  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.
