  We derive a general formalism to model the polariton states resulting from
the radiation-matter interaction between an arbitrary number of excitonic
transitions in semiconductor quantum dots and photon modes in a photonic
crystal structure in which the quantum dots are embedded. The Maxwell
equations, including the linear nonlocal susceptibility of the exciton
transitions in the quantum dots, are cast into an eigenvalue problem, which can
be applied to any structure whose photon modes can be computed with reliable
accuracy, and in addition naturally allows for disorder effects to be taken
into account. We compute realistic photon modes using Bloch-mode expansion. As
example systems, we study typical InGaAs quantum dots in a GaAs photonic
crystal structures -- an $Ln$ cavity or a $\mathit{W1}$ waveguide. For a single
dot, we reproduce known analytical results, while for the two dot case, we
study the radiative excitation transfer mechanism and characterize its
strength, the dependence on the detuning between quantum dot and photon modes,
and the dependence on inter-dot distance. We find in particular that the
inter-dot radiative coupling strength can reach $100 \mu{eV}$ in a short
cavity, and its decay with distance in longer cavities and waveguides is
determined by the group velocity of the exchanged photons and their radiative
lifetime. We also show that, for an $Ln$ cavity of increasing length, the
radiative excitation transfer mechanism is subject to a crossover from a regime
where a single photon mode is dominating, to a multi-mode regime -- occurring
around $\mathit{n}$ = 150 for the system under study.
