Energy or Mass and Interaction

Transcription

1 Energy or Mass and Interaction

2 ii

3 iii Energy or Mass and Interaction Gustavo R. González-Martín Professor, Physics Departent Sión Bolívar University Caracas

4 iv First English Edition, first published in 010 Gustavo R. González-Martín 010 An abridged english translation of Geoetría Fisica, Gustavo R. González Martín 1999, 010 Abridged and translated fro the Spanish by the author. Departaento de Física Universidad Sión Bolívar Valle de Sartenejas, Baruta, Estado Miranda Apartado 89000, Caracas 1080-A, Venezuela

5 v Mach felt that there was soething iportant about this concept of avoiding an inertial syste Not yet so clear in Rieann s concept of space. The first to see this clearly was Levi-Civita: absolute parallelis and a way to differentiate The representation of atter by a tensor was only a fill-in to ake it possible to do soething teporarily, a wooden nose in a snowan For ost people, special relativity, electroagnetis and gravitation are uniportant, to be added in at the end after everything else has been done. On the contrary, we have to take the into account fro the beginning Albert Einstein fro Albert Einstein s Last Lecture, 3 Relativity Seinar, Roo 307, Paler Physical Laboratory, Princeton University, April 14, 1954, according to notes taken by J. A. Wheeler. 3 J. A. Wheeler in: P. C. Eichelburg and R. U. Sexl (Eds.), Albert Einstein (Friedr. Vieweg & Sohn, Braunschweig) p. 01, (1979).

6 vi

7 Preface The physical ideas previously presented in Physical Geoetry are published in cited scientific journals. Nevertheless soe of our latest nuerical results, which are available as e-prints at have et resistance to publication by certain journals. This sees to be a reflection of opinions held by soe groups involved in research along soe presently canonical particle lines. These lines introduce odels, in ters of either a large nuber of diensions or a very large nuber of epirical paraeters, which are not clearly related to a fundaental underlying theoretical physical interaction. Our ideas represent a critique of these odels and indicate the need of a research thrust along a new direction. In this regard, the following general aspects should be pointed out: In first place, we have presented any new nuerical results which are not calculable fro known physical odels; In second place, these nubers arise fro the concept of energy and a generalized nonlinear electrodynaics, whose QFT should be used for perturbations corrections; In third place, there appears to be no physical experiental evidence contradicting our ideas. We also appreciate that there are difficulties understanding the results obtained in our work because we presented the in a geoetrically unified for. The essential objective is to start fro geoetry, which represents for and has developed fro observations of the universe during ore that 5 centuries. For these reasons there is a need to discuss these physical-geoetric ideas without the burden of a coplete atheatical treatent. Therefore, in this abridged book we ake ephasis on physical aspects of the theory, in particular at the start of each chapter. To help the reader a brief suary of the results is presented at the end of chapters. In particular, here we discus the following: 1- The connection between energy and nuerical asses; - The energy-ass classification of particles as fundaental representations of a relativity group action; 3- The unification of interactions, in particular the identification of short-distance nuclear effects with a fundaental electroagnetic su() subinteraction which provides very strong short-range attractive agnetic potentials and energy and predicts experiental nuclear data; 4- The existence of energy-ass ters in a generalized Einstein equation which appear to lead to dark atter and energy; 5- The need and relevance of classical statistics in a theory of icroscopic easureents. Caracas, Venezuela, February 4, 010. Gustavo R. González Martín vii Preface to Physical Geoetry The objective is to establish a foundations for unification of physical forces in order to give answers to fundaental questions: What are the relations aong the concepts of energy, ass, inertia and interaction forces? The fundaental ideas and results are published in the references. It is recognized that the action of atter defines these concepts and their relations, all of the capable of geoetrical representation. The ain aspects of the theory are the following: 1. The physical universe is described by atter equations associated to an evolution group.. The group is obtained fro geoetric algebraic tranforations of space and tie. 3. The interaction is represented by field equations and equations of otion in

