  Inspired by Polyakov's original formulation of quantum Liouville theory
through functional integral, we analyze perturbation expansion around a
classical solution. We show the validity of conformal Ward identities for
puncture operators and prove that their conformal dimension is given by the
classical expression. We also prove that total quantum correction to the
central charge of Liouville theory is given by one-loop contribution, which is
equal to 1. Applied to the bosonic string, this result ensures the vanishing of
total conformal anomaly along the lines different from those presented by KPZ
and Distler-Kawai.
