Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2). There is one n -dimensional irreducible representation of SU(2) for each dimension, though this representation is n -dimensional real for odd n and n -dimensional complex for even n (hence of real dimension 2 n). For a rotation by angle θ in the plane with normal vector θ θ ^ ^ , U = e − − i ℏ ℏ θ θ ⋅ ⋅ S , where θ θ = θ θ θ θ ^ ^ , and S is the vector of spin operators.