8 viii ters of potential and force tensors deterined by atter transforation currents. 4. Microscopic physics is seen as the study of linear geoetric excitations, which are representations of the group, characterized by a set of discrete nubers. 5. The equations deterine fundaental concepts of energy, ass and inertia which classify the interactions and particles. The results obtained indicate that gravitation and electroagnetis are unified in a nontrivial anner. There are additional generators that ay represent non classical interactions. Multipole equations of otion deterine the geodesic otion with the Lorentz force ter. If we restrict to the even part, we obtain the Einstein field equation and a purely geoetric energy oentu tensor that indicate the possibility of a geoetric internal solution. The constant curvature paraeter (geoetric energy density) of a hyperbolic syetric solution ay be related, in the newtonian liit, to the gravitational constant G. If the nonrieannian connection fields contribute to the scalar curvature, the paraeter G would be variable, diinishing with the field intensity. This effect ay be interpreted as the presence of dark atter or energy. In vacuu, the known gravitational solutions with a cosological constant are obtained. Electroagnetis is related to an SU() Q subgroup. If we exclusively restrict to a U(1) subgroup we obtain Maxwell s field equations. In general, the equation of otion is a geoetric generalization of Dirac s equation. In fact, it appears that this geoetry is the ger of quantu physics including its probabilistic aspects. The geoetric nature of Planck s constant h and of light speed c is deterined by their respective relations to the connection and the etric. The ass is defined in an invariant anner in ters of energy, depending on the connection and atter fraes. The geoetry shows a triple structure that deterines various physical triple structures. The geoetric excitations have quanta of charge, flux and spin that deterine the fractional quantu Hall effect. The quotient of bare asses of the three stable particles are calculated and leads us to a surprising geoetric expression for the proton electron ass ratio, previously known but physically unexplained. There are assive connection excitations whose asses correspond to the weak boson asses and allow a geoetric interpretation of Weinberg s angle.the geoetric equation of otion (a generalized Dirac equation) deterines the anoalous bare agnetic oents of the proton, the electron and the neutron. The strong electroagnetic SU() Q part, without the help of any other force, generates nuclear range attractive potentials which are sufficiently strong to deterine the binding energy of the deuteron and other light nuclides, coposed of protons and electrons. The bare asses of the leptons in the three failies are calculated as topological excitations of the electron. The asses of these excitations increase under the action of a strong connection (relativity of energy) and are related to eson asses. The geoetry deterines the geoetric excitation ass spectru, which for low asses, essentially agrees with the physical particle ass spectru. The proton shows a triple structure that ay be related to a quark structure. The cobinations of the three fundaental geoetric excitations (associated to the proton, the electron and the neutrino), foring other excitations, ay be used to represent particles and show a syetry under the group SU(3)SU()U(1). The alpha coupling constant is also deterined geoetrically. The first two chapters represent an introduction. In chapters 3 to 10 the fundaental geoetric ideas are developed. In chapters 11 to 18 the theory is applied to concrete cases. Caracas, Venezuela, February 4, 010. Gustavo R. González Martín

9 ix + E = E 3 q 3 Q E q 1 q E E ie - = 1 Geoetric Quantization of the SU() electroagnetic potential and the electric charge. The odd - E generators deterine attractive, short-range potentials which are sufficiently strong to sustain nuclear fusion. A geoetric universal action? Are neutron stars natural laboratories for these energy processes?

10 x

11 To Lourdes xi

12 xii Acknowledgents I have tried to give credit to those whose work give support to the ideas expressed in this book. Nevertheless, it appears ipossible to accoplish this copletely. At the oent of writing, it is uch what I owe to those fro which I have learned throughout the years. In this sense, I a grateful to the ebers physics counity of the Boston area, in particular to y professor, John Stachel. Specially I want to thank the colleagues with who I discussed these topics, even if I a unable to precisely deterine the contribution to the gerination and foration of the ideas; in particular, the senior faculty of the Caracas Relativity and Fields Seinar: Luis Herrera Coeta, Alvaro Restuccia, Sebastián Salaó and Carlos Aragone (R.I.P.). I also acknowledge the collaboration of research assistants and soe students in y relativity courses and special lectures on unification at the Sión Bolívar University, who served as stiulating test in the presentation and discussion of the geoetric hypothesis of physics: G. Salas, G. Sariento, V. Villalba, V. Varela, A. Mendoza, O. Rendón, E. Valdeblánquez, I. Taboada, V. Di Cleente, J. Díaz, J. González T., A. de Castro, A. Hernández and M. A. Lledó. Caracas, Venezuela, June 0, 000. Gustavo R. González Martín

13 xiii CONTENTS 1. Energy Extension of Relativity Energy and the Field Equation Inertial Effects and Mass The Classical Fields Results Quanta Induced Representations of the Structure Group G Relation Aong Quantu Nubers Spin, Charge and Flux Representations of a Subgroup P Magnetic Flux Quanta Magnetic Energy Levels Fractional Quantu Hall Effects Results Measureents and Motion Measureent of Geoetric Currents Geoetric Spin Geoetric Charge The Concept of Mass Invariant Mass Equation of Motion Agreeent with Standard Quantu Mechanics Results Masses Bare Inertial Masses for Frae Excitations or Ferions Syetric Cosets Volue of C Space Volue of K space Ratio of Geoetric Volues The p, e and n Mass Ratios The Equation for the Potential Excitations or Bosons Massive Particular Solutions Massive SU() Bosons Mass Values in Free Space Potential Excitations in a Lattice Results Nuclear Energy and Interaction Nonrelativistic Motion of an Excitation Magnetic Moents of p, e and n The Modified Pauli Equation The Proton-Electron-Proton Model for the Deuteron

14 xiv 5.5. Binding Energy of the Deuteron The Electron-Proton Model for the Neutron The Many Deuteron Model Nuclear Structure, Fusion and Fission Geoetric Weak Interations Relation with Feri s Theory Results Particles and Interactions Geoetric Classification of the Potential Algebraic Structure of Particles Interpretation as Particles and Interactions Topological Structure of Particles Geoetric Excitation Masses Leptonic Masses Mesonic Masses Barut s Model Relation with Particle Theory Results Gravitation and Geoetry Einstein s Equation Newtonian G and the Schwarzschild Mass Dark Matter Effects The Alpha Constant Syetric Space K Wyler s Measure on the K Space Value of the Geoetric Coefficient Results Quantu Fields Linearization of Fields Frae Solutions Potential Solutions Bracket as Derivation Geoetric Theory of Quantu Fields Product of Jacobi Operators in QED Geoetric Electrodynaics Free Particles and Currents The Interaction Hailtonian Statistical Interpretation Results Appendix INDEX References Bibliography on Geoetry and Relativity

