Classically, if a vector is parallel transported around a closed circuit on a curved space, it returns with its orientation altered. A drawing modified from the one at the a review of Berry’s Geometric Phase on the web page of Nicola Manini (assistant professor at Milan University) shows the classical effect:
In the drawing above, when the red vector is parallel transported through the 13 marked points around the sphere (all on a spherical triangle), the vector returns rotated.
If the vector is considered as a 2-d object in the tangent space of the surface of the sphere, the rotation of the vector can be given by an angle. The angle that the vector is rotated by is proportional to the surface area of the spherical triangle. If the spherical triangle is built from three 90 degree angles, it is clear that the vector is rotated by 90 degrees, and, under the (correct) assumption that the rotation angle is proportional to the area of the spherical triangle, this tells us that the ratio of spherical triangle area to rotation angle is given by 1. That is, if the spherical triangle has surface area A, the vector will be rotated by the angle A. A similar effect occurs in quantum mechanics when a spin state is sent through a sequence of orientations. This is kind of interesting in that we are turning a measure of spherical area into an angle.
In the usual spinor representation of quantum mechanics, the phase of a quantum state is arbitrary; one can always multiply a spinor by a complex phase and the new spinor represents the same state.
Density matrices are simpler in that they do not carry arbitrary phase information and consequently we can discuss Berry phase very elegantly and easily in density matrix formalism, which is the subject of this post.