Derivation of the geometrical berry phase

Abstract

We have seen that the equivalence of state vectors which differ by a phase, along with the scalar product, define the geometry of Hilbert space (i.e. the fiber bundle and connection). The geometry is non-trivial. It induces a U(1) holonomy in a normalized state vector which undergoes cyclic evolution. This induced phase is called the geometric phase. It depends only on the path in the space of physical states, not on the Hamiltonian which generates this path.

Physical effects of non-trivial geometries appeax in molecular physics [7]. These effects are described by the introduction of a vector potential (connection) into the molecular equations of motion [8]. The relative momenta coordinates P go into P - A. This change alters the canonical commutation relations [9], and may be of interest in a spectrum generating group approach [10].