15 xv Notation. Lower case Greek indices, corresponding to space tie, vary fro 0 to 3. Lower case Latin indices, correspond to a Lie algebra diension, usually fro 1 to 15; occasionally indicate the three diensional space, fro 1 to 3. Upper case Latin indices correspond to diensions of atrices or spinors, usually varying fro 1 to 4. Repetition of indices indicates suation over the diension of the corresponding space. The partial derivative is denoted by, the covariant derivative by, the exterior derivative by d and the covariant exterior derivative in a fiber bundle by D. The physical units are chosen geoetrically, so that c, and e are equal to 1. The space tie etric signature is 1, -1, -1, -1. Specialized atheatical notation is defined where needed, following the coon use in the bibliography.

16 xvi

17 1. Energy. Our physical notions of energy, space, ass, force and tie are local anifestations of a nonlinear physical geoetry. We are able to sense and experient this geoetry which appears to be generated by the evolution of atter currents. In any present day geoetric physical theories it is iplicitly assued that atter is linearly ade of particles related to points, strings, ebranes, etc. Nevertheless, this assuption ay not be sufficient nor necessary. Instead, it appears necessary to assue that the action of physical atter currents J nonlinearly deterines a physical geoetry which reacts back on the current. The reaction is locally experienced by atter as the action of an interaction potential A which ay be represented by a geoetric connection w associated to an interaction group [1] sufficient to account for the physical effects. The fundaental action-reaction dynaical process allows us to define the concept of (interaction) energy as the product J.A. The group generates infinitesial excitations of the geoetry which are representations of the group and behave as physical particles. We call this group the structure group of the physical theory. Starting fro space and tie we shall inquire the properties of this geoetry. In order to describe electroagnetis and relativistic otion we need a space-tie with 4 coordinates. Measureents require standard units which introduce a etric. Relativity allows us to say that space-tie is locally an orthonoral space with an invariant etric under the Lorentz group SO(3,1). The use of the Lorentz group as structure group of a curved space-tie leads to gravitation. We should require, at least, that the structure group also represent electrodynaics [1]. It is known that the structure group of electroagnetis is U(1). Since this group is coplex we felt, in a first attept, that in order to approach unification it was desirable to work with the spin group SL(,C), rather than the Lorentz group itself SO(3,1). A gravitation theory related to SL(,C) was discussed by Carelli []. The siplest way to enlarge the group, apparently, was to use the group U(1) SL(,C) which is the group that preserves the etric associated to a tetrad induced fro a spinor base. It was known to Infeld and Van der Waerden [3, 4, 5], when using this group, that there appeared arbitrary fields which aditted interpretation as electroagnetic potentials because they obeyed the necessary field equations. To adit this interpretation we further required that the electrodynaic Lorentz force equation be a consequence of the field equations. Otherwise the equation of otion, necessarily iplied by the nonlinear theory, contradicts the experientally wellestablished otion of charged particles and the theory should be rejected. This attept [6], using U(1) SL(,C) as the group, led to a negative result, because the equations of otion depend on the coutators of the gravitational and electroagnetic parts which coute. This eans that a charged particle would follow the sae path followed by a neutral particle. This proves that it is not possible, without contradictions, to consider that the U(1) part represents electroagnetis as suggested by Infeld and Van der Waerden. This also eans

18 ENERGY OR MASS AND INTERACTION Chapter 1 that to obtain the correct otion we ust enlarge the chosen group in such a way that the electroagnetic generators do not coute with the gravitational ones. It is not true that any structure group which contains SL(,C) U(1) as a subgroup gives a unified theory without contradicting the electrodynaic equations of otion. The correct classical otion is a fundaental requireent of a unified theory. In addition to this coutation physical proble there are two atheatical probles due to the SO(3,1) group. The second proble is that orthogonal groups are double valued due to their quadratic etric. The third proble is the definition of the square root of unit vectors associated to the etric signature. Clifford algebras were developed to solve the third proble for general orthonoral spaces R,n. The geoetric eleents in the algebras include real nubers R, coplex nubers C or R 0,1, anticouting quaternions H or R 0,, Clifford operators R 3,1 or Cl(3,1)... Cl(,n), etc. The groups generated by Cl(,n) solve the three probles. They are also useful in taking the square root of operators and have a richer atheatical structure which deterines a higher predictive power. Therefore, we should use the axial group of Clifford nuber transforations which preserve the etric and nuerical structures of the associated spaces. The construction of the theory was accoplished [7, 8, 9] by taking this group, which is essentially SL(4,), as the structure group G of a generalized curved electroagnetic theory. This construction appears to be sufficient to obtain a structure group which describes all known physical energy interactions and corresponds to Hilbert s sixth proble [10] Extension of Relativity. Associated to any orthonoral flat space there is a Clifford geoetric algebra [11]. There are inclusion appings k of the orthonoral space into the algebra, apping orthonoral vector bases to orthonoral sets of the algebra. The different iages of a base deterine a subspace of the algebra. The geoetrical reason for the introduction of these algebras is to obtain geoetric objects whose square is the negative of the scalar product of a vector x with itself, ( k ( )) =-. =- (, ) x xxi g x x I. (1.1.1) In a sense, this is a generalization of the introduction of iaginary nubers for the real line. These algebras are useful in defining square roots of operators. For tridiensional euclidian space, the even Clifford subalgebra also has the structure of the Lie algebra of the SU() group, to 1 hooorphic to the rotation group. SU() transforations by p and 4p are different but associated to a rotation by p. Furtherore, it is known that a rotation by 4p is not geoetrically equivalent to a rotation by p when its orientation entangleent relation with its surrounding is considered [1]. To preserve this geoetric difference in a space-tie subspace we ust require the use of, at least, the even geoetric subalgebra for the treatent of a relativistic space-tie. When the coplete algebra is defined for Minkowski space-tie, the observer orthonoral tetrads are apped to orthonoral sets of the algebra. Now the nu-

