Dirac's Monopole and the Hopf Map
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概要
- 論文の詳細を見る
A regular electromagnetic potential A_μ(x) is found on a 3-sphere which may be regarded as describing the Dirac magnetic monopole in the sense that the field tensor derived from A_μ(x) gives the monopole strength tensor on a 2-sphere when pulled back by the Hopf map: S^3→S^2. Relationship of A_μ(x) with the singular potentials is clarified. To represent the magnetic charge, an integral formula over S^3 found by Whitehead is employed, which alternatively decomposes into consecutive integrals over S^2 and U(1). An SO(3) gauge potential is also constructed from the viewpoint that the gauge group associated with A_μ(x) be the little group U(1) to which SO(3) is spontaneously broken back by the "Higgs vacuum". The resulting SO(3)-field is the one on a 3-sphere and proves to be identified with the instanton field at large distances. No dynamical consideration and so no information near the origin are given within the scope of this paper. It is shown however that the Whitehead integral which originally expresses the Hopf invariant as an integer turns out to be the surface integral which expresses the Pontrjagin index of the instanton.
- 理論物理学刊行会の論文
- 1979-10-25
著者
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MINAMI Masatsugu
Research Institute for Mathematical Sciences Kyoto University
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MINAMI Masatsugu
Research Institute for Mathematical Sciences, Kyoto University
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