부산대학교 도서관

In this paper the authors discuss the accumulation property of the geometric phase along curves in quantum spin state space. Based on the Hilbert space of a spin-$s$ quantum system, a complex projective space is defined. Then the projection operator $\varphi$ is defined in equation (2). Meanwhile the complex unitary group is assumed to act on the complex manifold of dimension $n=2s$, and thus a corresponding Lie algebra can be considered. Based on these elements the tangent space, the complex structure $J$ and covariant derivatives are derived. Following those the differential equation of $\varphi$ is deduced, just like the Schrödinger equation, which defines curves in quantum space. Finally the geometric phase is discussed in the tangent space of the manifold, interpreted by using the covariant derivatives. They find an analytical solution for the brachistophase problem. Some open questions are also discussed.