19 Energy ber of possible orthonoral sets in the algebra is uch larger than the possible orthonoral tetrads in space-tie. There are operations, within the algebra, which transfor all possible orthonoral sets aong each other. These are the inner autoorphiss of the algebra. Geoetrically this eans that the algebra space contains any copies of the orthonoral Minkowski space. A relativistic observer ay be ibedded in the algebra in any equivalent ways. It ay be said that a noral space-tie observer is algebraically blind. Usually the algebra is restricted to its even part, when the syetry is extended fro the Lorentz group (autoorphiss of space-tie) to the corresponding spin group SL(,C) [13], (autoorphiss of the even subalgebra). In this anner a fixed copy of Minkowski space is chosen within the geoetric algebra. This copy reains invariant under the spin group. The situation is siilar to the ibedding of a three diensional observer carrying a spatial triad into tetradiensional space-tie. This ibedding is not unique, depending on the relative state of otion of the observer. There are any spatial tridiensional spaces in space-tie, defining the concept of siultaneity which is different for observers with different constant velocities. These possible physical observers ay all be transfored into each other by the group of autoorphiss of space-tie, the Lorentz group. This siilarity allows the extension of the principle of relativity [1] by taking as structure group the group of correlated autoorphiss of the space-tie geoetric algebra spinors instead of the group of autoorphiss of space-tie itself or only the autoorphiss of the even subalgebra spinors. A relativistic observer carrying a space-tie tetrad is ibedded in the geoetric algebra space in a nonunique way, depending on soe bias related to the orientation of a tetradiensional space-tie subspace of the sixteen diensional algebra. As Dirac once pointed out, we should let the geoetrical structure itself lead to its physical eaning. We ay conceive coplete observers which are not algebraically blind. These observers should be associated to different but equivalent orthonoral sets in the algebra. Transforations aong coplete observers should produce algebra autoorphiss, preserving the algebraic structure. This is the sae situation of special relativity for space-tie observers and Lorentz transforations. In particular the inner autoorphiss of the algebra are of the for - a = g 1 ag, (1.1.) where g is an eleent of the largest subspace contained in the algebra which constitutes a group. This action corresponds to the adjoint group acting on the algebra. For the Minkowski orthonoral space, denoted by R 3,1, the Clifford algebra Cl(3,1)=R 3,1 is (), where ay be called the ring of pseudoquaternions [9] and the corresponding group is GL(,). The adjoint of the center of this group, acting on the algebra, corresponds to the identity. The quotient by its noral subgroup R + is the siple group SL(,). Therefore, the siple group nontrivially transforing the coplete observers aong each other is SL(,). This group is precisely the group G of correlated autoorphiss of the spinor space associ- 3

