open access

https://doi.org/10.14943/83417

https://hdl.handle.net/2115/69021

Gauge theory on a non-simply-connected domain and representations of canonical commutation relations

英語

1994-11-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

270

2-18

A 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.

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