Geometric algebras in physics: Eigenspinors and Dirac theory

Abstract

The foundations of quantum theory are closely tied to a formulation of classical relativistic physics. In Clifford's geometric algebra classical relativistic physics has a spinorial formulation that is closely related to the standard Dirac equation. The algebra of physical space, APS, gives clear insight into the quantum/classical interface. Here, APS is compared to other formulations of relativistic quantum theory, especially the Dirac equation. These formulations are shown to be effectively equivalent to each other and to the standard theory, as demonstrated by establishing several isomorphisms. Dirac spinors are four-component complex entities, and so must be represented by objects containing 8 real degrees of freedom in the standard treatment (or 7 if a normalization constant is added). The relation [special characters omitted]indicates that the 8-dimensional even subalgebra [special characters omitted] of the Space-time algebra, STA is isomorphic to APS Cl 3, which is isomorphic to complex quaternions [special characters omitted]. The complex quaternions should not be confused with the biquaternions, a name sometimes used for them. The biquaternions are more generally elements of the algebra [special characters omitted]. The algebras Cl1,3 and Cl 3,1 are not isomorphic but their even sub-algebras are[1]. The Klein paradox is resolved in APS by considering Feynman's picture of antiparticles as negative energy solutions traveling backward in time. It is also shown that the algebra of physical space can naturally describe an extended version of the De Broglie-Bohm approach to quantum theory. A relativistic causal account of a spin measurement in APS is given. The Stern-Gerlach magnet acts on the eigenspinor Λ field of a charged particle in a way that is analogous to the interaction of a birefringent medium acts on a beam of light. Then we introduce a covariant interpretation of complex algebra of physical space, CAPS, the complex extension of APS. This is done to solve a problem in that the space-time inversion, PT transformation, when P is parity inversion and T time reversal, although it is a proper transformation is not physical, yet, it has the form of a physical rotation in the traditional Dirac theory. The CAPS form of the PT transformation does not have the form of a physical rotation. A further problem is that the explicit form of the time inversion, T and charge conjugation, C transformations depends on the matrix representation chosen. A representation-independent form would be more fundamental. In CAPS, the complex extension of APS these problems are resolved.