Usually equations of motion containing a time convolution arise when a subsystem is projected out. A standard procedure is to use the Zwanzig projection operator method to obtain a generalised master equation. But it can also happen that such an equation of motion is directly given. This happens with the phenomenological Maxwell's equations for dispersive and absorptive systems. Then a "reverse" procedure can be followed. A new, auxiliary, field is introduced and the combined set of fields obey a unitary time evolution which can then be quantised. A posteriory the new field can here be identified as a polarisation field. References are
A. Tip: Phys. Rev. A 57, 1998, p. 4818.
B. Gralak and A. Tip: j.Math.Phys 51, 2010, p. 052902.
The method should also work for other linear cases with a time convolution term.
It is interesting to note that one can start from a unitary time evolution, use the Zwanzig method to obtain a time evolution with a convolution term and then use the reversed method to obtain a new unitary time evolution. The latter is not the same as the original one. It essentially picks up only stuff that is needed to describe the projected system. Note further that the method is exact. No approximations are made.