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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