A Laboratory-Frame View of the Dirac Electron Field Using Geometric Algebra

Gene McClellan, Applied Reseach Associates, Inc., USA

The geometric algebra and calculus of 3-D Euclidean space in a laboratory frame is sufficient to formulate and solve the Dirac equation as well as the Maxwell equations. The Dirac electron field at a given point in space is expressed as a linear combination of the algebraic basis elements of the geometric algebra of the tangent space at that point. The eight independent basis elements of the geometric algebra are sufficient to substitute for the eight independent quantities in the four complex components of the traditional Dirac spinor, providing a geometric interpretation of the Dirac field as first proposed by Hestenes. Most authors have worked with covariant formulations of the Maxwell and Dirac equations expressed in spacetime algebra (STA); however, a spacetime split of these equations yields equivalent equations for any given laboratory frame. Expressing the Dirac field in the laboratory frame greatly facilitates visualization of solutions of the Dirac equation and provides insight into the nature of spin up vs. spin down states and the distinction between electron and positron states. This presentation will include animations illustrating the Dirac fields for states at rest. Comparison with electromagnetic plane waves suggests physical insight into electron-positron annihilation, a basic process of the Standard Model.