Preface: This is a collection of papers intended as contributions
to the development and application of a Universal Geometric Calculus (UGC).
Many of them adopt a polemical tone, because a UGC is impossible
without revising common mathematical practices and opinions.
Of course, entrenched opinions are seldom changed, but the case for a UGC must be
made for those who are willing to listen.
Section I is concerned with purely algebraic matters. The sequence of papers on projective geometry, linear algebra and Lie groups make important improvements and extensions of the concepts and methods in the book Clifford Algebra to Geometric Calculus (CA to GC). Applications to computational geometry, robotics and computer graphics are discussed in another section. The third paper is especially important, as it provides a general framework for computations in linear algebra without matrices. It thus provides a new framework for Lie groups and their representations.
Section II is concerned with the extension of CA to GC, especially vector derivatives and directed integrals. The basic ideas were originally set forth in the papers Multivector Calculus and Multivector Functions -- subsequently elaborated in CA to GC. These ideas provide the foundation for "Clifford Analysis," a new branch of mathematics emerging during the last two decades. The first paper in Section II explains that the crucial synthesis of Clifford algebra with differential forms that opens this branch was made independently by several investigators. However, the crux of the synthesis is seldom understood, so more commentary is needed. It is no small irony that Clifford algebra and differential forms emerged from the work of Grassmann, but are combined in a kind of hybrid in most accounts of Clifford analysis. In line with Grassmann's vision, Geometric Calculus reconstructs the theory of differential forms in terms of GA from the ground up. The consequence, of course, is greater simplicity, clarity and efficiency. The paper on Hamiltonian mechanics opens up a new approach to the subject. An important by-product is that the characterization of symplectic relations given there provided a key to the whole development of Lie groups as spin groups.
Section III explores implications of GA for the interpretation of quantum mechanics. The basic fact is that GA reveals a hidden geometric structure in the Dirac electron theory, including a geometric intepretation of the unit imaginary that links it to spin. A "zitterbewegung interpretation" of the Dirac theory is proposed to explain the geometric structure as an expression of particle kinematics. It is argued that a point particle model of the electron is completely consistent with all features of the Dirac theory. Analogous arguments by Bohm and others, that a point particle interpretation is completely consistent with Schroedinger's theory have gained wider currency among physicists recently. Of course, the Dirac theory tells us a lot more about the electron than Schroedinger's theory. Several papers speculate on the possibility of a deeper particle theory of the electron to which the Dirac theory gives clues.
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