The Berry phase for spin in the Majorana representation

A continuous cyclic sequence of quantum states has zn associated geometric, or Berry, phase i phi(psi\d psi). For spin J, such a sequence is described by a cyclic change in the 2J + 1 coefficients a(m) of the basis states \J, m]. The Berry phase is analysed here for the general case-that is, the coefficients a(m) are allowed to vary in an arbitrary cyclic manner. The result is expressed in geometric terms, specifically in the democratic representation due to Majorana. This uniquely characterizes the spin state \psi], up to overall phase, by the positions of 2J dots on the unit sphere of directions in real space. If the positions are denoted by unit vectors u(k), where 1 less than or equal to k less than or equal to 2J, each traces out a parametrized loop on the sphere, and the Berry phase is given by an integral of combinations of these vectors.