Physical Geometry

     Summary of the theory.

    Classically, multipole aproximations give the geodesic motion with the Lorentz force, showing there is no contradiction with the experimental motion. Electromagnetism is related to an SU(2)Q subgroup of G=SL(4,R). Due to this SU(2)Q the geometry shows a triple structure which determines various physical triple structures in the classification of fields and particles. In particular SU(2)Q also produces gauge weak interactions. A complete set of three fundamental equations is derived from a variational principle: the field equation, the equation of motion and the energy equation. If we restrict to an even U(1) group we obtain the classical Maxwell's field equations. The additional odd generator sector determines the other interactions. The G group action determines Dirac's quantum equation of motion.  

  The group G generators have fundamental quanta of electric charge, magnetic flux and angular momentum. The measurement process of an excitation property is a functional of the geometric current generating the excitation. The measurable results are the eigenvalues of the spin and charge generators. The measurable quanta of charge e, spin h/2 and flux h/2e explain the fractional quantum Hall effect. 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 geometry has a bracket operation for generalized Jacobi vector fields which determines fermionic and bosonic operator fields end their rules of quantization (QFT). In fact, it appears that this geometry is the germ of quantum physics (QED) determining its probabilistic aspects.   

    If we restrict to the geometric gravitational part, the energy equation determines the Einstein equation with a cosmological constant and a geometric energy momentum tensor which indicates a geometric internal solution. In vacuum, the known gravitational solutions are obtained. There is a hyperbolic symmetric solution (the substratum) with constant curvature parameter (geometric energy density) which is related to the concept of inertia, mass and the gravitational constant G in the newtonian limit. In general the gravitational parameter G is variable under nonriemannian fields. These effects may be interpreted as the presence of dark matter. The fine structure alpha coupling constant is a geometric coefficient related to the odd coset G/G+ which is calculated to be 1/137.03608245.

    The mass may be defined in an invariant manner in terms of energy, depending on the connection and matter frames. It also provides a definition of a finite  geometric bare inertial mass for a generalized Dirac equation of motion without infinities and renormalization. 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 with masses corresponding to the weak W, Z boson masses and allow a geometric interpretation of Weinberg's angle in Weak Interactions.

    The geometric equation of motion (a generalized Dirac equation) determines the anomalous bare magnetic moment of both proton and neutron. The first QED correction for the proton gives 2(2.7797) for the Landé g-factor. The SU(2)Q magnetic "strong"part, without the help of any other force, generates short range attractive magnetic 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 lepton and meson masses may be calculated as topological excitations of the electron, giving 107.5927 Mev for the muon and 1802.7 Mev for the tau. The associated neutrinos have energy due to the gravitational geometric curvature energy on their spinor complex null cone, in the same manner as the photon has gravitational curvature energy on its vector null cone near the sun. The corresponding neutrino states may be taken as gravitational energy states of the standard theory of neutrino oscillations.

   The proton shows a dual triple structure (quarks?). 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 book "Physical Geometry" and its abridged version "Energy or Mass and Interaction" were born from a series of lectures over unified theories given at Universidad Simón Bolívar. Really, they are coherent recollections of scattered publications on the geometric unification of physics, including unpublished works. A short summary and introduction to the books, without equations, is given in "The Meaning of a Physical Geometry".  The research papers are available at articles.

   See Oil painting illustration of Energy and  Matter over Space Time: "Mots" by Marisela Hernández