Abstract
As the previous discussion indicates, it is impossible to realize the phenomenon of (spontaneous) breaking of a continuous symmetry in classical mechanical systems with a finite number of degrees of freedom. We are thus led to consider infinite dimensional systems, like classical fields.
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References
See, e.g., G.B. Whitham, Linear and Non-Linear Waves, J. Wiley, New York 1974; R. Rajaraman, Phys. Rep. 21 C, 227 (1975); S. Coleman, Aspects of Symmetry, Cambridge Univ. Press 1985, Chap. 6
For an extensive review on the mathematical problems of the non-linear wave equation see M. Reed, Abstract non-linear wave equation, Springer-Verlag, Heidelberg 1976. For the solution of the Cauchy problem for initial data not vanishing at infinity, a crucial ingredient for discussing spontaneous symmetry breaking, see C. Parenti, F. Strocchi and G. Velo, Phys. Lett. 59B, 157 (1975); Ann. Scuola Norm. Sup. (Pisa), III, 443 (1976), hereafter referred as I. A simple account with some addition is given in F. Strocchi, in Topics in Functional Analysis 1980–81, Scuola Normale Superiore Pisa, 1982. For a beautiful review of the recent developments see W. Strauss, Nonlinear Wave Equations, Am. Math. Soc. 1989.
For the discussion of more general symmetries see C. Parenti, F. Strocchi and G. Velo, Comm. Math. Phys. 53, 65 (1977), hereafter referred as II; Phys. Lett. 62B, 83 (1976).
F. Strocchi and G. Velo, Comm. Math. Phys. 53, 65 (1977) Ref. II (see footnote 5).
E. Noether, Nachr. d. Kgl. Ges. d. Wiss. Göttingen (1918), p. 235.
See e.g. H. Goldstein, Classical Mechanics, 2nd. ed., Addison-Wesley 1980; E. L. Hill, Rev. Mod. Phys. 23, 253 (1951); N.N. Bogoljubov and D.V. Shirkov, Introduction to the theory of quantized fields, Interscience 1958, Sect. 2.5.
For the representations of Lie groups and their generators in classical systems, see D.G. Currie, T.F. Jordan and E.C.G. Sudarshan, Rev. Mod. Phys. 35, 350 (1963); E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective, J. Wiley and Sons 1974.
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Strocchi, F. (2008). Symmetries in Classical Field Theory. In: Symmetry Breaking. Lecture Notes in Physics, vol 732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73593-9_3
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