A topological investigation of the quantum adiabatic phase. (English) Zbl 0625.55014
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 nonabelian) is studied and it is shown that the phase appears as a result of homotopically nontrivial 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.
MSC:
| 55S40 | Sectioning fiber spaces and bundles in algebraic topology |
| 53C05 | Connections (general theory) |
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
first Chern class; principal bundles over spheres; nth-level Hilbert subbundle; Schrödinger equation; obstructions to a continuous definition of the phase of a transported wave-function; adiabatic transport; universal U(n)-bundles; sections of U(n)-bundles over \({bbfC}P(n)\); homotopy lifting property; transport of an n-fold degenerate level; quantum adiabatic phase; gauge bundles; Hamiltonian in the space of wave-functions; field theory anomalies; Nielsen-Ninomiya theoremReferences:
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