  By extending the original Anderson singular gauge transformation for static
vortices to two mutual flux-attaching singular gauge transformations for moving
vortices, we derive an effective action describing the zero temperature quantum
phase transition from d-wave superconductor to underdoped regime. Neglecting
the charge fluctuation first, we find that the mutual statistical interaction
is exactly marginal. In the underdoped regime, the quasi-particles are
described by 2+1 dimensional QED; in the superconducting regime, they are
essentially free. However, putting back the charge fluctuation changes the
physical picture dramatically: both the dynamic Doppler shift term and the
mutual statistical interaction become {\em irrelevant} short-ranged
interactions on both sides of the quantum critical point. There are no
spin-charge separation and {\em no} dynamic gapless gauge field in the
Cooper-pair picture. The formalism developed at $ T=0 $ is applied to study
thermally generated vortices in the vortex plasma regime near the finite
temperature KT transition.
