2019-10-17T00:00:02Zhttps://eprints.lib.hokudai.ac.jp/dspace-oai/requestoai:eprints.lib.hokudai.ac.jp:2115/690212018-10-02T02:25:33Zhdl_2115_45007hdl_2115_116Gauge theory on a non-simply-connected domain and representations of canonical commutation relationsArai, A.open access410A quantum system of a particle interacting with a (non-Abelian) gauge field on the nonsimply-connected domain M = R2 \ { an}=l is considered, where an, n = 1, · · · , N, are fixed isolated points in R 2• The gauge potential A of the gauge field is a p x p anti-Hermitian matrix-valued 1-form on Mand may be strongly singular at the points an, n = 1, · · · , N. If A is fl.at, then the (non-canonical) momentum and the position operators {Pi, qi lJ= 1 of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert space L2(R2; CP). A necessary and sufficient condition for this representation to be the Schrodinger 2-system is given in terms of the Wilson loops of the rectangles not intersecting an, n = 1, · · · , N. This gives also a characterization for the representaion to be non-Schrodinger. It is proven that, for a class of gauge potentials, which is not necessarily fl.at, Pi is essentially self-adjoint. Moreover, an example, which gives a class of non-Schrodinger represenations of the CCR with two degrees of freedom, is discussed in some detail.1994-11-01engdepartmental bulletin paperVoRhttps://doi.org/10.14943/83417http://hdl.handle.net/2115/6902110.14943/83417Hokkaido University Preprint Series in Mathematics270218https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/69021/1/pre270.pdfapplication/pdf735.85 KB1994-11-01