  We analyze the constraints of general coordinate invariance for quantum
theories possessing conformal symmetry in four dimensions. The character of
these constraints simplifies enormously on the Einstein universe $R \times
S^3$. The $SO(4,2)$ global conformal symmetry algebra of this space determines
uniquely a finite shift in the Hamiltonian constraint from its classical value.
In other words, the global Wheeler-De Witt equation is {\it modified} at the
quantum level in a well-defined way in this case. We argue that the higher
moments of $T^{00}$ should not be imposed on the physical states {\it a priori}
either, but only the weaker condition $\langle \dot T^{00} \rangle = 0$. We
present an explicit example of the quantization and diffeomorphism constraints
on $R \times S^3$ for a free conformal scalar field.
