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It is well-known that coupling a spin $\frac32$-field to a gravitational orelectromagnetic background leads to potential problems both in the classicaland in the quantum theory. Various solutions to these problems have beenproposed so far, which are all restricted to a limited class of backgrounds. Onthe other hand, negative results for general gravitational backgrounds havebeen reported only for a limited set of couplings to the background to date.Hence, to our knowledge, a comprehensive analysis of all possible couplings tothe gravitational field and general gravitational backgrounds includingoff-shell ones has not been performed so far. In this work we analyse whetherit is possible to couple a spin $\frac32$-field to a gravitational field insuch a way that the resulting quantum theory is consistent on arbitrarygravitational backgrounds. We find that this is impossible as all couplingsrequire the background to be an Einstein spacetime for consistency. Thisenforces the widespread belief that supergravity theories are the onlymeaningful models which contain spin $\frac32$ fields as in these models suchrestrictions of the gravitational background appear naturally as on-shellconditions.

A No-Go Theorem for the Consistent Quantization of Spin 3/2 Fields on General Curved Spacetimes