  Conformal symmetry is taken as an attribute of theories of massless fields in
manifolds with specific dimensionalities. This paper shows that this is not an
absolute truth; it is a consequence of the mathematical representation used for
the physical interactions. It introduces a new kind of representation where the
propagation of massive (invariant mass) and massless interactions are unifiedly
described by a single conformally symmetric Green's function. Sources and
fields are treated at a same footing, symmetrically, as discrete fields - the
fields in this new representation - fields defined with support on straight
lines embedded in a (3+1)-Minkowski manifold. The discrete field turns out to
be a point in phase space. It is finite everywhere. With a finite number of
degrees of freedom it does not share the well known problems faced by the
standard continuous formalism which can be retrieved from the discrete one by
an integration over a hypersurface. The passage from discrete to continuous
fields illuminates the physical meaning and origins of their properties and
problems. The price for having massive discrete field with conformal symmetry
is of hiding its mass and time-like velocity behind its non-constant
proper-time.
