Covariant and gauge-invariant calculation of higher-order corrections in WKB expansions on Riemannian manifolds

We show how to compute explicitly and in a covariant way the higher-order corrections to the WKB approximation for propagators in curved spaces. We treat the quantum propagator corresponding to the classical system with Lagrangian $L_{\mathrm{cl}}=\left(\frac{1}{2}\right)g_{\mu \nu }{\dot{q}}^{\mu }{\dot{q}}^{\nu }+{\dot{q}}^{\mu }{A}_{\mu }\mathrm{}\left(q\right)-V\left(q\right)\mathrm{}$. The method uses discrete techniques of phase-space functional integrals and involves normal coordinates around the classical path. The gauge invariance of the corrections is also explicitly exhibited.