A Global Theory of Parametrized Quantum Mechanics

We study the dependence of energy eigenvalues and eigenfunctions on parameters in the Hamiltonian from a global perspective. This dependence can be realized by topological quantum numbers in general. A general conservation law is found for these additive topological charges in the metamorphoses of energy bands. Many topological properties of “accidental” degeneracies are revealed with these results. In particular, global constraints can make degeneracies more generic (smaller codi-mension). Non-trivial topological charges can also tie energy bands together and make them inseparable. Generally, non-abelian structures arise due to metamorphoses of energy bands. These conclusions are very useful in understanding the meaning of higher cocycles in gauge theories.