  The use of cavity quantum electrodynamical effects, i.e., of vacuum
electromagnetic fields, to modify material properties in cavities has rapidly
gained popularity and interest in the last few years. However, there is still a
scarcity of general results that provide guidelines for intuitive understanding
and limitations of what kind of effects can be achieved. We provide such a
result for the effective interactions between low-energy matter excitations
induced either directly by their mutual coupling to the cavity electromagnetic
(EM) field or indirectly through coupling to mediator modes that couple to the
EM field. We demonstrate that the induced interactions are purely electrostatic
in nature and are thus fully described by the EM Green's function evaluated at
zero frequency. Our findings imply that reduced models with one or a few cavity
modes can easily give misleading results.
