Generators of the Poincar\'e Group for arbitrary tensors and spinor-tensors

Abstract

In this work, we systematically derive explicit expressions for the Poincar\'e Group generators on arbitrary-rank tensors and spinor-tensors in $D=3+1$ and $D=2+1$ spacetimes, thus generalizing previous works in the literature for the groups $ISO(3,1)$ and $ISO(2,1)$. From the Casimir eigenvalue equations, we demonstrate in a model-independent way the Fierz-Pauli constraints for massive particles for spins $\mathfrak{s}=5/2$, 3, and 4 in $D=3+1$ and helicities $\alpha=5/2$, 3, and 4 in $D=2+1$ dimensions. We also comment on the demonstration of the Fierz-Pauli constraints for the general case of arbitrary spin (helicity).