  We make a general derivation for the magnetic dipole-dipole interaction based
on the mediation of the quantized electro-magnetic field. Due to the
interaction with the dipoles, the dynamics of the field is added by a dipole
field, which finally gives rise to the dipole-dipole interaction. Different
from previous studies, the rotating-wave-approximation is no longer needed
throughout this derivation, and our result naturally gives the interaction for
non-resonant dipoles. Moreover, our derivation also gives the counter-rotating
interaction terms, and even the mixed interaction terms between the permanent
and transition dipoles. We notice that this field-induced interaction is
associated with the interference of the virtual/real photons emitted from the
two dipoles, thus the interaction strength could be influenced by the frequency
difference of the two dipoles.
