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The thing about Skyrmions [I] that is surely hardest to understand is how a lump-like solution (soliton) of a classical scalar field theory can, and in some cases even must, be quantized as a fermion. How can you add integers together and get a half left over?

Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison


The motto in this section is: the higher the level of abstraction, the better.

Why is it interesting?


Tony Hilton Royle Skyrme was born in England in 1922 in the house of his maternal grandparents, c2~ His maternal great-grandfather Edward Robert knew and admired Kelvin, and was associated with the construc- tion of the Tidal Predictor under the direction of Kelvin and Tait. This machine was for predicting tides worldwide. The admiration led to his naming his son Herbert William Thomson Roberts. This machine was in the house where Tony was born. Tony has said in a speech(3~that he was greatly impressed by the ingenuity of its mechanism. Tony grew up in a world beset with increasing turbulence. In 1943, after Cambridge, he joined the British war effort in making the atomic bomb. It was only in 1946 that he began fundamental research. During 1946-61, he was associated with Cambridge, Birmingham, and Harwell and was engaged in wide-ranging investigations in nuclear physics. It was this work, especially the work on nuclear matter and the fluid drop model which eventually culminated in his beautiful proposition that nucleons are solitons made of pions. In the speech of Tony I mentioned above, he has described the reasons behind his extraordinary suggestions. He knew of Kelvin from the Tidal Predictor, and was vaguely aware of Kelvin's vortex atoms. Like Kelvin, he too desired a model of the nucleon which was visualizable and extended. He felt that fermions can emerge from self-interacting Bose fields just as bosons arise as bound states of fermions. Moved by these imprecise and intuitive desires, Tony began his work on nonlinear field theories, the sine- Gordon equation, and the chiral model, and was led to the proposition that nucleons are twisted topological lumps of pion fields. 16~ In his papers, Tony also had initiated ideas on bosonization, vertex opertors, and quantum theories on multiply connected spaces, all years and years ahead of his time, and all topics of central interest today. David Finkelstein had a grasp of topology and differential geometry which was exceptional for physicists in the 60's. Like Skyrme, he had under- stood that solitons can acquire spin 1/2 from the topology of the configura- tion space. He must have realized what little role relativistic quantum field theory played in the theory of solitons, and struck by the fact that all existing proofs of the spin-statistics theorem relied on relativistic quantum field theory. But solitons can acquire spin 1/2 in nonrelativistic models and can- not always be described by relativistic quantum fields. He and Rubinstein, I suppose, were led by such thoughts to seek and find an alternative proof of the spin-statistics theorem. Their proof was published in 1968.~71It is this proof which is important for chiral solitons. There are grounds to expect that it is the Finkelstein-Rubinstein approach which will be found signifi- cant in quantum gravity as well. An absolutely fundamental result, namely the spin-statistics theorem, is getting topologized. Even more striking, it is still not properly understood. TOPOLOGY IN PHYSICS - A PERSPECTIVE by A.P. Balachandran

advanced_notions/skyrmions.txt · Last modified: 2018/12/07 09:00 by