  It is shown that by imposing the relativistic symmetries on the cutoff in
field theories one rules out all known non-perturbative regulators except the
point splitting. The relativistic cutoff dynamics is non-local in time and
thereby unstable, bringing the very existence of relativistic field theory into
question. It is argued within the context of a scalar field theory model that
the combination of the point splitting with a sufficiently fast suppression in
the field amplitude may lead to stable semiclassical dynamics and to a new kind
of realistic vacuum structure.