20 4 ENERGY OR MASS AND INTERACTION Chapter 1 ated to the geoetric algebra. We ay associate a spinor base to a coplete observer. A transforation by G of a coplete observer into another produces an adjoint transforation of the algebra and, consequently, a transforation of an orthonoral set onto an equivalent set in a different Minkowski subspace of the algebra. The etric in the equivalent Minkowski spaces in the algebra is the sae. These transforations preserve the scalar products of space-tie vectors apped into the algebra. The subgroup of G whose Ad(G), in addition, leaves invariant the original Minkowski subspace, is known as the Clifford group. The Spin subgroup of the Clifford group is used in physics, in a standard anner, to extend the relativity principle fro vectors to spinors. Since there are any copies of the spin group L in SL(,), in our extension we have to choose a particular copy by specifying an inclusion ap i. Apart fro choosing an eleent of L, a standard vector observer, we ust choose i, thus defining a coplete spinor observer This coplete observer, associated to a spinor base, carries, not only spacetie inforation but also soe other internal inforation related to the algebra [14]. The group SL(,) of transforations of these coplete observers transfors the observations ade by the. The observations are relative. The special relativity principle ay be extended to this situation. Since coplete observers are theselves physical sytes we state the generalized principle as follows: All observers are equivalent under structure group transforations for stating the physical laws of natural systes and are defined by spinor bases associated to the geoetric algebra of space-tie. The nonuniqueness of the orthonoral set has been known in geoetry for a long tie. We have given physical eaning to the orthonoral sets by associating the to physical observers. This iplies that the physically allowable transforations are those apping the algebra to itself by its own operations. We also have given a relativistic eaning to these transforations. Furtherore, we should point out that our algebra is isoorphic to the usual Dirac algebra as a vector space but not as an algebra. The algebras correspond to space-ties of opposite signature. The requireent to use a tielike interval to paraetrize the tielike world line of an observer deterines that the appropriate algebra is not Dirac s algebra R 1,3 but instead the algebra R 3,1, indicated here. The ain practical difference is the appearance of a second copact subgroup related to electroagnetis and charge quantization as will be seen in the following sections and chapters.the algebra R 3,1 is discussed in the appendix. The equation of otion of atter is the integrability condition of the field equation. It ay be interpreted as a generalized Dirac equation with potentials given by the generators of the structure group SL(,) or its covering group SL(1,LQ). The equation for the frae using the other K ring in the group SL(,Q) does not lead to Schrödinger s equation for a particle [15] as shown in [9, chapter 3, section 3.5.]. Fro this point on, we shall use the notation SL(,) or its hooorphic SL(4,R) to indicate these groups or the covering group, unless otherwise explicitly stated when it is convenient to distinguish the. In general relativity [16] the space-tie anifold is peritted to have curvature, special relativity is required to be valid locally and local observer fraes

21 Energy 5 are introduced, depending on their positions on space-tie. In this anner we get fields of orthonoral tetrads on a curved anifold. The geoetry of the anifold deterines the otion, introducing accelerations of inertial and gravitational nature. Siilarly in our case, in order to include accelerated systes, we let spacetie have curvature and introduce local coplete observers which depend on their positions. But now, these observers are represented by general spinor fraes which are subject to transforations beyond relativity (Ultra relativity). In this anner we get fields of spinor bases (fraes) which geoetrically are local sections of a fiber bundle with a curved base space. The geoetry deterines the evolution of atter, but now we have, in addition to inertial and gravitational accelerations, other possible accelerations due to other fields of force represented by the additional generators. These algebraic observers are accelerated observers. In other words we now get a geoetrically unified theory with extra interactions (nuclear and particle) whose properties ust be investigated. 1.. Energy and the Field Equation. The group SL(4,R) is known not to preserve the corresponding etric. But, if we think of general relativity as linked to general coordinate transforations changing the for of the etric, it would be in the sae spirit to use such a group. Instead of coordinate transforations whose physical eaning is associated with a change of observers, we have transforations belonging to the structure group whose physical eaning should be associated with a change of spinors related to observers. Representations of this group would be linked to atter fields. If we restrict to the even part of the group, taken as a subset of the Clifford algebra, we get the group SL 1 (,C), used in spinor physics. Since SL(4,R) is larger (higher diensional) than SL(,C) it gives us an opportunity to associate the extra generators with energy interactions apart fro gravitation and electroagnetis. The generator which plays the role of the electroagnetic generator ust be consistent with its use in other equations of physics. The physical eaning of the reaining generators should be identified. The field equations should relate the interaction connection w or the physically equivalent interaction potential A to a geoetric object representing atter. We expect that atter is represented by a current J() function of points on a space-tie anifold M, valued in the group Lie algebra, rather than the nongeoetric stress energy tensor T. The siplest object of this type is a generalized curvature W of w or the physical generalized Maxwell tensor F. This generalized tensor obeys the Bianchi identity, which we write indicating the covariant exterior derivative by D, DF º DW =0. (1..1) The next siple object is constructed using Hodge duality. In siilarity with the linear Maxwell s theory, we postulate the corresponding nonlinear field equation for the curvature as the generalized geoetrical electrodynaic equation,

22 6 ENERGY OR MASS AND INTERACTION Chapter 1 * * * DF DW kj º =, (1..) where atter is represented by the current *J, which ust be a 3-for valued in the algebra, and k is a constant to be identified later. Because of the geoetrical structure of the theory the source current ust be a geoetrical object copatible with the field equation and the geoetry. The structure of J, of course, is given in ters of soe geoetric objects acted upon by the potential. The geoetrical properties of the curvature and the field equations deterine that J obeys an integrability condition, DD * F = éf, * F ù êë úû = 0, (1..3) DJ * = 0. (1..4) This relation being an integrability condition on the field equations, includes all self reaction ters of the atter on itself. A physical syste would be represented by atter fields and interaction fields which are solutions to this set of nonlinear siultaneous equations. There should be no worries about infinities produced by self-reaction ters. As in the EIH ethod in general relativity [17], when a perturbation is perfored on the nonlinear equations, for exaple to obtain linearity of the equations, the splitting of the equations into equations of different order brings in the concepts of field produced by the source, force produced by the field and therefore the self-reaction ters. These ters, not present in the original nonlinear syste, are a proble introduced by this particular ethod of solution. In the zeroth order a classical particle oves as a test particle without self-reaction. In the first order the field produced by the particle produces a self correction to the otion. Enlarging the group of the potential not only unifies satisfactorily gravitation and electroagnetis [8, 9], but requires other fields [14] and it appears to give a gravitational theory which differs, in principle, fro Einstein s theory and resebles Yang s theory [18]. This ay be seen fro the field equations of the theory, which relate the derivatives of the Ehresann curvature to a current source J. The product of the interaction potential A by the atter current J has units of energy or inverse length. We ay naturally define a fundaental unified geoetric energy M associated to the geoetry, which defines the concept of ass and appears to be related to the concepts of inertia, M ( ) tr( J ( ) A ( ) ). (1..5) = 1 4 This unification of the concept of energy and ass leads to iportant physical results. A theory of connections without any other objects is incoplete fro a geoetrical point of view. A connection on a principal bundle is related to the structure group and the base space of the bundle. Representations of the group provide a natural vector fiber for an associated vector bundle on which the connection ay be ade to act. The geoetric eaning of the physical potential is related to parallel translation of the eleents of the fiber at different points

