Abstract: 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.
Abstract: A linear form with an N-elements basis set {e i ; i = 1,...,N} generates an algebra which is that of multivectors, provided some commutation relation is defined to give a meaning to the outer product of the basis vectors. If, moreover, an inner product of sets of K basis vectors is also introduced, for a mapping producing a 0-form, a geometric algebra is obtained. The algebra has thus two basic numbers to define its dimension: the dimension N of the basis set and the dimension K of the number of elements to be multiplied together to obtain a scalar. If the dimension K refers to the order of the power of [e i ] K to obtain the scalar we will say that we have a K-atic algebra, the best known example is when the scalar form is a quadratic expression; these algebras are said to have a metric which in general is either diagonal or at least symmetric. Otherwise if the dimension K refers again to the number of different basis vectors to be multiplied together in (with j ≠i and in general all subindexes different) then we obtain a simplectic algebra where the best known case is also when K = 2 and the metric in this case is antisymmetric. In the present paper we define these sets of algebras, give the commutation relations for the algebras with a K-atic scalar form and relate the results to the best known examples of current use in the literature.
Mathematics Subject Classification (2000). 11E88 - 15A66
PACS. 02.10.De - 02.10.Ud - 02.10.Xm
Abstract: Gravitation can be casted as a physical theory by itself. The steps are: to define the interacting bodies, to define
the interaction and its properties, to define the energies and forces involved, to show the self consistency of the
mathematical structure involved, and a discussion of the (proportionality) relation between gravitational and
inertial mass. The mathematical frame for this development is a 5-D quadratic space reference manifold: space,
time and action (START). The principles, that is concepts defined through their use in the theory, are presented.
