Field equations from Killing spinors
Abstract
From the Killing spinor equation and the equations satisfied by their bilinears we deduce some well known bosonic and fermionic field equations of mathematical physics. Aside from the trivially satisfied Dirac equation, these relativistic wave equations in curved spacetimes respectively are Klein-Gordon, Maxwell, Proca, Duffin-Kemmer-Petiau, K\"ahler, twistor and Rarita-Schwinger equations. This result shows that, besides being special kinds of Dirac fermions, Killing fermions can be regarded as physically fundamental. For the Maxwell case the problem of motion is analysed in a reverse manner with respect to the works of Einstein-Groemer-Infeld-Hoffmann and Jean Marie Souriau. In the analysis of the gravitino field a generalised 3- rule is found which is termed the vanishing trace constraint.
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