Physical Geometry

It is generally recognized that there exist problems in Physics. In particular, A. Einstein remained critical of some theoretical ideas in modern physics until his death. He was unsatisfied with his own gravitation and advocated the need for a fundamental unified theory which would include relativity, electromagnetism and gravitation from the beginning. He stated that the concepts of parallelism and differentiation are important. Furthermore, present models for Particle Physics are expressed in terms of a very large number of empirical parameters and a high number of dimensions. They are not clearly related to a fundamental underlying physical interaction. Gravitation and Electrodynamics are faced with serious theoretical challenges, particularly in Cosmic Physics due to the possible existence of dark matter and energy. Differential geometry offers the appropriate mathematical structure to accomplish Einstein's physical ideas. The key is "how to differentiate." In the book we use an Ehresmann connection, a concept related to groups. The group is determined from electrodynamics and relativity as the automorphisms of the space-time geometric algebra. Matter and fields are represented by generator-valued tensors. Unification is given by the fundamental nonlinear equation associated to the group. Microscopic physics is the study of linear excitations, which are group representations characterized by discrete numbers. Results obtained indicate that gravitation and electromagnetism are unified in a nontrivial manner. Odd generators represent non classical interactions. The equation of motion determines a geodesic motion with the Lorentz force term. The Einstein field equation with a purely geometric energy momentum tensor is obtained. In the newtonian limit, the constant geometric energy density of a hyperbolic symmetric solution is related to the gravitational constant G. Far from the limit the parameter G would be variable. This effect may be interpreted as dark matter. In vacuum, the known gravitational solutions are obtained. Electromagnetism is related to an SU(2) subgroup. If we restrict to the even U(1) subgroup we obtain Maxwell's equations. The equation of motion is a geometric generalization of Dirac's equation. This geometry is the germ of quantum physics including its probabilistic aspects. The geometric nature of Planck's constant h and of light speed c is determined by their relations to the connection and the metric. The alpha coupling constant is also determined geometrically. Mass is defined in terms of energy. The geometry shows various physical triple structures. The geometric excitations have quanta of charge, flux and spin which determine the fractional quantum Hall effect. The bare-mass quotients of the three stable particles are calculated giving a geometric expression for the proton-electron mass ratio, previously known but physically unexplained. There are connection excitations whose masses correspond to the weak boson masses and allow a geometric interpretation of Weinberg's angle. The equation of motion determines the anomalous bare magnetic moments of the proton, the electron and the neutron. The "strong" electromagnetic odd SU(2) part, generates short range attractive magnetic potentials which are sufficiently strong to determine the binding energy of the deuteron and other light nuclides. The bare masses of the leptons in the three families are calculated as topological excitations of the electron. The masses of these excitations increase under the action of a strong connection and are related to meson and quark masses. The mass spectrum for low masses agrees with the physical particle mass spectrum. The proton shows a triple structure that may be related to a quark structure. The combinations of the three fundamental geometric excitations (proton, electron and neutrino) form other excitations and may be used to represent particles showing a symmetry under the group SU(3)xSU(2)XU(1).