Let $(M,g)$ be a spin manifold, the $n+\frac{1}{2}$ particules are:

$$\tilde \psi =\sum_a \psi^a \otimes e_1^a \otimes \ldots \otimes e_n^a $$

with $\psi^a $ spinors, and $e_i^a$ vectors, such that

$$\sum_a \prod_i e_i^a . \psi^a =0$$

with permutations of the $e_i^a$. The vectors act:

$$X.\tilde \psi=\sum_a X.\psi^a \otimes e_1^a \otimes \ldots \otimes e_n^a$$

The connection is:

$$\nabla^{n+1/2}=\nabla^{1/2} \otimes 1+ 1\otimes \nabla^n$$

with $\nabla^{1/2}$ the spinorial connection.

Then, the Rarita-Schwinger operator can be defined such that:

$${\cal D}^{RS} \tilde \psi = \sum_i e_i .\nabla^{n+1/2}_{e_i} \tilde \psi$$

with $(e_i)$ an orthonormal basis of the vectors.

Is the mass of the particule the first proper value of the Rarita-Schwinger operator?