# Field equations from Killing spinors

## TL;DR

This paper derives various fundamental bosonic and fermionic field equations from Killing spinor equations, highlighting the physical significance of Killing fermions and analyzing their implications in curved spacetimes.

## Contribution

It demonstrates that Killing spinors lead to well-known field equations and introduces the vanishing trace constraint in gravitino field analysis.

## Key findings

- Killing spinors generate Klein-Gordon, Maxwell, Proca, and other equations.
- Killing fermions are physically fundamental beyond Dirac fermions.
- A generalized vanishing trace constraint is identified for gravitino fields.

## 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\"{a}hler, 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 $3-\psi$ rule is found which is termed the vanishing trace constraint.