  We analyze properties of the Sp(2M) conformally invariant field equations in
the recently proposed generalized $\half M(M+1)$-dimensional space-time $\M_M$
with matrix coordinates. It is shown that classical solutions of these field
equations define a causal structure in $\M_M$ and admit a well-defined
decomposition into positive and negative frequency solutions that allows
consistent quantization in a positive definite Hilbert space. The effect of
constraints on the localizability of fields in the generalized space-time is
analyzed. Usual d-dimensional Minkowski space-time is identified with the
subspace of the matrix space $\M_M$ that allows true localization of the
dynamical fields. Minkowski coordinates are argued to be associated with some
Clifford algebra in the matrix space $\M_M$. The dynamics of a conformal scalar
and spinor in $\M_2$ and $\M_4$ is shown to be equivalent, respectively, to the
usual conformal field dynamics of a scalar and spinor in the 3d Minkowski
space-time and the dynamics of massless fields of all spins in the 4d Minkowski
space-time. An extension of the electro-magnetic duality transformations to all
spins is identified with a particular generalized Lorentz transformation in
$\M_4$. The M=8 case is shown to correspond to a 6d chiral higher spin theory.
The cases of M=16 (d=10) and M=32 (d=11) are discussed briefly.
