Representations of the Three-Dimensional Rotation Group in Terms of Direction and Angle of Rotation

We find the irreducible representations of the three-dimensional pure rotation group by Weyl's method, which makes use of the homomorphism of the special unitary group of order two onto the whole three-dimensional rotation group. The representations are realized, however, in terms of the angle of rotation in a specified direction and the spherical angles of the direction of the rotation rather than in terms of the familiar Euler angles. The results are then compared with those obtained by different methods and the advantages of the present technique are pointed out. We also derive the differential operators corresponding to infinitesimal rotations about the coordinate axis in terms of the new variables.