  Spin wave theory is a key ingredient in our comprehension of quantum spin
systems, and is used successfully for understanding a wide range of magnetic
phenomena, including magnon condensation and stability of patterns in dipolar
systems. Nevertheless, several decades of research failed to establish the
validity of spin wave theory rigorously, even for the simplest models of
quantum spins. A rigorous justification of the method for the three-dimensional
quantum Heisenberg ferromagnet at low temperatures is presented here. We derive
sharp bounds on its free energy by combining a bosonic formulation of the model
introduced by Holstein and Primakoff with probabilistic estimates and operator
inequalities.
