A topological investigation of the Quantum Adiabatic Phase

Using algebraic topology, the appearance of the Quantum Adiabatic Phase over various parameter manifolds is investigated. The relation with nontrivial gauge bundles (both abelian and non-abelian) is studied and it is shown that the phase appears as a result of homotopically non-trivial mappings, induced by the Hamiltonian in the space of wave-functions. The cohomological picture is developed and some topological considerations concerning field theory anomalies in the Hamiltonian picture are presented. A proof of the Nielsen-Ninomiya theorem is given inspired from the notion of the adiabatic phase.Work supported in part by the U.S. Department of Energy under contract DEAC 03-81-ER 40050