The Berwald-Moor metric in nilpotent Dirac spinor space

Abstract

The nilpotent version of the Dirac equation can be constructed on the basis of the algebra of a double vector space or complexified double quaternions. This algebra is isomorphic to the standard gamma matrix algebra, with 64 units that can be produced by just 5 generators. The H4 algebra used in the Berwald-Moor metric is a distinct subalgebra of this 64-part algebra. The creation of the 5 generators requires the rotation symmetry of one of the two-component vector spaces to be preserved while the other is broken. It is convenient to identify the respective spaces as an observable real space and an unobservable ‘vacuum’ space, with corresponding physical properties. In combination, the 5 generators produce a nilpotent structure which can be identified as a fermionic wavefunction or solution of the Dirac equation. The spinors required to generate the 4 components of the wavefunction can be derived from the first principles and have exactly the same form as the four components of the Berwald- Moor metric. They also incorporate the units of the H4 algebra in an identical way. The spinors produce a zero product which can be interpreted in terms of a fermionic singularity arising from the distortion introduced into the vacuum (or spinor) space by the application of a nilpotent condition.