  We discuss a Lorentz covariant space-time uncertainty relation, which agrees
with that of Karolyhazy-Ng-van Dam when an observational time period delta t is
larger than the Planck time lp. At delta t < lp, this uncertainty relation
takes roughly the form delta t delta x > lp^2, which can be derived from the
condition prohibiting the multi-production of probes to a geometry. We show
that there exists a minimal area rather than a minimal length in the
four-dimensional case. We study also a three-dimensional free field theory on a
non-commutative space-time realizing the uncertainty relation. We derive the
algebra among the coordinate and momentum operators and define a
positive-definite norm of the representation space. In four-dimensional
space-time, the Jacobi identity should be violated in the algebraic
representation of the uncertainty relation.
