This paper surveys the application of `geometric algebra’ to the physics of electrons. The mathematical ideas underlying geometric algebra were discovered jointly by Clifford and Grassmann in the late 19th century. Their discoveries were made during a period in which … Continued
A new gauge-theory description of gravity is presented, employing gauge fields in a flat background spacetime. These fields ensure that all physical relations are independent of the position and orientation of the matter fields in this background. The language of … Continued
Clifford algebras have been studied for many years and their algebraic properties are well known. In particular, all Clifford algebras have been classified as matrix algebras over one of the three division algebras. But Clifford Algebras are far more interesting … Continued
This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a `geometric product’ of vectors in 2- and 3-dimensional space provides precise geometrical interpretations of the imaginary … Continued
We present a reformulation of Grassmann calculus in terms of geometric algebra – a unified language for physics based on Clifford algebra. In this reformulation, Grassmann generators are replaced by vectors, so that every product of generators has a natural … Continued
It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available to simplify the analysis and applications … Continued
This paper employs the ideas of geometric algebra to investigate the physical content of Dirac’s electron theory. The basis is Hestenes’ discovery of the geometric significance of the Dirac spinor, which now represents a Lorentz transformation in spacetime. This transformation … Continued
A new calculus, based upon the multivector derivative, is developed for Lagrangian mechanics and field theory, providing streamlined and rigorous derivations of the Euler-Lagrange equations. A more general form of Noether’s theorem is found which is appropriate to both discrete … Continued
The spacetime algebra (STA) is the natural, representation-free language for Dirac’s theory of the electron. Conventional Pauli, Dirac, Weyl and Majorana spinors are replaced by spacetime multivectors, and the quantum sigma- and gamma-matrices are replaced by two-sided multivector operations. The … Continued
We present a new treatment of 2-spinors and twistors, using the spacetime algebra. The key rôle of bilinear covariants is emphasized. As a by-product, an explicit representation is found, composed entirely of real spacetime vectors, for the Grassmann entities of … Continued