Let (M,g) be a spin manifold, the n+12 particules are:
˜ψ=∑aψa⊗ea1⊗…⊗ean
with ψa spinors, and eai vectors, such that
∑a∏ieai.ψa=0
with permutations of the eai. The vectors act:
X.˜ψ=∑aX.ψa⊗ea1⊗…⊗ean
The connection is:
∇n+1/2=∇1/2⊗1+1⊗∇n
with ∇1/2 the spinorial connection.
Then, the Rarita-Schwinger operator can be defined such that:
DRS˜ψ=∑iei.∇n+1/2ei˜ψ
with (ei) an orthonormal basis of the vectors.
Is the mass of the particule the first proper value of the Rarita-Schwinger operator?