  We study the fluctuation-induced interaction between two impurities in a
weakly-interacting one-dimensional Bose gas using the field theoretical
approach. At separations between impurities shorter and of the order of the
healing length of the system, the induced interaction has a classical origin
and behaves exponentially. At separations longer than the healing length, the
interaction is of a quantum origin and scales as the third power of the inverse
distance. Finite temperature destroys the quasi-long-range order of the Bose
gas and, accordingly, the induced interaction becomes exponentially suppressed
beyond the thermal length. We obtain analytical expressions for the induced
interaction at zero and finite temperature that are valid at arbitrary
distances. We discuss experimental realizations as well as possible formation
of bound states of two impurities, known as bipolarons.
