  During the last 70 years, the quantum theory of angular momentum has been
successfully applied to describing the properties of nuclei, atoms, and
molecules, their interactions with each other as well as with external fields.
Due to the properties of quantum rotations, the angular momentum algebra can be
of tremendous complexity even for a few interacting particles, such as valence
electrons of an atom, not to mention larger many-particle systems. In this
work, we study an example of the latter: a rotating quantum impurity coupled to
a many-body bosonic bath. In the regime of strong impurity-bath couplings the
problem involves addition of an infinite number of angular momenta which
renders it intractable using currently available techniques. Here, we introduce
a novel canonical transformation which allows to eliminate the complex angular
momentum algebra from such a class of many-body problems. In addition, the
transformation exposes the problem's constants of motion, and renders it
solvable exactly in the limit of a slowly-rotating impurity. We exemplify the
technique by showing that there exists a critical rotational speed at which the
impurity suddenly acquires one quantum of angular momentum from the
many-particle bath. Such an instability is accompanied by the deformation of
the phonon density in the frame rotating along with the impurity.
