Any vector object which is parallel-transported along a path back to the original place, may acquire an angle with respect to its initial direction prior to transport. This angle is a geometric property. An intuitive classical example of such situation [see B. Goss Levi, Phys. Today 46, 17 (1993)] is the parallel transport of a vector along a loop on a sphere (you may think of a compass needle carried in a ship or car traveling on the surface of the planet). In the figure at the right, both vectors stay tangential to the curved surface at all times. They start from the red point above (the "north pole"), and get transported along the path, remaining as parallel as possible to the direction they were pointing at before each infinitesimal displacement (the long vector points "South" and the short one points "East" all the time). After completing the closed path, or loop, the vectors go back to the original point, but they find themselves rotated with respect to the directions they were pointing at when the journey started. Note that the vectors "rotate" despite the fact we have been careful to keep them parallel during transport. Had the path been "smaller", i.e. including a smaller part of the spherical surface, the rotation angle would have been smaller. In our example, where the loop surrounds one eighth of the sphere, the rotation angle amounts to 90°. A larger path surrounding for example one quarter of the sphere, would rotate the parallel-transported vectors by 180°. An even larger loop, a diameter, surrounding (on both sides) half the sphere gives a rotation angle of 360°, i.e. no rotation at all.
The reason for this rotation is purely geometrical-topological. In fact, it is connected to the intrinsic curvature of the sphere. No such phenomenon would appear if vectors are parallel-transported along a flat manifold, such as a plane or a cylinder. The rotation angle is in fact related to the integral of the curvature on the surface bounded by the loop.
Such rotation angles of geometrical origin are known as Berry phases. "Phase" is used meaning just "angle" for whatever possible argument of sin(·) or cos(·) or exp(i ·). Several specific cases had been recognized for many years, in particular the well-studied Aharonov-Bohm effect and Pancharatnam's work on the geometric phases in optics. M. Berry published in 1984 a very influential formal systematization of the closed-path geometric phase in quantum mechanical problems. see http://www.mi.infm.it/manini/berryphase.html for referenced graphics
The Berry phase (Berry 1984) is a crucial concept in many quantum mechanical effects, including quantum computing. For example, it modifies the motion of vortices in superconductors and the motion of electrons in nanoscale electronic devices. As a result of wave-particle duality, all particles have wavelike properties, but the Berry phase is a special type of phase that a particle acquires if it is forced to slowly rotate. Berry Phase
See Also
9.26 - Orbital Phases
12.11 - Eighteen Attributes or Dimensions
coupling
Figure 8.2 - Compression Wave Phase Illustration
Figure 8.10 - Each Phase of a Wave as Discrete Steps
Figure 8.11 - Four Fundamental Phases of a Wave
Figure 9.10 - Phases of a Wave as series of Expansions and Contractions
Figure 9.14 - Wave Flow and Phase as function of Particle Rotation
Figure 9.5 - Phases of a Wave as series of Expansions and Contractions
Keelys Secrets - Part 2 - One Phase of Keelys Discovery in Its Relation to the Cure of Disease
phase
phase conjugation
Rotational-vibrational coupling
Rovibrational coupling
Vibronic coupling