Anticipations of the geometric phase

Some early studies, seen as anticipations and examples of the geometric phase, are discussed. An early work was about quantum systems forced round a cycle by a slow circuit of parameters that govern them. This work gave rise to a number of applications such as parallel transport, coiled light, polarization cycles, degeneracy, and curved surfaces. The first of these applications considers the geometric state as anholonomic for the parallel transport of quantum states which are represented mathematically by complex unit vectors in Hilbert space. The geometric state for the coiled light application is represented by a rotation of the direction of polarized light after it has traveled along a coiled optical fiber. In a different application to optics, called polarization circles, light is traveling in a fixed direction with a slowly changing state of polarization. The geometric phase can be considered as an expression of a simple property of families of matrices that depend on parameters. Their eigenvectors are not single-valued when parallel-transported via changes of parameters and they do not return to their original values (this is called degeneracy).