LETTER TO THE EDITOR: The Berry phase for spin in the Majorana representation

A continuous cyclic sequence of quantum states has an associated geometric, or Berry, phase . For spin J, such a sequence is described by a cyclic change in the 2J+1 coefficients of the basis states . The Berry phase is analysed here for the general case - that is, the coefficients 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 , 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 , where , each traces out a parametrized loop on the sphere, and the Berry phase is given by an integral of combinations of these vectors.

[1]  S. Pancharatnam,et al.  Generalized theory of interference, and its applications , 1956 .

[2]  X. Gao,et al.  Formally exact solution and geometric phase for the spin-j system☆ , 1991 .

[3]  J. Hannay,et al.  Chaotic analytic zero points: exact statistics for those of a random spin state , 1996 .

[4]  Huang,et al.  Cyclic quantum evolution and Aharonov-Anandan geometric phases in SU(2) spin-coherent states. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[5]  R. Penrose,et al.  Shadows of the Mind , 1994 .

[6]  G. Gibbons,et al.  Non-integrable quantum phase in the evolution of a spin-1 system : a physical consequence of the non-trivial topology of the quantum state-space , 1988 .

[7]  M. Berry.  The Adiabatic Phase and Pancharatnam's Phase for Polarized Light , 1987 .

[8]  M. Berry.  Quantal phase factors accompanying adiabatic changes , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[9]  Aharonov,et al.  Phase change during a cyclic quantum evolution. , 1987, Physical review letters.

[10]  A geometric phase for m = 0 spins , 1994 .

[11]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[12]  P. Leboeuf.  Phase space approach to quantum dynamics , 1991 .

[13]  Michael Barr,et al.  The Emperor's New Mind , 1989 .

[14]  Classification of cyclic initial states and geometric phase for the spin-j system , 1994 .

[15]  Ettore Majorana.  Atomi orientati in campo magnetico variabile , 1932 .