quantum mechanical Dirac theory of the electron
by
Adewole Kayode Ajileye
Institute of Applied Mathematics

Geometric Algebra in Quantum Mechanics These papers analyze the quantum mechanical Dirac theory of the electron with respect to its geometric structure as revealed by reformulation in terms of Spacetime Algebra. The main result is that the Dirac wave function psi can be decomposed into the invariant operator form while the unit imaginary in the Dirac equation is necessarily identified with electron spin. This striking result was first derived [1] from a formulation in the book STA, which, incidentally, already showed that imaginary scalars are superfluous in the Dirac theory. Alternative derivations more directly related to the standard matrix formulation are given in [3] and an appendix to [2]. The method employed in [2] makes it transparently clear that the socalled "Fierz identities for bilinear covariants" are trivial consequences of the above invariant form for the wave function. Paper [2] provides a compact and complete formulation and analysis of local conservation laws in the one-particle Dirac theory. Comparable derivations by standard matrix and tensor methods are nearly ten times longer, as can be seen in the work of Takabayashi referenced in [2]. An analogous treatment of local conservation laws in Schroedinger's theory plays an essential role in the Bohmian interpretation of quantum mechanics. Paper [2] makes explicit the complications of extending Bohm's approach to relativistic QM.

The nonrelativistic treatment of local conservation laws including spin is given in [5] and further discussed in [6]. The main message of these papers is that standard interpretations of quantum mechanics (including Bohm's) fail to take account of the relation between spin and imaginary numbers that is inherent in Dirac theory. The necessary connections between Dirac, Pauli and Schroedinger theories are derived in [4], where inconsistencies among standard interpretations are pointed out.

Paper [3] emphasizes the point that common interpretations of Pauli and Dirac matrices as quantum mechanical operators are unjustified and ill-conceived. GA makes it absolutely clear that these matrices represent directions in space and spacetime, with no implications about spin whatsoever. Indeed, contrary to Dirac's claim and popular belief, spin is not introduced into the Dirac theory by gamma matrices but by the definition of energy-momentum operators.

Date received: January 13, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cake-14.