I only did one QFT course so there may be something obvious I'm missing. Studying quantum fields, some of the easiest examples are bosonic scalar $\phi$ and vector $A_\mu$ fields, and fermionic spinor $\psi$ field. On the book from Peskin and Schroeder, it also mentions the (probable) bosonic tensor $g_{\mu\nu}$ for gravity. Are there theories regarding higher-order fermionic fields, for example with spin 3/2? If not, why not? If yes, how would one build such a theory?
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If not, why not?
Because we have not yet observed a fundamental particle with $3/2$ spin.
If yes, how would one build such a theory?
SuperCiocia
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3/2 spin particles are studied in the framework of Supergravity. So textbooks and lectures on Supergravity will certainly contain a study on 3/2-fermions since the supersymmetric partner(s) of the graviton, the gravitino(s) are 3/2-spin particles. On the other hand, as such particles are not part of the Standard Model (or variants of it) and Supergravity is not an experimentally confirmed theory, most textbooks and lectures on QFT don't study them (but sometimes in exercises). Actually, there is a book without referring to Supergravity has a section on higher spin particles, that is Landau & Lifshitz volume IV on Relativistic Quantum Theory. But anyway, if you wish to know more about 3/2-fermions, learn something about Supergravity, for instance the book from Freedman & Van Proeyen is a good source.
Lagrangian
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There is a theorem by Weinberg that spin-1 massless particles can only enter an interacting theory as gauge bosons and spin-2 massless particles as gravitons. This was extended by Grisaru Pendleton and Nieuwenhuizen to show that the only renormalizable interactions for a light spin-3/2 particle are via the supercurrent of supergravity. So if you don't want supersymmetry and gravity, you can't have light fundamental spin 3/2 particles. https://inspirehep.net/literature/109586
Guy Moore
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