  It is argued that the familiar algebra of the non-commutative space-time with
$c$-number $\theta^{\mu\nu}$ is inconsistent from a theoretical point of view.
Consistent algebras are obtained by promoting $\theta^{\mu\nu}$ to an
anti-symmetric tensor operator ${\hat\theta}^{\mu\nu}$. The simplest among them
is Doplicher-Fredenhagen-Roberts (DFR) algebra in which the triple commutator
among the coordinate operators is assumed to vanish. This allows us to define
the Lorentz-covariant operator fields on the DFR algebra as operators diagonal
in the 6-dimensional $\theta$-space of the hermitian operators,
${\hat\theta}^{\mu\nu}$. It is shown that we then recover Carlson-Carone-Zobin
(CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no
need of compactification of the extra 6 dimensions. It is also pointed out that
a general argument concerning the normalizability of the weight function in the
Lorentz metric leads to a division of the $\theta$-space into two disjoint
spaces not connected by any Lorentz transformation so that the CCZ covariant
moment formula holds true in each space, separately. A non-commutative
generalization of Connes' two-sheeted Minkowski space-time is also proposed.
Two simple models of quantum field theory are reformulated on $M_4\times Z_2$
obtained in the commutative limit.
