  We introduce a formalism to solve the problem of photon scattering from a
system of multi-level quantum emitters. Our approach provides a direct solution
of the scattering dynamics. As such the formalism gives the scattered fields
amplitudes in the limit of a weak incident intensity. Our formalism is equipped
to treat both multi-emitter and multi-level emitter systems, and is applicable
to a plethora of photon scattering problems including conditional state
preparation by photo-detection. In this paper, we develop the general formalism
for an arbitrary geometry. In the following paper (part II), we reduce the
general photon scattering formalism to a form that is applicable to
$1$-dimensional waveguides, and show its applicability by considering explicit
examples with various emitter configurations.

Quantum Physics;Optics