The Rationale for the Present Book Perhaps the most critical problem facing present-day particle physicistsis to delineate the relationship between classical and quantum systems. This relationship has many facets. Particle-waveduality is one. The concept of the point particle is another. And theconcept of particle mass is yet another. The electron, as the lightest of the charged particles, represents a fundamental "ground state",and many of the essential problems in the murky area between the domainsofclassical and quantum physics can be brought into focus by studyingjust this one particle. Thus the present book is centered on questions that arise in connection with the electron, and in particular with its mass, which has remained an unsolved, and indeed almost unexplored, mystery. Each student ofphysics, beginner and professional alike, has to fashion for himselfa way of thinking about the electron. If, after reading this book, the reader views this topic somewhat differently than before, the efforts of the author will have been amply rewarded. When physicists were confronted with the properties of the electron, they made a conceptualleap into the unknown: they concluded that the electron does not obey classical laws with respect to mechanics (as connected to the spin of the electron), and also with respect to electrodynamics (as connected to the magnetic moment of the electron).

This book offers a new look at the electron. It was the first elementary particle discovered, is probably one of the simplest, and yet is possibly one of the most misunderstood. The author presents here a straightforward classical model that accurately reproduces the main spectroscopic features of the electron, and also its principal quantum aspects. The key to this model is the relativistically spinning sphere, RSS, which has been clamoring for recognition for decades. Although its electrical charge is point-like, the electron itself is Compton-sized, and is composed mainly of non-electromagnetic "mechanical" matter. The bridge between the electron and the other elementary particles is provided by the fine structure constant alpha 1/137, as manifested in the factor-of-137 spacings between the classical electron radius, electron Compton radius, and Bohr orbit radius. An expanded form of the constant alpha leads to equations that define the transformation of electromagnetic energy into electron mass/energy, and, via the electron doorway, to the formation of higher-mass lepton and hadron ground states. An alpha-quantized mass-generation grid extends accurately from the electron all the way to the top quark t, and leads to a corresponding alpha-quantized particle lifetime grid. The mathematics used in these studies is standard, and the calculations are guided by fits to the experimental elementary particle data. This book is written for all scientists who are interested in recent developments in fundamental particle physics.

The electron, discovered in 1897, was found to be a constituent of all atoms. While the nucleus of the atom remains fixed, the electrons are free to move with different amounts of energy. When supplied with more energy, by physical or mechanical means, light is produced when the original energy state is reached. Electrons can easily be removed altogether from the atom as in the case of electric current. This has given rise to our electrical and electronic industries. The associated magnetic field allowed motors and dynamos to be developed. Rapid movement of electrons results in the production of electromagnetic waves, from the longest wavelengths (radio waves) to the shortest wavelengths (gamma rays). This has had a huge impact on our lives in the fields of medicine and telecommunications. A beam of electrons can be directed in the same way as a beam of light. As light can show wave/particle duality so can an electron beam. Its measured wavelength is about the same as X-rays. This means electrons can be diffracted. The famous 'double-slit' experiment where a single electron appears to 'interfere with itself' cannot be explained by classical physics and so we enter the strange world of quantum mechanics. The birth of the quantum computer is not far away and will be much faster than existing computers. Finally, all chemical reactions are the result of electron movement between reactants.

The Rationale for the Present Book Perhaps the most critical problem facing present-day particle physicistsis to delineate the relationship between classical and quantum systems. This relationship has many facets. Particle-waveduality is one. The concept of the point particle is another. And theconcept of particle mass is yet another. The electron, as the lightest of the charged particles, represents a fundamental "ground state",and many of the essential problems in the murky area between the domainsofclassical and quantum physics can be brought into focus by studyingjust this one particle. Thus the present book is centered on questions that arise in connection with the electron, and in particular with its mass, which has remained an unsolved, and indeed almost unexplored, mystery. Each student ofphysics, beginner and professional alike, has to fashion for himselfa way of thinking about the electron. If, after reading this book, the reader views this topic somewhat differently than before, the efforts of the author will have been amply rewarded. When physicists were confronted with the properties of the electron, they made a conceptualleap into the unknown: they concluded that the electron does not obey classical laws with respect to mechanics (as connected to the spin of the electron), and also with respect to electrodynamics (as connected to the magnetic moment of the electron).

Presenting a realistic interpretation of quantum mechanics and, in particular, a realistic view of quantum waves, this book defends, with one exception, Schrodinger's views on quantum mechanics. Johansson goes on to defend the view that the collapse of a wave function during a measurement is a real physical collapse of a wave and argues that the collapse is a consequence of quantisation of interaction. Lastly Johansson argues for a revised principle of individuation in the quantum domain and that this principle enables a sort of explanation of non-local phenomena.

