  Synchronization is a phenomenon where interacting particles lock their motion
and display non-trivial dynamics. Despite intense efforts studying
synchronization in systems without clear classical limits, no comprehensive
theory has been found. We develop such a general theory based on novel
necessary and sufficient algebraic criteria for persistently oscillating
eigenmodes (limit cycles) of time-independent quantum master equations. We show
these eigenmodes must be quantum coherent and give an exact analytical solution
for all such dynamics in terms of a dynamical symmetry algebra. Using our
theory, we study both stable synchronization and metastable/transient
synchronization. We use our theory to fully characterise spontaneous
synchronization of autonomous systems. Moreover, we give compact algebraic
criteria that may be used to prove absence of synchronization. We demonstrate
synchronization in several systems relevant for various fermionic cold atom
experiments.
