The concept of trajectory for a quantum particle is an elusive
one: in the path-integral formulation it is possible to prove
that path which are continuous but nowhere differentiable
give the dominant contribution; Abbot and Wise also proved
that this properties can be formulated in terms of a path
with Hausdorff dimension 2.
Starting from the functional
Schroedinger equation for a string that we developed elsewhere,
we show that an analogous properties holds for the world-sheet
of a string: the fact that nowhere differentiable world-sheets
give the dominant contribution to the path-integral sum for
the string is then interpreted in terms of a Hausdorff dimension
3 for the string world-sheet. We show how it is possible to
understand the classical -> quantum transition in
terms of the resolution of the detecting apparatus and we
interpret these considerations in connection with an uncertainty
principle for the string. This paves the way to the discussion
of the small scale properties of spacetime, which is the subject
of another project.
