Quantization of geodesics

Question

The geodesics minimize the length of paths. The Lagrangian is: 

$L= \sqrt {g_{ij} d/ds(x^i)d/ds(x^j)}$

I propose to make a Legendre transform. I obtain for a geodesic $p_i=  g_{ij} d/ds (x^j)$. The Hamiltonian is:

$H= x^i p_i -L$

The quantization of the Hamiltonian is:

$\hat H (\psi )= -i h x^j (d/dx^j)(\psi ) - D \psi$

With $D$ the Dirac operator. Does it correspond to  something in Quantum Gravity?