Evolution of Fixed-End Strings and the Off-Shell Disk Amplitude

An exact integral expression is found for the amplitude of a Bosonic string with ends separated by a fixed distance $R$ evolving over a time $T$ between arbitrary initial and final configurations. It is impossible to make a covariant subtraction of a covariant quantity which would render the amplitude non-zero. It is suggested that this fact (and not the tachyon) is responsible for the lack of a continuum limit of regularized random-surface models with target-space dimension greater than one. It appears consistent, however, to remove this quantity by hand. The static potential of Alvarez and Arvis $V(R)$ is recovered from the resulting finite amplitude for $R>R_c$. For $R<R_c$, we find $V(R)=-\mathrm{\infty }$, instead of the usual tachyonic result. A rotation-invariant expression is proposed for special cases of the off-shell disk amplitude. {\it None} of the finite amplitudes discussed are Nambu or Polyakov functional integrals, except through an unphysical analytic continuation. We argue that the Liouville field does not decouple in off-shell amplitudes, even when the space-time dimension is twenty-six.