23 Energy throughout the base space. This is, essentially, a process of coparison of eleents at different events. A vector fiber space of this type has a base and the effect of the potential is naturally defined on the base. Fro a geoetrical point of view the potential should be copleented by a vector base. It is well known that Einstein s gravitation theory ay be expressed using an orthonoral tetrad instead of the etric [19]. In this theory we have taken this idea one step further, introducing a spinor base e on the fiber space of an associated vector bundle S, in addition to the base on the fiber of the tangent space. In other words, we work with the base of the square root space of the usual flat space. The potential, which represents the gravitational and electroagnetic fields, depends on a current source ter. We also postulate that this source current is built fro fundaental atter fields which have the geoetrical interpretation of foring a base e on the fiber of the associated vector bundle and defines an orthonoral subset k of the geoetric Clifford algebra, J * = 1 a b g eabge k edx dx dx 3!. (1..6) This base e, when arranged as a atrix with the vectors of the base as coluns, is related to an eleent of the group of the principal bundle. It is natural to expect that a base field e (a section in geoetric language), which we shall call a frae e, should obey equations of otion which naturally depend on the potential field. In fact, it will be seen that a particular solution of the integrability condition, the covariant conservation equation (1..4) of J, is obtained fro the equation k e =0. (1..7) This equation ay be interpreted as a generalized Dirac equation since the structure group is SL(4,R) or its covering group. We should note that whenever we have an sl(,c) potential, there is a canonical coupling of standard gravitation to spin ½ particles obtained by postulating a Dirac equation which depends on a spin frae [0, 1, ]. Nevertheless, strictly this does not represent a real unification. Our field equation iplies integrability conditions in ters of J. Together with the geoetric structure of J, our conditions iply the generalized Dirac equation which, therefore, is not required to be separately postulated, as in the previously entioned nonunified case. The theory under discussion is not a ere pasting together of canonical gravitation and canonical electroagnetis for spin ½ particles. Rather, it is the introduction of a generalized geoetric structure which nontrivially odifies both canonical theories and their coupling. Actually, the nonlinear field equation for the potential and the siplest geoetric structure of the current are sufficient to predict this generalized Dirac equation and provide a unified concept of energy and ass. If we introduce a variational principle [8,9] to obtain the two fundaental equations, (1..) and (1..4), the principle deterines a third related fundaental equation, 7

24 8 ENERGY OR MASS AND INTERACTION Chapter 1 é ù é ( tr Fˆ F n u ˆF kl -1 F ktr e u ˆ) e u -1 êë 4 ˆe u n rn - r kl úû = ê4 i - r i n e ë r ú (1..8) û which represents the total generalized field energy. It has been shown [8, 9] that the latter equation leads to the Einstein equation of gravitation in General Relativity with a geoetric tensor source T. Soe of the features of the theory depend only on its geoetry and not on a particular field equation and ay be seen directly. For exaple, atter ust evolve as a representation of SL(4,R) instead of the Lorentz group. It follows that atter states are characterized by three quantu nubers corresponding to the discrete nubers characterizing the states of a representation of SL(4,R). One of these nubers is spin, another is associated to the electroagnetic SU(). This gives us the opportunity to recognize the latter as the electric charge [3]. Perhaps we should realize that the idea of a quantu entered Modern Physics by the experiental deterination of the discreteness of electric charge. Later atoic easureents were explained by quantu theory by assuing the quantu of spin, but quantu theory was not given the burden to quantize electric charge. The possibility of obtaining the quantu of charge as explained before, ay indicate that present day quantu theory is an incoplete theory as Dirac indicated [4]. As a bonus, this theory provides a third quantu nuber for a atter state, which ay be recognized as a quantu of agnetic flux, providing a plausible fundaental explanation to the fractional quantu Hall effect [3]. As in general relativity, the integrability conditions iply the equations of otion for a classical particle, without knowledge of the detailed for of the source J, if we assue that J has a ultipole structure. The desired classical Lorentz equations of otion were obtained [1, 7]. Nevertheless the ain objective at present, is not to describe the classical otion of atter exhaustively but rather to construct the geoetrical theory and to show that it is copatible with the classical otion of the sources and with odern ideas in quantu theory. In particular, it appears, as first objective, to exploit the opportunity provided by the theory to give a geoetrical interpretation to the source current in ters of fundaental geoetric atter field objects. With the geoetric structure given to J, the first stage in the construction of the unified concept of energy is copleted Inertial Effects and Mass. The proposed nonlinear equation and its integrability condition have peculiar aspects which distinguish the fro standard equations in classical physics. Norally coupled field equations and equations of otion, for exaple Maxwell and Lorentz equations, in presence of a current source do not provide, by theselves, a static internal solution for a source which ay represent a particle or object under the influence of its own field. Use of delta functions for point particles avoid the proble rather than solve it, and ay introduce self accelerated solutions [5, 6]. The choice of current density in the theory, together with the ù

