Field Theories with a Degenerate Vacuum

  • Bruno Zumino


One of the outstanding problems in elementary particle physics is the occurence in nature of approximate symmetries. The best known example is the invariance of the strong interactions with respect to isotopic spin rotations and the violation of this invariance by the electromagnetic and weak interactions. In trying to gain a theoretical understanding of these approximate invariance laws one can take two quite different points of view. The first is to consider an approximate invariance as a property of the solutions and not as a basic symmetry of the underlying equations. The approximate invariance appears then as a sort of dynamical accident; if one adopts this attitude it is very difficult to see how a knowledge of the approximate invariances occuring in nature could be used as a guide in an effort to discover the underlying equations. An interesting model which can be used to illustrate this first point of view has been given by van Hove [1].


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    L. Van Hove, Physica 25, 365 (1959).CrossRefGoogle Scholar
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    H. P. DÜRr, W. Heisenberg, H. Mitter, S. Schlieder and K. Yamazaki, Z. Naturforschung 14a 441 (1959). In this paper one can find references to earlier papers on the non-linear spinor theory.Google Scholar
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    Theories of this type have been investigated also by J. Goldstone, Nuovo Cimento 19, 154 (1961). One of the arguments used by Goldstone in order to establish the degeneracy of the solution is essentially a correspondence argument. In this way the source of the degeneracy of the quantum solution is traced to a degeneracy of the corresponding classical solution. This sort of arguments seems to be somewhat dangerous, since it is well known that quantization often removes the degeneracy present in the classical solution.Google Scholar
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    Proceedings of the International Congress on Many-Particle Problems, Utrecht 1960, in Supplement to Physica, 26 (1960). We refer in particular to the articles of N. N. B000lubov, J. R. Schrieffer and S. T. Beliaev.Google Scholar
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    Anyway, the model of Section 3 should not be taken too seriously and was given only because it provides a simple illustrative example of the perturbation technique. A more detailed study shows that ultraviolet divergences are still present, in perturbation theory, in spite of the restriction to a two-dimensional space time. Furthermore, the model possesses an exact solution with a non degenerate vacuum. This fact makes it possible to argue that the solution (26) is probably, for this model, a spurious result of perturbation theory.Google Scholar

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© Friedr.Vieweg & Sohn, Braunschweig 1961

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  • Bruno Zumino

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