  In previous report (J. Phys. A (1997) 30 4019-4029), I showed that the Dirac
field confined in a surface immersed in $R^3$ by means of a mass type potential
is governed by the Konopelchenko-Kenmotsu-Weierstrass-Enneper equation. In this
article, I quantized the Dirac field and calculated the gauge transformation
which exhibits the gauge freedom of the parameterization of the surface. Then
using the Ward-Takahashi identity, I showed that the expectation value of the
action of the Dirac field is expressed by the Willmore functional and area of
the surface.
