Relativistic Green functions in a plane-wave gravitational background

We consider a massive relativistic particle in the background of a gravitational plane wave. The corresponding Green functions for both spinless and spin-1 2 cases, previously computed by Barducci and Giachetti (2005 J. Phys. A: Math. Gen. 38 1615), are reobtained here by alternative methods, as for example, the Fock-Schwinger proper-time method and the algebraic method. In analogy with the electromagnetic case, we show that for a gravitational plane- wave background a semiclassical approach is also sufficient to provide the exact result, though the Lagrangian involved is far from being a quadratic one.