let be the Hamiltonian of a surface $ H= g_{a,b} p^{a}p^{b} $ (Einstein summation assumed) my question is if although the space time is curved then can we use the WKB approximation to get the quantum energies from the momenta

$ \oint _{C} p_{a}dq_{a}=2\pi \hbar (n_ {a}+ \mu _{a})$ and

$ \oint _{C} p_{b}dq_{b}=2\pi \hbar (n_ {b}+ \mu _{b})$

for example for the hyperbolic metric $ ds^{2} = \frac{dx^{2} +dy^{2}}{y^{2}} $ with Hamiltonian $ H= -y^{2}( \partial _{x}^{2}+ \partial _{y}^{2}) $

Let be the Hamiltonian of a surface $ H= g_{a,b} p^{a}p^{b} $ (Einstein summation assumed). My question is if although the space time is curved then can we use the WKB approximation to get the quantum energies from the momenta

$ \oint _{C} p_{a}dq_{a}=2\pi \hbar (n_ {a}+ \mu _{a})$ and

$ \oint _{C} p_{b}dq_{b}=2\pi \hbar (n_ {b}+ \mu _{b})$

For example for the hyperbolic metric $ ds^{2} = \frac{dx^{2} +dy^{2}}{y^{2}} $ with Hamiltonian $ H= -y^{2}( \partial _{x}^{2}+ \partial _{y}^{2}) $.