  In this work we write down a classical (not quantum) action for the surface
height for the Kardar-Parisi-Zhang (KPZ) equation for surface growth. We do so
starting with the regular Martin-Siggia-Rose (MSR) action (which is quantum -
contains the constraint field) and integrate out the quantum constraint field
exactly. We analyze the classical action, we thereby obtain, within the
gaussian and one loop approximations various instabilities to rough surfaces.
The gaussian analysis predicts instabilities to rough surfaces below two
dimensions while one loop analysis predicts stronger surface stability to
roughness and instabilities to roughness below one dimension rather than two
dimensions. The one loop analysis shows that the KPZ action is incomplete and
that we generate additional terms (not found in the initial classical action)
in the action perturbatively. In the supplement we also modify the KPZ equation
for growing surfaces to include the effect of surface tilt on the noise (that
is have two sources of noise one of which is multiplicatively coupled noise).
