Spin-1/2 "bosons'' with mass dimension 3/2 and fermions with mass dimension 1 cannot represent physical particle states

We delve into the first principles of quantum field theory to prove that the so-called spin-1/2 ''bosons'' and the fermions with mass dimension 1, including ELKO, cannot represent physical particle states with spin $1/2$. Specifically, we first demonstrate that both aforementioned fields are not invariant under rotational symmetry, which implies that the particles created for these fields are not eigenstates of the spin operator in the $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ representation of the Lorentz group, nor is it possible to construct a Hamiltonian density scalar under the rotational group from them. Furthermore, following Weinberg's approach to local causal fields, we prove that regardless of any discrete symmetry or adjoint structure, the relativistic fields in the $(\frac{1}{2}, 0) \oplus (0,\frac{1}{2})$ representation satisfy the Fermi-Dirac statistics in complete agreement with the well-established spin-statistics theorem and experimental results.Comment: 9 page