomalous bare magnetic moment of both proton and neuctron . The first QED correction for the proton gives 2(2.7797) for the Landé g-factor. The "strong" electromagnetic SU(2)Q part, without the help of any other force, generates short range atractive nuclear potentials which are sufficiently strong to determine the binding energy of the deuteron (-2.20 Mev.), the alpha particle and other light nuclides . The bare masses of the lepton families and mesons may be calculated as topological excitations of the electron, giving 107.5927 Mev for the muon and 1802.7 Mev for the tau. The neutrino mass energy and its oscillations are calculated from the geometric curvature. The proton shows a triple structure. The combinations of the three fundamental geometric excitations (associated to the proton, the electron and the neutrino), forming other excitations, may be used to classify particles and show a symmetry under the group SU(3)xSU(2)xU(1). The alpha coupling constant is a geometric coefficient which is calculated to be 1/137.03608245.

The Universe: inexorable geometric action?

Super Nova 1987a, Hubble Telescope, Space Telescope Science Institute; www.aip.org (7/7/99).

Universe infrared radiation, COBE Spacecraft, Space Telescope Science Institute; www.aip.org (7/7/99).

…Mach felt that there was something important about this concept of avoiding an inertial system… Not yet so clear in Riemann's concept of space. The first to see this clearly was Levi-Civita: absolute parallelism and a way to differentiate…

…The representation of matter by a tensor was only a fill-in to make it possible to do something temporarily, a wooden nose in a snowman…

…For most people, special relativity, electromagnetism and gravitation are unimportant, to be added in at the end after everything else has been done. On the contrary, we have to take them into account from the beginning…

Albert Einstein

from Albert Einstein's Last Lecture1, Relativity Seminar, Room 307, Palmer Physical Laboratory, Princeton University, April 14, 1954, according to notes taken by J. A. Wheeler.

1 J. A. Wheeler in: P. C. Eichelburg and R. U. Sexl (Eds.), Albert Einstein (Friedr. Vieweg & Sohn, Braunschweig) p. 201, (1979).

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