The Dirac algebra is widely recognized as a Clifford algebra and there have been numerous representations of the Dirac equation using a single Cl(3,0) for the momentum term while retaining the conventional gamma matrices to separate the various terms in the equation. It is suggested here, however, that a particularly powerful result occurs when we use a double Clifford algebra to remove the whole of the gamma matrix construction. With a relatively simple transformation the equation then takes on a form in which the spinor part of the wavefunction can be interpreted as either idempotent or nilpotent. The latter case creates a universal form of the Dirac spinor for a fermion in any state and subject to any kind of force. Many things then become immediate consequences of the algebraic structure, including Pauli exclusion, bosons, baryons, C. P and T transformations, creation and annihilation operators, vacuum states, the structure of the equation itself and the basic symmetries of the Standard Model. The relative simplicity with which this is accomplished suggests that the most ’natural’ representation of the Dirac particle is not an idempotent or minimal left ideal – even though these can be extracted from the same equation – but a nilpotent.
Idempotent or nilpotent?
ICNPAA 2018 WORLD CONGRESS: 12th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences ; 2018 ; Yerevan, Armenia
AIP Conference Proceedings ; 2046 , 1
2018-12-04
9 pages
Conference paper
Electronic Resource
English
Algebraic description of data embedding basing on idempotent image transformations
British Library Online Contents | 2009
|Corrections to "Use of Idempotent Matrices to Validate Linear Systems Software"
Online Contents | 1995
|Solution of an Optimal Sensing and Interception Problem Using Idempotent Methods
Springer Verlag | 2013
|