25 Energy interpretation developed allows a discussion on different grounds. The frae e that enters in the current represents atter. Since a easureent is always a coparison between siilar objets, a easureent of e entails another frae e to which its coponents are referred. If we choose the referential e properly we ay find interesting solutions. In particular we can find a geoetric background solution, which we call the substratu, whose excitations behave as particles. The integrability condition of the nonlinear equation leads to a generalized Dirac equation for the otion of atter, with a paraeter that ay be identified with a ass defined in ters of energy [7, 8, 9]. The recognition of a single concept of ass is fundaental in General Relativity and erits discussion of possible solutions of the coupled equations. If we identify a geoetrical excitation with a physical particle, the Dirac equation for a linear excitation, which now would be the linear equation for a particle, contains paraeters provided by the curved (nonlinear) background solution which we shall call its substratu solution. Soe of the particle properties could be deterined by a substratu geoetry. In particular a ass paraeter arises for the frae excitation particle fro the ass-energy concept defined in ters of energy. It is clear that this paraeter is not calculable fro the linearized excitation equation but rather fro a nonlinear substratu solution.this appears interesting, but requires a knowledge of a substratu solution to the nonlinear field equation. Thus it is necessary to find a nonlinear solution, the sipler the better, which could illustrate this ideas. It is in this context that the following solution is presented. The nonlinear equations of the theory are applicable to an isolated physical syste interacting with itself. Of course the equations ust be expressed in ters of coponents with respect to an arbitrary reference frae. A reference frae adapted to an arbitrary observer introduces arbitrary fields which do not contain any inforation related to the physical syste in question. The only nonarbitrary reference frae is the frae defined by the physical syste itself. Any excitation ust be associated to a definite background solution or substratu solution. An arbitrary observation of an excitation property depends on both the excitation and its substratu, but the physical observer ust be the sae for both excitation and substratu. We ay use the freedo to select the reference frae to refer the excitation to the physical frae defined by its own substratu. We have chosen the current density 3-for J to be ˆ J ek a u aˆ e =, (1.3.1) in ters of the atter spinor frae e and the orthonoral space-tie tetrad u. Since we selected that the substratu be referred to itself, the substratu atter local frae e, referred to e r becoes the group identity I. Actually this generalizes cooving coordinates (coordinates adapted to dust atter geodesics) [7]. We adopt coordinates adapted to local substratu atter fraes (the only nonarbitrary frae is itself, as are the cooving coordinates). If the frae e becoes the identity I, the cooving substratu current density becoes a constant. Coparison of an object with itself gives trivial inforation. For exaple 9

26 10 ENERGY OR MASS AND INTERACTION Chapter 1 free atter or an observer are always at rest with theselves, no velocity, no acceleration, no self forces, etc. In its own reference frae these effects actually disappear. This substratu represents inert atter. Only constant self energy ters, deterined by the nonlinearity, ake sense and should be the origin of the constant bare inertial ass paraeter. At the sall distance l, characteristic of excitations, the eleents of the substratu, both connection and frae, appear syetric, independent of spacetie. We should reeber that space-tie M is, atheatically, a locally syetric space or hyperbolic anifold [8]. We recognize these as the necessary condition for the substratu to locally adit a axial set of Killing vectors [9] which should deterine the space-tie syetries of the connection (and curvature). This eans that there are space-tie Killing coordinates such that the connection is constant but nonzero in the sall region of particle interest. A flat connection does not satisfy the field equation. The excitations ay always be taken around a syetric nonzero connection or potential. In particular the nonlinear equation adits a local nonzero constant potential solution. This would be the potential deterined by an observer at rest with the atter frae. Of course, this solution is trivial but since the potential has units of energy, ass or inverse length, this actually introduces fundaental diensions in the theory. Furtherore, a constant nonzero solution assigns a constant ass paraeter to a fundaental particle excitation and allows the calculation of ass ratios of particles by integration of M on syetric spaces [9]. The result is obtained in ters of the diensionless coupling constant in eq. (1..). Therefore, we wish to find a constant nonzero inert (trivial) solution to the nonlinear field equation which we shall call the inert substratu solution. First we look into the left side of the field equation (1..), and notice that for a constant potential for A and a flat etric, the expression reduces to triple wedge products of A with itself [9], which ay be put in the for of a polynoial in the coponents of A. This cubic polynoial represents a self interaction of the potential field since it ay also be considered as a source for the differential operator. Rather than work with the whole group G we first restrict the group to the 10 diensional Sp(4,) subgroup. Furtherore we desire to look at the nongravitational part of the potential. Hence, we liit the coponents of the potential to the Minkowski subspace defined by the orthonoral set, which is the coset Sp(4,)/SL(,C). This is possible because if the potential is odd so is the triple product giving an odd current as required. The substratu solution is - ˆ As e ˆedx a = pa ka + e de=- 4 J + e de= L+ e de 3 M.(1.3.) where M is a constant deterining the equipartition of excitation energy. This inert potential is essentially proportional to the current, up to an autoorphis. It should be noted that in the expression for A, the ter containing the current J defines a potential tensorial for L. Its subtraction fro A gives an object, e -1 de, which transfors as a potential or connection. This solution ay be extended fro the subgroup to the whole 15 diensional SL(4,R) group using coplex coordinates on the coplex coset SL(4,R)/SL(,C). We call this solution the coplex inert substratu [9].

