UBC Theses and Dissertations
Holonomy in quantum physics Rutherford, Alexander R.
Holonomy in nonrelativistic quantum mechanics is examined in the context of the adiabatic theorem. This theorem is proven for sufficiently regular unbounded hamiltoni-ans. Then, simplifying to matrix hamiltonians, it is proven that the adiabatic theorem defines a connection on vector bundles constructed out of eigenspaces of the hamiltonian. Similar degeneracy regions, the natural base spaces for these bundles, are defined in terms of stratifications for the spaces of complex, hermitian matrices and real, symmetric matrices. The algebraic topology of similar degeneracy regions is studied in detail, and the results are used to classify and calculate all possible adiabatic phases for time-reversal invariant matrix hamiltonians in terms of the relevant topological data. It is shown how vector bundles may be used to impose transversality on the helicity vector of a photon. This is used to give a calculation, which is consistent with transversality, of quantum adiabatic phase for photons in a coiled optical fibre. As an additional application, the importance of quantum adiabatic in the dynamical Jahn-Teller effect is briefly explained. An introduction is given to some important aspects of algebraic topology, which are used herein. Moreover, a number of mathematical results for flag manifolds are obtained. These results are applied to quantum adiabatic holonomy.
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