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Media type:
E-Article
Title:
Non-integer flux: Why it does not work
Contributor:
Smilga, A. V.
Published:
AIP Publishing, 2012
Published in:
Journal of Mathematical Physics, 53 (2012) 4
Language:
English
DOI:
10.1063/1.3703127
ISSN:
0022-2488;
1089-7658
Origination:
Footnote:
Description:
We consider the Dirac operator on S2 without one point in the case of non-integer magnetic flux. We show that the spectral problem for \documentclass[12pt]{minimal}\begin{document}$H = \big/\!\!\!\!{\cal D}^2$\end{document}H=D̸2 can be well defined, if including in the Hilbert space \documentclass[12pt]{minimal}\begin{document}${\cal H}$\end{document}H only nonsingular on S2 wave functions. However, this Hilbert space is not invariant under the action of \documentclass[12pt]{minimal}\begin{document}$\big/\!\!\!\!{\cal D}$\end{document}D̸ — for certain \documentclass[12pt]{minimal}\begin{document}$\psi \in {\cal H}$\end{document}ψ∈H, \documentclass[12pt]{minimal}\begin{document}$\big/\!\!\!\!{\cal D} \psi$\end{document}D̸ψ does not belong to \documentclass[12pt]{minimal}\begin{document}${\cal H}$\end{document}H anymore. This breaks explicitly the supersymmetry of the spectrum. In the integer flux case, supersymmetry can be restored if extending the Hilbert space to include locally regular sections of the corresponding fiber bundle. For non-integer fluxes, such an extension is not possible.