27 Energy For any potential solution A or connection we can always define a new potential or connection by subtracting the tensorial potential for L corresponding to the substratu solution, eq. (1.3.), M Aº A- L = A+ J. (1.3.3) 4 In ters of the new potential defined by equation (1.3.3) the equation of otion (1..9), in induced representations, explicitly displays the ter depending on the substratu ass required by the Dirac equation. Using the algebraic relations aong the orthonoral subset k of the geoetric Clifford algebra we get, æ ö k e k ( e ea) k æ ö e ea e M = - = - + J ç è çè 4 ø ø M = k e+ k ke= k e- e= 0, 4 11 M (1.3.4) which explicitly shows the geoetric energy M as the ger of the Dirac particle ass. This equation is nonlinearly coupled to the field equation (1..) through the definition of J, eq. (1..6) The Classical Fields. The curvature of this geoetry is a generalized curvature associated to the group SL(4,R). Since it is known that the even subgroup of SL(4,R) is the Spin group related to the Lorentz group, we look for a liit theory to get this reduction. When ultra relativistic effects are sall, we expect that we can choose bases so that the odd part is sall of order e. This is accoplished atheatically by contracting the SL(4,R) group with respect to its odd subspace [30]. In the contracted group this odd subspace becoes an abelian subspace. Then the SL(4,R) curvature reduces to () F = F +O e, (1.4.1) + where F + is the curvature of the even subgroup SL 1 (,C) The result is that in this liit the curvature reduces to its even coponent which splits into an SL(,C) curvature and a separate couting U(1) curvature. It is known that an SL(,C) curvature ay represent gravitation [31] and a U(1) curvature ay represent electroagnetis [3, 33, 34]. If we take this liit U(1) as representing the standard physical electroagnetis we ust accept that, in the full theory, electroagnetis is related to the SU() subgroup of SL(4,R) obtained using the inner autoorphiss. Siilarly the SL(,C) of gravitation ay be transfored into an equivalent subgroup by an autoorphis. This abiguity of the subgroups within G represents a syetry of the interactions. Since the noncopact generators are equivalent to space-

28 1 ENERGY OR MASS AND INTERACTION Chapter 1 tie boosts, their generated syetry ay be considered external. The internal syetry is deterined by the copact nonrotational SU() sector. It is well known in special relativity, that otion produces a relativity of electric and agnetic fields. We find, since SL(4,R) acts on the curvature, an intrinsic relativity of the unified fields, altering the nonunified fields which are seen by an observer. Given the orthonoral set corresponding to an observer, the SL(4,R) curvature ay be decoposed in ters of a base generated by the set. The quadratic ters correspond to the SL(,C) curvature and its associated Rieann curvature seen by the observer. A field naed gravitation by an observer, ay appear different to another observer. These transforations disguise interactions into each other. The algebra associates soe generators to space-tie and siultaneously to soe interactions. This appears surprising, but on a closer look this is a natural association. In an experient, changes due to an interaction generator are interpreted by an observer as tie and distance which becoe paraeters of change. Then, it is natural that a reorientation, a gyre of space-tie within the algebra corresponds to a rearrangeent of interactions. A coplete observer has the capacity to sense forces not iputable to his space-tie rieannian curvature. He senses the as nongravitational, nonrieannian, forces. This capacity ay be interpreted as the capacity to carry soe generalized charge corresponding to the nongravitational interactions. Ultra relativity is essentially interpreted as an intrinsic relativity of energy interactions. We should separate the equations with respect to the even subalgebra or subgroup as indicated by eq. (1.4.1), because this part represents the classical fields. In other words, we have the sl 1 (,C) fors, as functions of its generators i, E, a A AI i G E a + = +, (1.4.) n n a F FiI R E a + = +. (1.4.3) It should be noted that the even curvature does not just arise fro the even part of the connection because it depends on the product of odd parts. The curvature F of the abelian even connection A corresponds to the Maxwell curvature tensor in electrodynaics and obeys D* F = d * F = k * j º4 pa * j, (1.4.4) DF = df =0, (1.4.5) which are exactly the standard Maxwell s equations if we define k in ters of the fine structure constant a. The standard j.a interaction is obtained back. The curvature for R of the G connection corresponds to the Rieann curvature with torsion, in standard spinor forulation. They obey the equations D* R= k * J, (1.4.6) + DR = 0, (1.4.7)