Complex Geometry and Dirac Equation

S. De Leo, WA Rodrigues, J. Vaz (Dpt Fisica,Lecce and IMECC-UNICAMP)

(Submitted on 17 May 1999)

Abstract: Complex geometry represents a fundamental ingredient in the formulation of the Dirac equation by the Clifford algebra. The choice of appropriate complex geometries is strictly related to the geometric interpretation of the complex imaginary unit $i=\sqrt{-1}$. We discuss {\em two} possibilities which appear in the multivector algebra approach: the $\sigma_{123}$ and $\sigma_{21}$ complex geometries. Our formalism permits to perform a set of rules which allows an immediate translation between the complex standard Dirac theory and its version within geometric algebra. The problem concerning a double geometric interpretation for the complex imaginary unit $i=\sqrt{-1}$ is also discussed.

Comments:	11 pages, RevTex
Subjects:	High Energy Physics – Theory (hep-th)
Journal reference:	Int.J.Theor.Phys. 37 (1998) 2479
Report number:	IMECC RP16/98
Cite as:	arXiv:hep-th/9905124v1

Pauli identity and vector products: geometric, dot, wedge, and cross

Let and be two vectors in the Clifford algebra :




The geometric (direct) product of and is





Using the orthonormality axiom, we can show that

,

where


.

In terms of the imaginary number (unit trivector) , we may rewrite the wedge product as

,

where



is the cross product of and . Substituting Eq. (5) back to Eq. (3), we obtain the familiar Pauli identity:

.