While dealing with General Sigma models $$ S = \frac{1}{2}\int dt g_{ij} \dot{X^i} \dot{X^j} $$ where the Riemann metric can be expanded as, $$g_{ij} = \delta_{ij} + C_{ijkl}X^kX^l$$ 
The Hamiltonian is given by, $$ H = \frac{1}{2} g^{ij} P_{i}P_{j}$$ 
The authors say that in quantum theory the above expression is ambiguous, because $X$ and $P$ don't commute. Hence there are many nonequivalent quantum choices for $H$ reduces to the same classical object. I am not able to figure this out. 


Also this Hamiltonian is related to Laplacian, which I am not able to understand, why ? This Hamiltonian can be related to Laplacian if $g^{ij}$ is the usual $\eta^{ij}$. Do the authors want to say that in some atlas we can always find a local coordinates which reduces to $\eta^{ij}$ or is there a general definition of Laplacian which I am unaware of ?