The Magnetic Monopole Fifty Years Later

  • Sidney Coleman


This is a jubilee year. In 1931, P. A. M. Dirac1 founded the theory of magnetic monopoles. In the fifty years since, no one has observed a monopole; nevertheless, interest in the subject has never been higher than it is now.


Gauge Field Magnetic Charge Topological Charge Grand Unify Theory Magnetic Monopole 
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Footnotes and References

  1. 1.
    P. A. M. Dirac, Proc. Roy. Soc. (London) Ser. A, 133, 60 (1931).Google Scholar
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    J. Preskill, Phys. Rev. Lett. 43, 1365 (1979).CrossRefGoogle Scholar
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    S. Coleman, “Classical Lumps and Their Quantum Descendants”, in New Phenomena in Subnuclear Physics, edited by A. Zichichi (Plenum, 1977).Google Scholar
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    Ref. 4 has a more extensive bibliography. See also the reviews of P. Goddard and D. Olive, Rep. Prog. Phys. 41, 1357 (1978), and E. Amaldi and N. Cabibbo, in Aspects of Quantum Theory, edited by A. Salam and E. Wigner (Cambridge Univ. Press, 1972).Google Scholar
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    Note that there is no 4π in the definition of magnetic charge.Google Scholar
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    Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).CrossRefGoogle Scholar
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    This is a truism for non-Abelian gauge theories, where everyone thinks of gauge transformations as labeled by functions from space-time into the gauge group. For the Abelian theory at hand, the gauge group is U(1), and the function into the group is exp(-i e Χ).Google Scholar
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    T. T. Wu and C. N. Yang, Nucl. Phys. B107, 365 (1976).CrossRefGoogle Scholar
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    This problem was solved very early on by I. Tamm, Z. Phys. 71, 141 (1931), and my results are the same as his, although my method is somewhat different. To my knowledge, the treatment in the literature closest to that given here is that of H. J. Lipkin, W. I. Weisberger, and M. Peshkin, Ann. of Phys. 53, 203 (1969).Google Scholar
  11. 11.
    These commutators are not so innocuous as they seem. If we attempt to verify the Jacobi identity for three D’s, we obtain, (3) instead of zero, a term proportional to δ (3) (r). This is no real problem, for two reasons. Firstly, we don’t believe the monopole field all the way down to the origin; it’s just the long-range part of something that is more complicated at short distances. (In Sec. 4 we’11 see what that something is.) Secondly, even if we believe the field all the way down to the origin, there is, as we shall see, a centrifugal barrier in all partial waves that keeps the particle away from the origin.Google Scholar
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    The analysis given here follows that of A. Goldhaber, Phys. Rev. Lett. 36, 1122 (1976). Goldhaber’s work was stimulated by investigations by R. Jackiw and C. Rebbi (ibid., 1116) and by P. Hassenfratz and G. ’t Hooft (ibid., 1119).Google Scholar
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    P. Goddard, J. Nuyts, and D. Olive, Nucl. Phys. B125, 1 (1977).CrossRefGoogle Scholar
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    I’m being very sloppy here about singularities. Here’s a more careful argument: We define a gauge-field configuration to be locally non-singular in some region if the region is the union of a family of open sets, such that in each set the gauge field is non-singular and such that in the intersection of any two sets the gauge fields in the two sets are connected by a non-singular gauge transformation. It is possible to show that a gauge-field configuration that is locally non-singular in all of space except for the origin is gauge-equivalent to one that is non-singular (in the ordinary sense) everywhere except for the south polar axis. (This theorem is proved in Ref. 4.) That is to say, if all singularities, other than ones at the origin, are gauge artifacts, then they can all be shoved onto the Dirac string by an appropriate choice of gauge.Google Scholar
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    E. Lubkin, Ann. Phys. (N.Y.) 23, 233 (1963). See especially Sec. XV.CrossRefGoogle Scholar
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    A somewhat longer (though still hopelessly vulgar) course can be found in Ref. 4, together with references to the mathematical literature.Google Scholar
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    This stability analysis was first done by R. Brandt and F. Neri, Nucl. Phys. B161, 253 (1979).Google Scholar
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    W. Nahm and D. Olive (private communication, summer 1979).Google Scholar
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    The work described here is drawn from many sources; for references, see Ref. 4.Google Scholar
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    E. Bogomol’nyi, Sov. J. Nucl. Phys. 24, 449 (1976). S. Coleman, S. Parke, A. Neveu, and C. Sommerfield, Phys. Rev. D15, 544 (1977).Google Scholar
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    M. Prasad and C. Sommerfield, Phys. Rev. Lett. 35, 760 (1975).CrossRefGoogle Scholar
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    Much of this section is blatant plagiarism from Ref. 4. (Copyright holder take note!)Google Scholar
  24. 24.
    Actually, one can avoid this computational horror by group-theoretic tricks, just as we did for the Laplace operator in Sec. 2.Google Scholar
  25. 25.
    This note is for experts only. The discussion in the text leaves the impression that the detailed computation of the isorotational spectrum is easier than it is. There are technical complications associated with gauge invariance. These can all be dealt with, but they make the calculation lengthier than it would be if they weren’t around. For example, in temporal gauge, there are many invariances of the Hamiltonian that do not leave the monopole solution unchanged, to wit, time-independent gauge transformations. No one in his right mind expects these to lead to isorotational levels. However, to demonstrate this, and to disentangle the spurious excitations from the genuine ones, requires fiddling around with the subsidiary condition that is the bane of temporal-gauge quantization. Subsidiary conditions can be avoided by working in Coulomb gauge, for example, where none are needed, but then the form of the Hamiltonian is more complicated, and this makes things messy. (For a careful Coulombgauge treatment of the dyons discussed immediately below, see E. Tomboulis and G. Woo, Nucl. Phys. B107, 221 (1976).Google Scholar
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    These dyons were first discovered by B. Julia and A. Zee, Phys. Rev. D11, 2227 (1975), using quite different methods from these.Google Scholar
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    This trick was discovered by G. ’t Hooft, Nucl. Phys. B105, 538 (1976) and by E. Corrigan, D. Olive, D. Fairlie, and J. Nuyts, Nucl. Phys. B106, 475 (1976).Google Scholar
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    J. Schwinger, Phys. Rev. 144, 1087; 151, 1048; 151, 1055 (1966).CrossRefGoogle Scholar
  31. 31.
    For example, see L. Landau and E. Lifshitz, Quantum Mechanics (3rd ed.) (Pergamon, 1977) p. 410.Google Scholar

Copyright information

© Plenum Press, New York 1983

Authors and Affiliations

  • Sidney Coleman
    • 1
  1. 1.Lyman Laboratory of PhysicsHarvard UniversityCambridgeUSA

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