This book brings together papers by a number of authors. More than ten different models of the electron are presented and more than twenty models are discussed briefly. Thus, the book gives a complete picture of contemporary theoretical thinking (traditional and new) about the physics of the electron.

In the first century after its discovery, the electron has come to be a fundamental element in the analysis of physical aspects of nature. This book is devoted to the construction of a deductive theory of the electron, starting from first principles and using a simple mathematical tool, geometric analysis. Its purpose is to present a comprehensive theory of the electron to the point where a connection can be made with the main approaches to the study of the electron in physics. The introduction describes the methodology. Chapter 2 presents the concept of space-time-action relativity theory and in chapter 3 the mathematical structures describing action are analyzed. Chapters 4, 5, and 6 deal with the theory of the electron in a series of aspects where the geometrical analysis is more relevant. Finally in chapter 7 the form of geometrical analysis used in the book is presented to elucidate the broad range of topics which are covered and the range of mathematical structures which are implicitly or explicitly included. The book is directed to two different audiences of graduate students and research scientists: primarily to theoretical physicists in the field of electron physics as well as those in the more general field of quantum mechanics, elementary particle physics, and general relativity; secondly, to mathematicians in the field of geometric analysis.

"I cannot define coincidence [in mathematics]. But 1 shall argue that coincidence can always be elevated or organized into a superstructure which perfonns a unification along the coincidental elements. The existence of a coincidence is strong evidence for the existence of a covering theory. " -Philip 1. Davis [Dav81] Alluding to the Thomas gyration, this book presents the Theory of gy rogroups and gyrovector spaces, taking the reader to the immensity of hyper bolic geometry that lies beyond the Einstein special theory of relativity. Soon after its introduction by Einstein in 1905 [Ein05], special relativity theory (as named by Einstein ten years later) became overshadowed by the ap pearance of general relativity. Subsequently, the exposition of special relativity followed the lines laid down by Minkowski, in which the role of hyperbolic ge ometry is not emphasized. This can doubtlessly be explained by the strangeness and unfamiliarity of hyperbolic geometry [Bar98]. The aim of this book is to reverse the trend of neglecting the role of hy perbolic geometry in the special theory of relativity, initiated by Minkowski, by emphasizing the central role that hyperbolic geometry plays in the theory.

The book consists of two Volumes. The first (the preceding volume) is devoted to the general nonlinear theory of the hierarchical dynamic oscillative–wave systems. This theory has been called the theory of hi- archical oscillations and waves. Here two aspects of the proposed theory are discussed. The first aspects concern the fundamental nature and the basic c- cepts and ideas of a new hierarchical approach to studying hierarchical dynamic systems. A new hierarchical paradigm is proposed as a - sis of a new point of view of such types of systems. In turn, a set of hierarchical principles is formulated as the fundamental basis of this paradigm. Therein the self-resemblance (holographic) principle plays a key role here. An adequate mathematic description (factorization) of the proposed paradigm is carried out. The concepts of structural and dynamic (functional) operators are put into the basis of this descr- tion. Electrodynamics is chosen as a convenient basis for an obvious demonstration of some key points of the proposed new theory. The second aspect has a purely mathematical nature. It is related to the form of factorization (i.e., mathematical description) of hier- chical types of dynamic models, and discussion of the methods of their mathematical analysis. A set of the hierarchical asymptotic analytical– numerical methods is given as an evidence of the practical effectiveness of the proposed version of hierarchical theory.

From a historical point of view, the theory we submit to the present study has its origins in the famous dissertation of P. Finsler from 1918 ([Fi]). In a the classical notion also conventional classification, Finsler geometry has besides a number of generalizations, which use the same work technique and which can be considered self-geometries: Lagrange and Hamilton spaces. Finsler geometry had a period of incubation long enough, so that few math ematicians (E. Cartan, L. Berwald, S.S. Chem, H. Rund) had the patience to penetrate into a universe of tensors, which made them compare it to a jungle. To aU of us, who study nowadays Finsler geometry, it is obvious that the qualitative leap was made in the 1970's by the crystallization of the nonlinear connection notion (a notion which is almost as old as Finsler space, [SZ4]) and by work-skills into its adapted frame fields. The results obtained by M. Matsumoto (coUected later, in 1986, in a monograph, [Ma3]) aroused interest not only in Japan, but also in other countries such as Romania, Hungary, Canada and the USA, where schools of Finsler geometry are founded and are presently widely recognized.