The Geometric Phase in Quantum

Berry's phase has been fashionable in many areas of physics and in chemistry and also among mathematicians and mathematical physicists. The mathematical people are attracted to this area because it is related to the beautiful mathematics of bre bundles which underlay gauge theories. In fact this is the most accessible example of a gauge theory for people who know just the elementary facts of nonrelativistic quantum mechanics. The chemists and physicists are interested because the geometric or Berry phase has observable consequences, which could not be explained before. It is this aspect which distinguishes the Berry phase from the many other fashions of mathematical physics. Though the Berry phase may turn out not to be as important as its present popularity suggests, it is a discovery which will remain forever. The surprising thing about it is that its importance has been realized 60 years too late. The reason for this was that many people (including myself) thought that phases are unimportant in quantum mechanics, because a quantum mechanical state is not described by a vector but by a ray or a projection operator j >< j and that phase factors could always be removed by a suitable phase or gauge transformation. That this is not always possible and under which conditions this is not possible was shown by Berry in his famous 1984 paper. 1) And the Berry phase fashion started when Simon 2) explained that Berry's phase is the holonomy (element of the holonomy group) for a bre bundle with a particular connection, the adiabatic connection. But the physical eeect of the Berry phase had been known for quite some time. It was observed as some anomalies in the spectra of molecules 3) and then, (1978), explained in a series of remarkable papers by C.A. Mead and Truhlar 4) by the introduction of a gauge potential, which is identical with the one derived by Berry and which is now called Berry connection. 5) This gauge potential emerges naturally from the Born-Oppenheimer procedure in molecular physics 6) if one does not make the drastic Born-Oppenheimer approximation. 7) The Born-Oppenheimer method is concerned with the study of complicated molecules by dividing them into two parts: the electronic motion described by a set of \fast" variables, and the collective motion described by a set of \slow" variables. Berry connection and Berry phase, therefore, arise in the dissection of complicated quantum physical systems …