# can we apply WKB method for curved space times

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 $H= -y^{2}( \partial _{x}^{2}+ \partial _{y}^{2})$

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yes. Why would you think not? –  Ron Maimon Jul 20 '12 at 19:23