  The manipulation of low-energy matter properties such as superconductivity,
ferromagnetism and ferroelectricity via cavity quantum electrodynamics
engineering has been suggested as a way to enhance these many-body collective
phenomena. In this work, we investigate the effective interactions between
low-energy matter excitations induced by the off-resonant coupling with cavity
electromagnetic modes. We extend previous work by going beyond the dipole
approximation accounting for the full polarization and magnetization densities
of matter. We further include the often neglected diamagnetic interaction and,
for the cavity, we consider general linear absorbing media with possibly
non-local and non-reciprocal response. We demonstrate that, even in this
general scenario, the effective cavity-induced interactions between the matter
degrees of freedom are of electrostatic and magnetostatic nature. This confirms
the necessity of a multimode description for cavity engineering of matter
systems where the low-energy assumption holds. Our findings provide a
theoretical framework for studying the influence of general optical
environments on extended low-energy matter excitations.
