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.


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    Title :

    Idempotent or nilpotent?


    Contributors:

    Conference:

    ICNPAA 2018 WORLD CONGRESS: 12th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences ; 2018 ; Yerevan, Armenia


    Published in:

    Publication date :

    2018-12-04


    Size :

    9 pages





    Type of media :

    Conference paper


    Type of material :

    Electronic Resource


    Language :

    English