LOW-DIMENSIONAL REDUCTION OF THE NON-ABELIAN QUANTUM SYNCHRONIZATION MODELS ON THE UNITARY GROUP

We present a low-dimensional reduction of the non-Abelian syn-chronization models on the unitary group, when the initial configuration has special structures. We first consider the microscopic model, and show that the dynamics can be reduced to the n-copies of the one-dimensional Kuramoto systems, if the initial data can be represented by simultaneously diagonalizable matrices. Then, we extend the dimensional reduction to the kinetic synchro-nization model on U(n), when the initial distribution belongs to a special family of functions called a class function. Under this condition, a solution to the kinetic equation can be reduced to a solution to the Kuramoto-like dynam-ics on the n-dimensional torus Tn. Using these dimensional reductions and the relationship between the Kuramoto models, we establish a novel general synchronization framework for the non-Abelian synchronization models.