Generalized Dirac Equation for All Fermions?

I don't have an answer, but I thought it was the case that the only fundamental fermions appearing in any current theory is spin-1/2 and spin-3/2 (which appears in supergravity). Higher-spin fermions are not used for anything, as far as I know. Of course, a composite particle can have an arbitrarily large spin.

 

The equivalent equation for spin-3/2 particles is the Rarita-Schwinger equation. I don't know if there is any sort of general equation to cover all half-integer spins.

 

It can be done for any particular value of the spin; Weinberg's QFT vol.1 explains how, in general, this works. Also, in string theory, you get infinite towers of fields with increasing spin (and mass).

 

The experimental part is an issue, there's no gravitino except on paper so far. Spin 5/2, 7/2, ... can't exist on paper, apparently, even though the general spin wave equation is as old as 1936 and the article by Dirac in PRSL. Then Pauli + Fierz, Bhabha, Duffin + Kemmer and last but not least Gel'fand and Yaglom.