  Let $\Theta$ be the Wigner time reversal operator for spin half and let
$\phi$ be a Weyl spinor. Then, for a left-transforming $\phi$, the construct
$\zeta_\lambda \Theta \phi^\ast$ yields a right-transforming spinor. If
instead, $\phi$ is a right-transforming spinor, then the construct $\zeta _\rho
\Theta \phi^\ast$ results in a left-transforming spinor ($\zeta_{\lambda,\rho}$
are phase factors). This allows us to introduce two sets of four-component
spinors. Setting $\zeta_\lambda$ and $\zeta_\rho$ to $\pm i$ render all eight
spinors as eigenspinor of the charge conjugation operator~$\mathcal{C}$ (called
ELKO). This allows us to introduce two quantum fields. A calculation of the
vacuum expectation value of the time-ordered product of the fields and their
adjoints reveals the mass dimension of the fields to be one. Both fields are
local in the canonical sense of quantum field theory. Interestingly, one of the
fields is fermionic and the other bosonic. The mass dimension of the introduced
fermionic fields and the matter fields of the Standard Model carry an intrinsic
mismatch. As such, they provide natural darkness for the new fields with
respect to the Standard Model doublets. The statistics and locality are
controlled by a set of phases. These are explicitly given. Then we observe that
in $p_\mu p^\mu = m^2$, Dirac took the simplest square root of the $4\times 4$
identity matrix $I$ (in $I \times m^2 $, while introducing $\gamma_\mu p^\mu$
as the square root of the left hand side of the dispersion relation), and as
such he implicitly ignored the remaining fifteen. When we examine the remaining
roots, we obtain additional bosonic and fermionic dark matter candidates of
spin half. We point out that by early nineteen seventies, Dirac had suspected
the existence of spin half bosons, in the same space as his fermions. Abstract
truncated.
